**Symmetry in Modeling and Analysis of Dynamic Systems**

Editor

**Jan Awrejcewicz**

MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin

*Editor* Jan Awrejcewicz Department of Automation, Biomechanics and Mechatronics Lodz University of Technology Łod´ z´ Poland

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This is a reprint of articles from the Special Issue published online in the open access journal *Symmetry* (ISSN 2073-8994) (available at: www.mdpi.com/journal/symmetry/special issues/ Symmetry Modeling Analysis Dynamic Systems).

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## **Contents**


## **About the Editor**

#### **Jan Awrejcewicz**

Jan Awrejcewicz is Full Professor employed at Lodz University of Technology. He is Head of the Department of Automation, Biomechanics and Mechatronics, and a Full Member of the Polish Academy of Sciences. He has won numerous national/international awards, including The Alexander von Humboldt Award for research and educational achievements as well as honorary doctorates from the Universities in Poland (Bielsko-Biala, Czestochowa, Kielce, Gdansk), Ukraine ´ (Kharkiv, Dnipro) and Romania (Timisoara).

His research covers mechanics, applied mathematics, material science, biomechanics, automation, physics and computer sciences, with his main focus being nonlinear processes.

To date, he has supervised 27 doctoral students. He is a contributor to 60 different research journals and to 308 conferences, including 80 Keynote/Plenary talks, published 53 monographs (including 27 in English and 1 in Russian) and over 920 papers in scientific journals and conference proceedings.

## **Preface to "Symmetry in Modeling and Analysis of Dynamic Systems"**

Real-world systems exhibit complex behavior, therefore novel mathematical approaches or modifications of classical ones have to be employed to precisely predict, monitor, and control complicated chaotic and stochastic processes. One of the most basic concepts that has to be taken into account while conducting research in all natural sciences is symmetry, and it is usually used to refer to an object that is invariant under some transformations including translation, reflection, rotation or scaling.

The following Special Issue is dedicated to investigations of the concept of dynamical symmetry in the modelling and analysis of dynamic features occurring in various branches of science like physics, chemistry, biology, and engineering, with special emphasis on research based on the mathematical models of nonlinear partial and ordinary differential equations. Addressed topics cover theories developed and employed under the concept of invariance of the global/local behavior of the points of spacetime, including temporal/spatiotemporal symmetries.

*Small Solutions of the Perturbed Nonlinear Partial Discrete Dirichlet Boundary Value Problems with (p,q)-Laplacian Operator* by Feng Xiong and Zhan Zhou [1] presents results of using critical point theory for study of perturbed partial discrete boundary value problems.

*Vibration Properties of a Concrete Structure with Symmetries Used in Civil Engineering* by Sorin Vlase, Marin Marin and Ovidiu Deaconu [2] is dedicated to identification of the eigenvalue and eigenmode properties of vibration for components of the concrete constructions' structure allowing for the simplification of their dynamic analysis.

*Evaluating the Impact of Different Symmetrical Models of Ambient Assisted Living Systems* by Wael Alosaimi, Md Tarique Jamal Ansari, Abdullah Alharbi, Hashem Alyami, Adil Hussain Seh, Abhishek Kumar Pandey, Alka Agrawal and Raees Ahmad Khan [3] provides results of investigation of the potential symmetrical models of Ambient Assisted Living systems and frameworks for the implementation of effective new installations enhancing the living standard for old-aged people.

*Dynamical Simulation of Effective Stem Cell Transplantation for Modulation of Microglia Responses in Stroke Treatment* by Awatif Jahman Alqarni, Azmin Sham Rambely and Ishak Hashim [4] presents studies of the stability of the mathematical model by using the current biological information on stem cell therapy as a possible treatment for inflammation from microglia during stroke.

In *Second-Order Non-Canonical Neutral Differential Equations with Mixed Type: Oscillatory Behavior* by Osama Moaaz, Amany Nabih, Hammad Alotaibi and Y. S. Hamed [5] new sufficient conditions for the oscillation of solutions of a class of second-order delay differential equations with a mixed neutral term under the non-canonical condition are discussed.

*Generalized Attracting Horseshoe in the R¨ossler Attractor* by Karthik Murthy, Ian Jordan, Parth Sojitra, Aminur Rahman and Denis Blackmore [6] presents the generalized attracting horseshoe and its trapping region obtained by using a chosen Poincare map of the R ´ ossler attractor for an electronic ¨ circuit.

*Multiple Solutions for a Class of Nonlinear Fourth-Order Boundary Value Problems* by Longfei Lin, Yansheng Liu and Daliang Zhao [7] considers multiple solutions for a class of nonlinear fourth-order boundary value problems obtained by constructing a special cone and applying fixed point index theory.

*Existence of Three Solutions for a Nonlinear Discrete Boundary Value Problem with* φc*-Laplacian* by Yanshan Chen and Zhan Zhou [8] is focused on the multiplicity of nontrivial solutions for a nonlinear discrete Dirichlet boundary value problem involving the mean curvature operator.

Last but not least, *On the Absolute Stable Difference Scheme for Third Order Delay Partial Differential Equations* by Allaberen Ashyralyev, Evren Hınc¸al and Suleiman Ibrahim [9] offers investigations of the third order delay differential equation in a Hilbert space with an unbounded operator.

#### **References**

1. Feng Xiong and Zhan Zhou, Small Solutions of the Perturbed Nonlinear Partial Discrete Dirichlet Boundary Value Problems with (p,q)-Laplacian Operator, *Symmetry* 2021, 13(7), 1207;

2. Sorin Vlase, Marin Marin and Ovidiu Deaconu, Vibration Properties of a Concrete Structure with Symmetries Used in Civil Engineering, *Symmetry* 2021, 13(4), 656;

3. Wael Alosaimi, Md Tarique Jamal Ansari, Abdullah Alharbi, Hashem Alyami, Adil Hussain Seh, Abhishek Kumar Pandey, Alka Agrawal and Raees Ahmad Khan

Evaluating the Impact of Different Symmetrical Models of Ambient Assisted Living Systems, *Symmetry* 2021, 13(3), 450;

4. Awatif Jahman Alqarni, Azmin Sham Rambely and Ishak Hashim, Dynamical Simulation of Effective Stem Cell Transplantation for Modulation of Microglia Responses in Stroke Treatment, *Symmetry* 2021, 13(3), 404;

5. Osama Moaaz, Amany Nabih, Hammad Alotaibi and Y. S. Hamed, Second-Order Non-Canonical Neutral Differential Equations with Mixed Type: Oscillatory Behavior, *Symmetry* 2021, 13(2), 318;

6. Karthik Murthy, Ian Jordan, Parth Sojitra, Aminur Rahman and Denis Blackmore, Generalized Attracting Horseshoe in the Rossler Attractor, ¨ *Symmetry* 2021, 13(1), 30;

7. Longfei Lin, Yansheng Liu and Daliang Zhao, Multiple Solutions for a Class of Nonlinear Fourth-Order Boundary Value Problems, *Symmetry* 2020, 12(12), 1989;

8. Yanshan Chen and Zhan Zhou, Existence of Three Solutions for a Nonlinear Discrete Boundary Value Problem with φc-Laplacian, *Symmetry* 2020, 12(11), 1839;

9. Allaberen Ashyralyev, Evren Hınc¸al and Suleiman Ibrahim, On the Absolute Stable Difference Scheme for Third Order Delay Partial Differential Equations, *Symmetry* 2020, 12(6), 1033.

> **Jan Awrejcewicz** *Editor*

## *Article* **Small Solutions of the Perturbed Nonlinear Partial Discrete Dirichlet Boundary Value Problems with** (*p***,** *q*)**-Laplacian Operator**

**Feng Xiong 1,2 and Zhan Zhou 1,2,\***


**Abstract:** In this paper, we consider a perturbed partial discrete Dirichlet problem with the (*p*, *q*)- Laplacian operator. Using critical point theory, we study the existence of infinitely many small solutions of boundary value problems. Without imposing the symmetry at the origin on the nonlinear term *f* , we obtain the sufficient conditions for the existence of infinitely many small solutions. As far as we know, this is the study of perturbed partial discrete boundary value problems. Finally, the results are exemplified by an example.

**Keywords:** boundary value problem; partial difference equation; infinitely many small solutions; (*p*, *q*)-Laplacian; critical point theory

#### **1. Introduction**

Let **Z** and **R** denote the sets of integers and real numbers, respectively. Denote **Z**(*a*, *b*) = {*a*, *a* + 1, · · · , *b*} when *a* ≤ *b*.

We consider the following partial discrete problem, namely (*Dλ*,*<sup>µ</sup>* )

−[∆1(*φp*(∆1*y*(*s* −1, *t*))) + ∆2(*φp*(∆2*y*(*s*, *t* −1)))] + *l*(*s*, *t*)*φq*(*y*(*s*, *t*)) = *λ f*((*s*, *t*), *y*(*s*, *t*)) +*µg*((*s*, *t*), *y*(*s*, *t*)),(*s*, *t*) ∈ **Z**(1, *a*) × **Z**(1, *b*), with Dirichlet boundary conditions as follows:

$$\begin{aligned} y(s,0) = y(s, b+1) &= 0, & s \in \mathbf{Z}(0, a+1), \\ y(0,t) = y(a+1, t) &= 0, & t \in \mathbf{Z}(0, b+1), \end{aligned} \tag{1}$$

where *a* and *b* are the given positive integers, *λ* and *µ* are the positive real parameters, ∆<sup>1</sup> and ∆<sup>2</sup> are the forward difference operators defined by ∆1*y*(*s*, *t*) = *y*(*s* + 1, *t*) − *y*(*s*, *t*) and ∆2*y*(*s*, *t*) = *y*(*s*, *t* + 1) − *y*(*s*, *t*), ∆ 2 1 *y*(*s*, *t*) = ∆1(∆1*y*(*s*, *t*)) and ∆ 2 2 *y*(*s*, *t*) = ∆2(∆2*y*(*s*, *t*)), *φr*(*y*) = |*y*| *<sup>r</sup>*−2*<sup>y</sup>* with *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>**, <sup>1</sup> <sup>&</sup>lt; *<sup>q</sup>* <sup>≤</sup> *<sup>p</sup>* <sup>&</sup>lt; <sup>+</sup>∞, *<sup>l</sup>*(*s*, *<sup>t</sup>*) <sup>≥</sup> <sup>0</sup> for all (*s*, *<sup>t</sup>*) <sup>∈</sup> **<sup>Z</sup>**(1, *<sup>a</sup>*) <sup>×</sup> **<sup>Z</sup>**(1, *<sup>b</sup>*), and *f*((*s*, *t*), ·), *g*((*s*, *t*), ·) ∈ *C*(**R**, **R**) for each (*s*, *t*) ∈ **Z**(1, *a*) × **Z**(1, *b*).

Difference equations are widely applied in diverse domains, including natural science, and biological neural networks, as shown in [1–4]. For the existence and multiplicity of solutions to boundary value problems, some authors derived a number of conclusions using nonlinear analysis methods, such as fixed point methods as well as the Brouwer degree [5–9]. In 2003, Yu and Guo [10] used firstly the critical point theory to study a class of difference equations. Since then, many mathematical researchers have explored difference equations and made great achievements, which include the results of periodic solutions [10,11], homoclinic solutions [12–18], boundary value problems [19–25] and so on. Bonanno et al. [20] in 2016 considered the following discrete Dirichlet problem:

$$\begin{cases} \Delta^2 y\_{h-1} + \lambda f(h, y\_h) = 0, & h \in \mathbf{Z}(1, N), \\ y\_0 = y\_{N+1} = 0, \end{cases} \tag{2}$$

and acquired at least two positive solutions of (2).


**Citation:** Xiong, F.; Zhou, Z. Small Solutions of the Perturbed Nonlinear Partial Discrete Dirichlet Boundary Value Problems with (*p*, *q*)-Laplacian Operator. *Symmetry* **2021**, *13*, 1207. https://doi.org/10.3390/sym13071207

Academic Editor: Jan Awrejcewicz

Received: 25 May 2021 Accepted: 3 July 2021 Published: 5 July 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Mawhin et al. [21] in 2017 studied the following boundary value problem:

$$\begin{cases} -\Delta(\phi\_p(\Delta y\_{h-1})) + q\_h \phi\_p(y\_h) = \lambda f(h, y\_h), & h \in \mathbf{Z}(1, N), \\ y\_0 = y\_{N+1} = 0, \end{cases} \tag{3}$$

extending the results in [20] with *p* = 2.

Nastasi et al. [22] in 2017 studied the discrete Dirichlet problem involving the (*p*, *q*)- Laplacian operator as follows:

$$\begin{cases} -\Delta(\phi\_p(\Delta y(h-1))) - \Delta(\phi\_\emptyset(\Delta y(h-1))) + a(h)\phi\_p(y(h)) + \beta(h)\phi\_\emptyset(y(h)) = \lambda g(h, y(h)), \ h \in \mathbf{Z}(1, N),\\ y(0) = y(N+1) = 0, \end{cases} \tag{4}$$

and obtained at least two positive solutions of (4).

In 2019, Ling and Zhou [24] considered the Dirichlet problem involving *φc*-Laplacian as follows:

$$\begin{cases} -\Delta(\phi\_{\varepsilon}(\Delta y\_{h-1})) + q\_h \phi\_{\varepsilon}(y\_h) = \lambda f(h, y\_h), \quad h \in \mathbb{Z}(1, N), \\ y\_0 = y\_{N+1} = 0. \end{cases} \tag{5}$$

They studied the existence of positive solutions of (5) when *q<sup>h</sup>* ≡ 0 in [23].

In 2020, Wang and Zhou [25] considered discrete Dirichlet boundary value problem as follows:

$$\begin{cases} \Delta(\phi\_{p,\mathcal{E}}(\Delta y\_{h-1})) + \lambda f(h, y\_h) = 0, & h \in \mathbf{Z}(1, N), \\ y\_0 = y\_{N+1} = 0. \end{cases} \tag{6}$$

The difference equations studied above involve only one variable. However, the difference equations containing two or more variables are less studied, and such difference equations are called partial difference equations. Recently, partial difference equations were widely used in many fields. Boundary value problems of partial difference equations seem to be challenging problem that has attracted many mathematical researchers [26,27].

In 2015, Heidarkhani and Imbesi [26] adopted two critical points theorems to establish multiple solutions of the partial discrete problem as shown below:

$$
\Delta\_1^2 y(s-1, t) + \Delta\_2^2 y(s, t-1) + \lambda f((s, t), y(s, t)) = 0, \quad (s, t) \in \mathbf{Z}(1, a) \times \mathbf{Z}(1, b), \tag{7}
$$

with Dirichlet boundary conditions (1).

Recently, in 2020, Du and Zhou [27] studied a partial discrete Dirichlet problem as follows:

$$
\Delta\_1(\phi\_p(\Delta\_1 y(s-1,t))) + \Delta\_2(\phi\_p(\Delta\_2 y(s,t-1))) + \lambda f((s,t), y(s,t)) = 0, \quad (s,t) \in \mathbf{Z}(1,a) \times \mathbf{Z}(1,b), \tag{8}
$$

with Dirichlet boundary conditions (1).

Inspired by the above research, we found that the perturbed partial difference equations had rarely been studied, so this paper aims at studying small solutions of the perturbed partial discrete Dirichlet problems with the (*p*, *q*)-Laplacian operator. Here, the perturbed partial difference equations mean that the term with the parameter *µ* in the right hand of the equation for the problem (*Dλ*,*<sup>µ</sup>* ) is very small. A solution *y*(*s*, *t*) of (*Dλ*,*<sup>µ</sup>* ) is called a small solution if the norm k*y*(*s*, *t*)k is small. In fact, without the symmetric assumption on the origin for the nonlinear term *f* , we can still verify that problem (*Dλ*,*<sup>µ</sup>* ) possesses a sequence of solutions which converges to zero by using the Lemma 2. Moreover, by Lemma 1, we can show that all of these solutions are positive. Furthermore, by truncation techniques, we obtain two sequences of constant-sign solutions, which converge to zero (with one being positive and the other being negative). As far as we know, our study takes the lead in addressing small solutions of the perturbed partial discrete Dirichlet problems with the (*p*, *q*)-Laplacian operator.

The rest of this paper is organized as follows. In Section 2, we establish the variational framework linked to (*Dλ*,*<sup>µ</sup>* ) and recall the abstract critical point theorem. In Section 3, we give the main results. In Section 4, we provide an example to demonstrate our results. We make a conclusion in the last section.

#### **2. Preliminaries**

The current section is the first one to establish the variational framework linked to (*Dλ*,*<sup>µ</sup>* ). We consider the *ab*-dimensional Banach space

*Y* = {*y* : **Z**(0, *a* + 1) × **Z**(0, *b* + 1) → **R** : *y*(*s*, 0) = *y*(*s*, *b* + 1) = 0,*s* ∈ **Z**(0, *a* + 1) and *y*(0, *t*) = *y*(*a* + 1, *t*) = 0, *t* ∈ **Z**(0, *b* + 1)}, endowed with the norm

$$\|y\|\| = \left(\sum\_{t=1}^{b} \sum\_{s=1}^{a+1} |\Delta\_1 y(s-1, t)|^p + \sum\_{s=1}^{a} \sum\_{t=1}^{b+1} |\Delta\_2 y(s, t-1)|^p\right)^{\frac{1}{p}}, \quad y \in \mathcal{Y}.$$

and k*y*k<sup>∞</sup> = max{|*y*(*s*, *t*)| : (*s*, *t*) ∈ **Z**(1, *a*) × **Z**(1, *b*)} is another norm in *Y*. Let *l*<sup>∗</sup> = min{*l*(*s*, *t*) : (*s*, *t*) ∈ **Z**(1, *a*) × **Z**(1, *b*)}.

**Proposition 1.** *The following inequality holds:*

$$\|y\|\_{\infty} \le \max\left\{ \left( \frac{p(a+b+2)^{p-1}}{4^p + l\_\*(a+b+2)^{p-1}} \right)^{1/q} \left( \frac{||y||^p}{p} + \frac{\sum\_{t=1}^b \frac{a}{s-1} l(s,t) |y(s,t)|^q}{q} \right)^{1/q} \right\}^{1/q}.$$

$$\left( \frac{p(a+b+2)^{p-1}}{4^p + l\_\*(a+b+2)^{p-1}} \right)^{1/p} \left( \frac{||y||^p}{p} + \frac{\sum\_{t=1}^b \frac{a}{s-1} l(s,t) |y(s,t)|^q}{q} \right)^{1/p} \right\}.$$

**Proof.** According to the result of ([27], Proposition 1), we have the following:

$$||y||\_{\infty}^{p} \le \frac{(a+b+2)^{p-1}}{4^{p}}||y||^{p}.\tag{9}$$

When k*y*k<sup>∞</sup> > 1, according to (9), we have the following:

$$\begin{split} \frac{(1+l\*\frac{(a+b+2)^{p-1}}{4^p})||y||\_{\infty}^q}{p} &\leq \frac{||y||\_{\infty}^p + \frac{(a+b+2)^{p-1}}{4^p}\sum\_{t=1}^b \sum\_{s=1}^d l(s,t)|y(s,t)|^q}{p} \\ &\leq \frac{(a+b+2)^{p-1}}{p4^p} \left(||y||^p + \sum\_{t=1}^b \sum\_{s=1}^d l(s,t)|y(s,t)|^q\right) \\ &\leq \frac{(a+b+2)^{p-1}}{p4^p} ||y||^p + \frac{(a+b+2)^{p-1}}{q4^p} \sum\_{t=1}^b \sum\_{s=1}^d l(s,t)|y(s,t)|^q, \end{split} \tag{10}$$

that is,

$$||y||\_{\infty} \le \left(\frac{p(a+b+2)^{p-1}}{4^p + l\_\*(a+b+2)^{p-1}}\right)^{1/q} \left(\frac{||y||^p}{p} + \frac{\sum\_{t=1}^b l(s,t)|y(s,t)|^q}{q}\right)^{1/q} \cdot \tag{11}$$

When k*y*k<sup>∞</sup> ≤ 1, according to (9), we have the following:

$$\begin{split} \frac{(1+l\_\*\frac{(a+b+2)^{p-1}}{4^p})\|y\|\_\infty^p}{p} &\leq \frac{\|y\|\_\infty^p + \frac{(a+b+2)^{p-1}}{4^p} \sum\_{t=1}^b \sum\_{s=1}^d l(s,t)|y(s,t)|^q}{p} \\ &\leq \frac{(a+b+2)^{p-1}}{p4^p} \|y\|^p + \frac{(a+b+2)^{p-1}}{q4^p} \sum\_{t=1}^b \sum\_{s=1}^d l(s,t)|y(s,t)|^q, \end{split} \tag{12}$$

that is,

$$\|y\|\_{\infty} \le \left(\frac{p(a+b+2)^{p-1}}{4^p + l\_\*(a+b+2)^{p-1}}\right)^{1/p} \left(\frac{||y||^p}{p} + \frac{\sum\_{t=1}^b \sum\_{s=1}^a l(s,t)|y(s,t)|^q}{q}\right)^{1/p} . \tag{13}$$

In summary, we have the following:

$$\|y\|\_{\infty} \le \max\left\{ \left( \frac{p(a+b+2)^{p-1}}{4^p + l\_\*(a+b+2)^{p-1}} \right)^{1/q} \left( \frac{||y||^p}{p} + \frac{\sum\_{t=1}^b \frac{a}{s-1} l(s,t) |y(s,t)|^q}{q} \right)^{1/q} \right\}^{1/q}.$$

$$\left( \frac{p(a+b+2)^{p-1}}{4^p + l\_\*(a+b+2)^{p-1}} \right)^{1/p} \left( \frac{||y||^p}{p} + \frac{\sum\_{t=1}^b \frac{a}{s-1} l(s,t) |y(s,t)|^q}{q} \right)^{1/p} \right).$$

Define

$$\Phi(y) = \Phi\_1(y) + \Phi\_2(y)\_\prime$$

$$\Psi(y) = \sum\_{t=1}^b \sum\_{s=1}^a \left( F((s, t), y(s, t)) + \frac{\mu}{\lambda} G((s, t), y(s, t)) \right)\_\prime$$

for every *<sup>y</sup>* <sup>∈</sup> *<sup>Y</sup>*, where <sup>Φ</sup>1(*y*) = <sup>k</sup>*y*<sup>k</sup> *p p* , Φ2(*y*) = *b* ∑ *t*=1 *a* ∑ *s*=1 *l*(*s*,*t*)|*y*(*s*,*t*)| *q q* , *F*((*s*, *t*), *y*) = R *<sup>y</sup>* 0 *f*((*s*, *t*), *τ*) *dτ*, *G*((*s*, *t*), *y*) = R *<sup>y</sup>* 0 *g*((*s*, *t*), *τ*)*dτ* for each ((*s*, *t*), *y*) ∈ **Z**(1, *a*) × **Z**(1, *b*) × **R**. Let

$$I\_{\lambda}(y) = \Phi(y) - \lambda \Psi(y)\_{\prime}$$

for any *y* ∈ *Y*. Obviously, Φ, Ψ ∈ *C* 1 (*Y*, *R*), that is, Φ1, Φ<sup>2</sup> and Ψ are continuously Fréchet differentiable in *Y*.

$$\begin{split} \Phi\_{1}^{\prime}(y)(v) &= \lim\_{t \to 0} \frac{\Phi\_{1}(y+tv) - \Phi\_{1}(y)}{t} \\ &= \sum\_{t=1}^{b} \sum\_{s=1}^{a+1} \Phi\_{p}(\Delta\_{1}y(s-1,t))\Delta\_{1}v(s-1,t) + \sum\_{s=1}^{a} \sum\_{t=1}^{b+1} \Phi\_{p}(\Delta\_{2}y(s,t-1))\Delta\_{2}v(s,t-1) \\ &= -\sum\_{t=1}^{b} \sum\_{s=1}^{a} \Delta\_{1}\Phi\_{p}(\Delta\_{1}y(s-1,t))v(s,t) - \sum\_{s=1}^{a} \sum\_{t=1}^{b} \Delta\_{2}\Phi\_{p}(\Delta\_{2}y(s,t-1))v(s,t), \\ &\qquad\qquad \Phi\_{2}^{\prime}(y)(v) = \sum\_{t=1}^{b} \sum\_{s=1}^{a} l(s,t)\Phi\_{q}(y(s,t))v(s,t), \end{split}$$

and

$$\Psi'(y)(v) = \lim\_{t \to 0} \frac{\Psi(y + tv) - \Psi(y)}{t} = \sum\_{t=1}^{b} \sum\_{s=1}^{a} \left( f((s, t), y(s, t)) + \frac{\mu}{\lambda} g((s, t), y(s, t)) \right) v(s, t),$$

for *y*, *v* ∈ *Y*.

Thus

$$\begin{aligned} \left[\Phi'(y) - \lambda \Psi'(y)\right](\upsilon) &= -\sum\_{l=1}^{b} \sum\_{s=1}^{d} \left\{\Delta\_1 \Phi\_{\mathcal{P}}(\Delta\_1 y(s-1, t)) + \Delta\_2 \Phi\_{\mathcal{P}}(\Delta\_2 y(s, t-1)) \right. \\ &\left. -l(s, t)\Phi\_{\mathcal{I}}(y(s, t)) + \lambda f((s, t), y(s, t)) + \mu g((s, t), y(s, t))\right\} \mathbf{v}(s, t) = 0, \quad \forall \mathbf{v}(s, t) \in \mathcal{Y}, \end{aligned} \tag{14}$$

is equivalent to

$$-\left[\Delta\_1(\phi\_p(\Delta\_1 y(s-1,t))) + \Delta\_2(\phi\_p(\Delta\_2 y(s,t-1)))\right] + l(s,t)\phi\_q(y(s,t)) = \lambda f((s,t), y(s,t)) + \lambda f((s,t), y(s,t))$$

*µg*((*s*, *t*), *y*(*s*, *t*)), for any (*s*, *t*) ∈ **Z**(1, *a*) × **Z**(1, *b*) with *y*(*s*, 0) = *y*(*s*, *b* + 1) = 0, *s* ∈ **Z**(0, *a* + 1), *y*(0, *t*) = *y*(*a* + 1, *t*) = 0, *t* ∈ **Z**(0, *b* + 1). Thus, we reduce the existence of the solutions of (*Dλ*,*<sup>µ</sup>* ) to the existence of the critical points of Φ − *λ*Ψ on *Y*.

**Lemma 1.** *Suppose that there exists y: Z*(0, *a* + 1) × *Z*(0, *b* + 1) → *R such that the following is true:*

$$y(\mathbf{s},t) > 0 \text{ or } -\Delta\_1(\phi\_p(\Delta\_1 y(\mathbf{s}-1,t))) - \Delta\_2(\phi\_p(\Delta\_2 y(\mathbf{s},t-1))) + l(\mathbf{s},t)\phi\_q(y(\mathbf{s},t)) \ge 0,\tag{15}$$

*for all* (*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*) *and y*(*s*, 0) = *y*(*s*, *b* + 1) = 0,*s* ∈ *Z*(0, *a* + 1)*, y*(0, *t*) = *y*(*a* + 1, *t*) = 0*, t* ∈ *Z*(0, *b* + 1)*.*

*Then, either y*(*s*, *t*) > 0 *for all* (*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*) *or y* ≡ 0*.*

**Proof.** Let *h* ∈ **Z**(1, *a*), *k* ∈ **Z**(1, *b*) and

$$y(h,k) = \min\{y(s,t) : s \in \mathbf{Z}(1,a), t \in \mathbf{Z}(1,b)\}.$$

If *y*(*h*, *k*) > 0, then it is clear that *y*(*s*, *t*) > 0 for all *s* ∈ **Z**(1, *a*), *t* ∈ **Z**(1, *b*). If *y*(*h*, *k*) ≤ 0, then *y*(*h*, *k*) = min{*y*(*s*, *t*) : *s* ∈ **Z**(0, *a* + 1), *t* ∈ **Z**(0, *b* + 1)}, since ∆1*y*(*h* − 1, *k*) = *y*(*h*, *k*) − *y*(*h* − 1, *k*) ≤ 0, ∆2*y*(*h*, *k* − 1) = *y*(*h*, *k*) − *y*(*h*, *k* − 1) ≤ 0, and ∆1*y*(*h*, *k*) = *y*(*h* + 1, *k*) − *y*(*h*, *k*) ≥ 0, ∆2*y*(*h*, *k*) = *y*(*h*, *k* + 1) − *y*(*h*, *k*) ≥ 0, *φp*(*s*) is increasing in *s*, and *φp*(0) = 0, we have

$$
\phi\_p(\Delta\_1 y(h,k)) \ge 0 \ge \phi\_p(\Delta\_1 y(h-1,k))\_\prime
$$

and

$$
\phi\_p(\Delta\_2 y(h,k)) \ge 0 \ge \phi\_p(\Delta\_2 y(h,k-1)).
$$

Owing to ∆1(*φp*(∆1*y*(*h* − 1, *k*))) = *φp*(∆1*y*(*h*, *k*)) − *φp*(∆1*y*(*h* − 1, *k*)) ≥ 0, ∆2(*φp*(∆2*y* (*h*, *k* − 1))) = *φp*(∆2*y*(*h*, *k*)) − *φp*(∆2*y*(*h*, *k* − 1)) ≥ 0. Thus, we have

$$
\Delta\_1(\phi\_p(\Delta\_1 y(h-1,k))) + \Delta\_2(\phi\_p(\Delta\_2 y(h,k-1))) \ge 0. \tag{16}
$$

By (15), we have the following:

$$-\Delta\_1(\phi\_p(\Delta\_1 y(h-1,k))) - \Delta\_2(\phi\_p(\Delta\_2 y(h,k-1))) \ge -l(h,k)\phi\_q(y(h,k)) \ge 0,$$

that is

$$
\Delta\_1(\phi\_p(\Delta\_1 y(h-1,k))) + \Delta\_2(\phi\_p(\Delta\_2 y(h,k-1))) \le 0. \tag{17}
$$

By combining (16) with (17), we have the following:

$$
\Delta\_1(\phi\_p(\Delta\_1 y(h-1,k))) + \Delta\_2(\phi\_p(\Delta\_2 y(h,k-1))) = 0,
$$

namely,

$$\begin{cases} \phi\_p(\Delta\_1 y(h,k)) = \phi\_p(\Delta\_1 y(h-1,k)) = 0, \\ \phi\_p(\Delta\_2 y(h,k)) = \phi\_p(\Delta\_2 y(h,k-1)) = 0. \end{cases}$$

Therefore,

$$\begin{cases} y(h+1,k) = y(h,k) = y(h-1,k), \\ y(h,k+1) = y(h,k) = y(h,k-1). \end{cases}$$

If *h* + 1 = *a* + 1, we have *y*(*h*, *k*) = 0. Otherwise, (*h* + 1) ∈ **Z**(1, *a*). Replacing *h* by *h* + 1, we get *y*(*h* + 2, *k*) = *y*(*h* + 1, *k*). Continuing this process (*a* + 1 − *h*) times, we have *y*(*h*, *k*) = *y*(*h* + 1, *k*) = *y*(*h* + 2, *k*) = · · · = *y*(*a* + 1, *k*) = 0. Similarly, we have *y*(*h*, *k*) = *y*(*h* − 1, *k*) = *y*(*h* − 2, *k*) = · · · = *y*(0, *k*) = 0. Hence, *y*(*s*, *k*) = 0 for each *s* ∈ **Z**(1, *a*). In the same way, we can prove that *y* ≡ 0, and the proof is completed.

From Lemma 1, we have the following:

**Corollary 1.** *Suppose that there exists y: Z*(0, *a* + 1) × *Z*(0, *b* + 1) → *R such that*

$$\text{log}(s, t) < 0 \text{ or } -\Delta\_1(\phi\_p(\Delta\_1 y(s-1, t))) - \Delta\_2(\phi\_p(\Delta\_2 y(s, t-1))) + l(s, t)\phi\_q(y(s, t)) \le 0,\tag{18}$$

*for all* (*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*) *and y*(*s*, 0) = *y*(*s*, *b* + 1) = 0,*s* ∈ *Z*(0, *a* + 1)*, y*(0, *t*) = *y*(*a* + 1, *t*) = 0, *t* ∈ *Z*(0, *b* + 1)*.*

*Then, either y*(*s*, *t*) < 0 *for all* (*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*) *or y* ≡ 0*.*

*The existence of constant-sign solutions is discussed by truncation techniques. So, we introduce the following truncations of the functions f*((*s*, *t*), *ξ*) *and g*((*s*, *t*), *ξ*) *for every* (*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*)*.*

*If f*((*s*, *t*), 0) ≥ 0 *and g*((*s*, *t*), 0) ≥ 0 *for every* (*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*)*, let*

$$f^+((s,t),\xi) := \begin{cases} f((s,t),\xi), & \text{if } \xi \ge 0, \\ f((s,t),0), & \text{if } \xi < 0, \end{cases} \quad \mathcal{g}^+((s,t),\xi) := \begin{cases} \mathcal{g}((s,t),\xi), & \text{if } \xi \ge 0, \\ \mathcal{g}((s,t),0), & \text{if } \xi < 0. \end{cases}$$

*Define problem* (*Dλ*,*<sup>µ</sup>* + ) *as follows:*

− h <sup>∆</sup>1(*φp*(∆1*y*(*<sup>s</sup>* <sup>−</sup> 1, *<sup>t</sup>*))) + <sup>∆</sup>2(*φp*(∆2*y*(*s*, *<sup>t</sup>* <sup>−</sup> <sup>1</sup>)))<sup>i</sup> + *l*(*s*, *t*)*φq*(*y*(*s*, *t*)) = *λ f* <sup>+</sup>((*s*, *t*)*, y*(*s*, *t*)) + *µg* <sup>+</sup>((*s*, *<sup>t</sup>*), *<sup>y</sup>*(*s*, *<sup>t</sup>*)),(*s*, *<sup>t</sup>*) <sup>∈</sup> *<sup>Z</sup>*(1, *<sup>a</sup>*) <sup>×</sup> *<sup>Z</sup>*(1, *<sup>b</sup>*)*, with Dirichlet boundary conditions (1).*

*Obviously, f* <sup>+</sup>((*s*, *<sup>t</sup>*), ·) *and <sup>g</sup>* <sup>+</sup>((*s*, *<sup>t</sup>*), ·) *are also continuous for every* (*s*, *<sup>t</sup>*) <sup>∈</sup> *<sup>Z</sup>*(1, *<sup>a</sup>*) <sup>×</sup> *Z*(1, *b*)*. By Lemma 1, the solutions of problem* (*Dλ*,*<sup>µ</sup>* + ) *are also those of problem* (*Dλ*,*<sup>µ</sup>* )*. Therefore, when problem* (*Dλ*,*<sup>µ</sup>* + ) *has non-zero solutions, then problem* (*Dλ*,*<sup>µ</sup>* ) *possesses positive solutions. If f*((*s*, *t*), 0) ≤ 0 *and g*((*s*, *t*), 0) ≤ 0 *for every* (*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*)*, let*

$$f^-((s,t),\xi) := \begin{cases} \ f((s,t),0), & \text{if } \xi > 0, \\ \ f((s,t),\xi), & \text{if } \xi \le 0, \end{cases} \quad \text{g}^-((s,t),\xi) := \begin{cases} \ g((s,t),0), & \text{if } \xi > 0, \\ \ g((s,t),\xi), & \text{if } \xi \le 0. \end{cases}$$

*Define problem* (*Dλ*,*<sup>µ</sup>* − ) *as follows:*

− h <sup>∆</sup>1(*φp*(∆1*y*(*<sup>s</sup>* <sup>−</sup> 1, *<sup>t</sup>*))) + <sup>∆</sup>2(*φp*(∆2*y*(*s*, *<sup>t</sup>* <sup>−</sup> <sup>1</sup>)))<sup>i</sup> + *l*(*s*, *t*)*φq*(*y*(*s*, *t*)) = *λ f* −((*s*, *t*)*, y*(*s*, *t*)) + *µg* <sup>−</sup>((*s*, *t*), *y*(*s*, *t*)),(*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*)*, with Dirichlet boundary conditions (1).*

By Corollary 1, the solutions of problem (*Dλ*,*<sup>µ</sup>* − ) are also those of problem (*Dλ*,*<sup>µ</sup>* ). Therefore, when problem (*Dλ*,*<sup>µ</sup>* − ) has non-zero solutions, then problem (*Dλ*,*<sup>µ</sup>* ) possesses negative solutions.

Here, we present the main tools used in this paper.

**Lemma 2** (Theorem 4.3 of [28])**.** *Let X be a finite dimensional Banach space and let I<sup>λ</sup>* : *X* → *R be a function satisfying the following structure hypothesis:*

(*H*) *Iλ*(*u*) := Φ(*u*) − *λ*Ψ(*u*) *for all u* ∈ *X, where* Φ, Ψ : *X* → *R be two continuously Gâteux differentiable functions with* Φ *coercive, i.e.,* lim k*u*k→+∞ Φ(*u*) = +∞*, and such that* inf *X* Φ = Φ(0) =

$$\Psi(0) = 0.$$

*For all r* > 0*, put the following:*

$$\varrho(r) := \frac{\sup\_{v \in \Phi^{-1}[0,r]} \Psi(v)}{r}, \quad \varrho\_0 := \liminf\_{r \to 0^+} \varrho(r).$$

*Assume that <sup>ϕ</sup>*<sup>0</sup> <sup>&</sup>lt; <sup>+</sup><sup>∞</sup> *and for every <sup>λ</sup>* <sup>∈</sup> (0, <sup>1</sup> *ϕ*0 )*, 0 is not a local minima of functional Iλ. Then, there is a sequence* {*un*} *of pairwise distinct critical points (local minima) of I<sup>λ</sup> such that* lim *<sup>n</sup>*→+<sup>∞</sup> *u<sup>n</sup>* = 0*.*

#### **3. Main Results**

In this section, the existence of constant-sign solutions of problem (*Dλ*,*<sup>µ</sup>* ) is discussed. Our aim is to use Lemma 2 for the function *I* ± *λ* : *X* → *R*,

$$I\_{\lambda}^{\pm}(y) := \Phi(y) - \lambda \Psi^{\pm}(y)\_{\prime}$$

where

$$\Psi^{\pm}(y) = \sum\_{t=1}^{b} \sum\_{s=1}^{d} (F^{\pm}((s,t), y(s,t)) + \frac{\mu}{\lambda} G^{\pm}((s,t), y(s,t))),$$

$$F^{\pm}((s,t), y) := \int\_{0}^{y} f^{\pm}((s,t), \tau) d\tau,\ \ G^{\pm}((s,t), y) := \int\_{0}^{y} g^{\pm}((s,t), \tau) d\tau,$$

for each (*s*, *t*) ∈ **Z**(1, *a*) × **Z**(1, *b*). Then, we apply Lemma 1 or Corollary 1 to obtain our results.

Let

$$A\_{0\*} = \liminf\_{c \to 0^{+}} \frac{\sum\_{t=1}^{b} \sum\_{s=1}^{a} \max\_{0 \le m \le c} F((s, t), \pm m)}{c^p}, \quad B^{0\*} = \limsup\_{c \to 0^{+}} \frac{\sum\_{t=1}^{b} \sum\_{s=1}^{a} F((s, t), c)}{c^p}.$$

$$\mathbb{C}^{0\*} = \limsup\_{c \to 0^{+}} \frac{\sum\_{t=1}^{b} \sum\_{s=1}^{a} \max\_{0 \le m \le c} G((s, t), \pm m)}{c^p}, \quad I = \sum\_{t=1}^{b} \sum\_{s=1}^{a} l(s, t).$$

In addition, put the following:

$$
\mu\_\lambda^\* := \frac{1}{C^{0\*}} \left( \frac{4^p + l\_\*(a+b+2)^{p-1}}{p(a+b+2)^{p-1}} - \lambda A\_{0\*} \right).
$$

It should be pointed out that if the denominator is 0, we regard <sup>1</sup> 0 as +∞.

**Theorem 1.** *Let f*((*s*, *t*), *y*) *be a continuous function of y, and f*((*s*, *t*), 0) ≥ 0*, g*((*s*, *t*), ·) ∈ *C*(*R*, *R*) *for every* (*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*)*. Suppose the following:*


*has a sequence of positive solutions, which converges to zero.*

˜

*,*

**Proof.** We take *X* = *Y*, Φ1, Φ<sup>2</sup> and Ψ as in Section 2. Obviously, for each (*s*, *t*) ∈ **Z**(1, *a*) × **Z**(1, *b*), *g*((*s*, *t*), 0) ≥ 0.

Now, we consider the auxiliary problem (*Dλ*,*<sup>µ</sup>* + ). Clearly Φ and Ψ<sup>+</sup> satisfy the hypothesis required in Lemma 2. Let

$$r = \frac{4^p + l\_\*(a+b+2)^{p-1}}{p(a+b+2)^{p-1}} \min\{c^q, c^p\}, \quad \text{for } c > 0.1$$

Assume *y* ∈ *Y*, and the following:

$$\Phi(y) = \frac{||y||^p}{p} + \frac{\sum\_{t=1}^{b} \sum\_{s=1}^{a} l(s,t)|y(s,t)|^q}{q} \le r.$$

If *r* = 4 *<sup>p</sup>*+*l*∗(*a*+*b*+2) *p*−1 *p*(*a*+*b*+2) *p*−1 *c q* , it means that *c* ≥ 1. According to Proposition 1, we have the following:

$$\begin{aligned} \|y\|\_{\infty} &\le \max \left\{ \left( \frac{p(a+b+2)^{p-1}}{4^p + l\_\*(a+b+2)^{p-1}} \right)^{1/q} r^{1/q} \right\} \\ &\left( \frac{p(a+b+2)^{p-1}}{4^p + l\_\*(a+b+2)^{p-1}} \right)^{1/p} r^{1/p} \, \Big\} \\ &= \max \{ c, c^{q/p} \} = c. \end{aligned}$$

If *r* = 4 *<sup>p</sup>*+*l*∗(*a*+*b*+2) *p*−1 *p*(*a*+*b*+2) *p*−1 *c p* , we know 0 < *c* < 1, then

$$\begin{aligned} ||y||\_\infty &\le \max\left\{ \left( \frac{p(a+b+2)^{p-1}}{4^p + l\_\*(a+b+2)^{p-1}} \right)^{1/q} r^{1/q} \right\} \\ &\left( \frac{p(a+b+2)^{p-1}}{4^p + l\_\*(a+b+2)^{p-1}} \right)^{1/p} r^{1/p} \right\} \\ &= \max\{c^{p/q}, c\} = c\_\prime \end{aligned}$$

and

*c p r* = *p*(*a* + *b* + 2) *p*−1 4 *<sup>p</sup>* + *<sup>l</sup>*∗(*<sup>a</sup>* + *<sup>b</sup>* + <sup>2</sup>) *p*−1 . (19)

Therefore, we have Φ−<sup>1</sup> [0,*r*] ⊆ {*y* ∈ *Y* : k*y*k<sup>∞</sup> ≤ *c*}. By the definition of *ϕ*, we have the following:

$$\begin{split} \sup\_{\begin{subarray}{c} v \in \Phi^{-1}[0,r] \\ \displaystyle \displaystyle \end{subarray}} & \Psi^{+}(v) \\ & \leq \frac{1}{r} \sup\_{\begin{subarray}{c} \|y\|\_{\infty} \leq c \ t = 1 \ s = 1 \end{subarray}} \sum\_{s=1}^{a} \left( F^{+}((s,t),y(s,t)) + \frac{\mu}{\lambda} G^{+}((s,t),y(s,t)) \right) \\ & \leq \frac{c^{p}}{r} \left( \frac{\sum\_{t=1}^{b} \sum\_{s=1}^{a} \max\_{0 \leq m \leq c} F((s,t),m)}{c^{p}} + \frac{\mu}{\lambda} \frac{\sum\_{t=1}^{b} \max\_{s=1} G((s,t),m)}{c^{p}} \right). \end{split}$$

According to condition (*i*1), (*g*1) and (19), we have the following:

$$\varphi\_0 \le \frac{p(a+b+2)^{p-1}}{4^p + l\_\*(a+b+2)^{p-1}} (A\_{0^+} + \frac{\mu}{\lambda} \mathbb{C}^{0^+}) < +\infty.$$

We assert that if *λ* ∈ 2*bq*+2*aq*+*p* ˜ *l pqB*0<sup>+</sup> , 4 *<sup>p</sup>*+*l*∗(*a*+*b*+2) *p*−1 *p*(*a*+*b*+2) *<sup>p</sup>*−1*A*0<sup>+</sup> and *µ* ∈ [0, *µ*¯ + *λ* ), then *<sup>λ</sup>* <sup>∈</sup> (0, <sup>1</sup> *ϕ*0 ). In fact, for *λ* ∈ Λ, we have *λ* > 0. When *C* 0 + = 0, then

$$
\varphi\_0 \le \frac{p(a+b+2)^{p-1}}{4^p + l\_\*(a+b+2)^{p-1}} A\_{0^+} < \frac{1}{\lambda}.
$$

When *C* 0 + > 0, then

$$\begin{split} \mathcal{O}\_{0} &< \frac{p(a+b+2)^{p-1}}{4^{p}+l\_{\ast}(a+b+2)^{p-1}} \Biggl( A\_{0^{+}} + \frac{\overline{\mu}\_{\lambda}^{+}}{\lambda} \mathcal{C}^{0^{+}} \Biggr) \\ &= \frac{p(a+b+2)^{p-1}}{4^{p}+l\_{\ast}(a+b+2)^{p-1}} \Bigl( A\_{0^{+}} + \frac{1}{\lambda} \frac{1}{\mathcal{C}^{0^{+}}} \left( \frac{4^{p}+l\_{\ast}(a+b+2)^{p-1}}{p(a+b+2)^{p-1}} - \lambda A\_{0^{+}} \right) \mathcal{C}^{0^{+}} \Biggr) \\ &= \frac{1}{\lambda} .\end{split}$$

Clearly, (0, 0, · · · , 0) ∈ *Y* is a global minima of Φ.

Next, we need to prove that (0, 0, · · · , 0) is not a local minima of *I* + *λ* . Let us prove this in two cases: *B* 0 + = +∞ and *B* 0 + < +∞.

Firstly, when *B* 0 + = +∞, fix *M* such that *M* > 2*aq*+2*bq*+˜ *lp pq* and there exists a sequence of positive numbers {*cn*} such that lim *<sup>n</sup>*→+<sup>∞</sup> *c<sup>n</sup>* = 0, and

$$\sum\_{t=1}^{b} \sum\_{s=1}^{a} F^+((s, t), c\_n) = \sum\_{t=1}^{b} \sum\_{s=1}^{a} F((s, t), c\_n) \ge \frac{\mathcal{M}c\_n^q}{\lambda}, \quad \text{for } n \in \mathbb{Z}(1).$$

Define a sequence {*ηn*} in *Y* with the following:

$$\eta\_n(\mathbf{s}, t) = \begin{cases} \mathbf{c}\_{\text{ll}} \text{ if } (\mathbf{s}, t) \in \mathbf{Z}(1, a) \times \mathbf{Z}(1, b), \\\ \mathbf{0}, \text{ if } \begin{aligned} \mathbf{s} = \mathbf{0}, t \in \mathbf{Z}(0, b + 1), \text{ or, } \mathbf{s} = a + 1, t \in \mathbf{Z}(0, b + 1), \\\ \mathbf{0}, \text{ if } \ t = \mathbf{0}, \mathbf{s} \in \mathbf{Z}(0, a + 1), \text{ or, } \mathbf{t} = b + 1, s \in \mathbf{Z}(0, a + 1). \end{aligned} \end{cases}$$

According to *G* <sup>+</sup>((*s*, *<sup>t</sup>*), *<sup>η</sup>n*(*s*, *<sup>t</sup>*)) = *<sup>G</sup>*((*s*, *<sup>t</sup>*), *<sup>η</sup>n*(*s*, *<sup>t</sup>*)) <sup>≥</sup> 0, (*s*, *<sup>t</sup>*) <sup>∈</sup> **<sup>Z</sup>**(1, *<sup>a</sup>*) <sup>×</sup> **<sup>Z</sup>**(1, *<sup>b</sup>*), we acquire the following:

$$\begin{aligned} I\_\lambda^+(\eta\_n) &\le \left(\frac{2a+2b}{p}\right) c\_n^p + \frac{\tilde{l}}{q} c\_n^q - \lambda \left(\sum\_{t=1}^b \sum\_{s=1}^a F((s,t),c\_n)\right) \\ &\le \left(\frac{2a+2b}{p}\right) c\_n^q + \frac{\tilde{l}}{q} c\_n^q - \mathcal{M} c\_n^q \\ &= \left(\frac{2a+2b}{p} + \frac{\tilde{l}}{q} - \mathcal{M}\right) c\_n^q \\ &< 0. \end{aligned}$$

Secondly, when *B* 0 + < +∞, let *λ* ∈ 2*bq*+2*aq*+*p* ˜ *l pqB*0<sup>+</sup> , 4 *<sup>p</sup>*+*l*∗(*a*+*b*+2) *p*−1 *p*(*a*+*b*+2) *<sup>p</sup>*−1*A*0<sup>+</sup> , choose *ε*<sup>0</sup> > 0 such that ˜ +

$$\frac{2a+2b}{p} + \frac{\bar{l}}{q} - \lambda (B^{0^+} - \varepsilon\_0) < 0.$$

Then, there is a positive sequence {*cn*} ⊂ (0, *<sup>δ</sup>*) such that lim *<sup>n</sup>*→+<sup>∞</sup> *c<sup>n</sup>* = 0 and

$$(\mathcal{B}^{0^+} - \varepsilon\_0)c\_n^q \le \sum\_{t=1}^b \sum\_{s=1}^a F^+((s, t), c\_n) = \sum\_{t=1}^b \sum\_{s=1}^a F((s, t), c\_n) \le (\mathcal{B}^{0^+} + \varepsilon\_0)c\_n^q.$$

By the definition of the sequence {*ηn*} in *Y* being the same as the case where *B* 0 + = +∞, we have the following:

$$\begin{aligned} I\_{\lambda}^{+}(\eta\_{n}) &\leq \left(\frac{2a+2b}{p}\right)c\_{n}^{p} + \frac{\mathsf{I}}{q}c\_{n}^{q} - \lambda \left(\sum\_{t=1}^{b}\sum\_{s=1}^{a}F((s,t),c\_{n})\right) \\ &\leq \left(\frac{2a+2b}{p}\right)c\_{n}^{p} + \frac{\mathsf{I}}{q}c\_{n}^{q} - \lambda (B^{0^{+}} - \varepsilon\_{0})c\_{n}^{q} \\ &\leq \left(\frac{2a+2b}{p} + \frac{\mathsf{I}}{q} - \lambda (B^{0^{+}} - \varepsilon\_{0})\right)c\_{n}^{q} \\ &< 0. \end{aligned}$$

According to the above discussion, we have *I* + *λ* (*ηn*) < 0.

Since *I* + *λ* (0, 0, · · · , 0) = <sup>0</sup> and lim*n*→<sup>∞</sup> *η<sup>n</sup>* = (0, 0, · · · , 0). By combining the above two cases, we obtain that (0, 0, · · · , 0) ∈ *Y* is a global minima of Φ but (0, 0, · · · , 0) is not a local minima of *I* + *λ* .

Through the above discussion, *I* + *λ* satisfies every condition of Lemma 2. According to Lemma 2, there exists a sequence {*un*} of pairwise distinct critical points (local minima) of *I* + *λ* such that lim *<sup>n</sup>*→+<sup>∞</sup> *u<sup>n</sup>* = 0. So, (*s*, *t*) ∈ **Z**(1, *a*) × **Z**(1, *b*), *y*(*s*, *t*) is a non-zero solution of problem (*Dλ*,*<sup>µ</sup>* + ), by Lemma 1, *y*(*s*, *t*) is a positive solution of problem (*Dλ*,*<sup>µ</sup>* ). Therefore, the proof of Theorem 1 is completed.

**Remark 1.** *When the nonlinear terms f and g are symmetric on the origin, i.e., f*(·, −*y*) = −*f*(·, *y*)*, g*(·, −*y*) = −*g*(·, *y*)*, it is easy to obtain infinitely many small solutions to problem* (*Dλ*,*<sup>µ</sup>* ) *by using the critical point theory with symmetries. However, in this paper, we obtain infinitely many small solutions to problem* (*Dλ*,*<sup>µ</sup>* ) *without the symmetry on f .*

When *λ* = 1, according to Theorem 1, we obtain the following.

*,*

**Corollary 2.** *Let f*((*s*, *t*), *y*) *is a continuous function of y, and f*((*s*, *t*), 0) ≥ 0*, g*((*s*, *t*), ·) ∈ *C*(*R*, *R*) *for every* (*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*)*. Suppose the following:*

$$\left(i\_2\right)\_{\frac{p\left(a+b+2\right)p^{-1}A\_{0^+}}{4^p+l\_\*\left(a+b+2\right)^{p-1}}} < 1 < \frac{pqB^{0^+}}{2bq+2aq+p\bar{l}}$$

*<sup>p</sup>*+*l*∗(*a*+*b*+2) (*g*1) *there exsits <sup>δ</sup>* <sup>&</sup>gt; <sup>0</sup> *such that at* [0, *<sup>δ</sup>*]*, G*((*s*, *<sup>t</sup>*), *<sup>y</sup>*) <sup>≥</sup> <sup>0</sup> *and C*<sup>0</sup> + < +∞*. Then, for each µ* ∈ [0, *µ*¯ + 1 )*, the following problem* (*D<sup>µ</sup>* )

$$-\left[\Delta\_1(\phi\_p(\Delta\_1 y(s-1,t)))+\Delta\_2(\phi\_p(\Delta\_2 y(s,t-1)))\right]+l(s,t)\phi\_q(y(s,t))=f((s,t),t)$$
  $\Delta\_1(\phi\_p(\Delta\_2 y(s,t)),(s,t))$   $\Delta\_2(\Delta\_1 y(s,t))\Delta\_1(\Delta\_2 y(s,t))$ 

*y*(*s*, *t*)) + *µg*((*s*, *t*), *y*(*s*, *t*)),(*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*)*, with Dirichlet boundary conditions (1), has a sequence of positive solutions which converges to zero.*

Similarly, we obtain the following results.

**Theorem 2.** *Let f*((*s*, *t*), *y*) *is a continuous function of y, and f*((*s*, *t*), 0) ≤ 0*, g*((*s*, *t*), ·) ∈ *C*(*R*, *R*) *for every* (*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*)*. Suppose the following:*


*Then, for every λ* ∈ 2*bq*+2*aq*+*p* ˜ *l pqB*0<sup>−</sup> , 4 *<sup>p</sup>*+*l*∗(*a*+*b*+2) *p*−1 *p*(*a*+*b*+2) *<sup>p</sup>*−1*A*0<sup>−</sup> *and µ* ∈ [0, *µ*¯ − *λ* )*, problem* (*Dλ*,*<sup>µ</sup>* ) *has a sequence of negative solutions which converges to zero.*

When *λ* = 1, according to Theorem 2, we obtain the following.

**Corollary 3.** *Let f*((*s*, *t*), *y*) *is a continuous function of y, and f*((*s*, *t*), 0) ≤ 0*, g*((*s*, *t*), ·) ∈ *C*(*R*, *R*) *for every* (*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*)*. Suppose the following:*

(*i*4) *p*(*a*+*b*+2) *<sup>p</sup>*−1*A*0<sup>−</sup> 4 *<sup>p</sup>*+*l*∗(*a*+*b*+2) *<sup>p</sup>*−<sup>1</sup> < 1 < *pqB*0<sup>−</sup> 2*bq*+2*aq*+*p*

(*g*2) *there exsits <sup>δ</sup>* <sup>&</sup>gt; <sup>0</sup> *such that at* [−*δ*, 0]*, G*((*s*, *<sup>t</sup>*), *<sup>y</sup>*) <sup>≥</sup> <sup>0</sup> *and C*0<sup>−</sup> <sup>&</sup>lt; <sup>+</sup>∞*. Then, for each µ* ∈ [0, *µ*¯ − 1 )*, problem* (*D<sup>µ</sup>* ) *has a sequence of negative solutions, which converges to zero.*

Combining Theorem 1 with Theorem 2, we have the following.

˜ *l ,*

**Theorem 3.** *Let f*((*s*, *t*), *y*) *is a continuous function of y, and f*((*s*, *t*), 0) = 0*, g*((*s*, *t*), ·) ∈ *C*(*R*, *R*) *for every* (*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*)*. Suppose that*


*,*

,

<sup>0</sup><sup>−</sup> } *<sup>p</sup>*−<sup>1</sup> max{*A*0<sup>+</sup> ,*A*0<sup>−</sup> } *µ*¯ − *λ* })*, problem* (*Dλ*,*<sup>µ</sup>* ) *has two sequences of constant-sign solutions, which converge to zero (with one being positive and the other being negative).*

When *λ* = 1, according to Theorem 3, we acquire the following.

**Corollary 4.** *Let f*((*s*, *t*), *y*) *is a continuous function of y, and f*((*s*, *t*), 0) = 0*, g*((*s*, *t*), ·) ∈ *C*(*R*, *R*) *for every* (*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*)*. Suppose the following:*


*Then, for each µ* ∈ [0, min{*µ*¯ + 1 , *µ*¯ − 1 })*, problem* (*D<sup>µ</sup>* ) *has two sequences of constant-sign solutions which converge to zero (with one being positive and the other being negative).*

**Remark 2.** *As a special case of Theorem 1, when µ* = 0*.*

*Considering the following problem, namely* (*D<sup>λ</sup>* )

−[∆1(*φp*(∆1*y*(*s* −1, *t*))) + ∆2(*φp*(∆2*y*(*s*, *t* −1)))] + *l*(*s*, *t*)*φq*(*y*(*s*, *t*)) = *λ f*((*s*, *t*), *y*(*s*, *t*)), (*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*)*, with Dirichlet boundary conditions (1).*

**Theorem 4.** *Let f*((*s*, *t*), *y*) *be a continuous function of y, and f*((*s*, *t*), 0) ≥ 0 *for every* (*s*, *t*) ∈ *Z*(1, *a*) × *Z*(1, *b*)*. Suppose the following:*

(*i*7) *p*(*a*+*b*+2) *<sup>p</sup>*−1*A*0<sup>+</sup> 4 *<sup>p</sup>*+*l*∗(*a*+*b*+2) *<sup>p</sup>*−<sup>1</sup> < *pqB*0<sup>+</sup> 2*bq*+2*aq*+*p* ˜ *l .*

*Then, for each λ* ∈ 2*bq*+2*aq*+*p* ˜ *l pqB*0<sup>+</sup> , 4 *<sup>p</sup>*+*l*∗(*a*+*b*+2) *p*−1 *p*(*a*+*b*+2) *<sup>p</sup>*−1*A*0<sup>+</sup> *, problem* (*D<sup>λ</sup>* ) *has a sequence of positive solutions, which converges to zero.*

#### **4. Example**

We provide an example to illustrate our Theorem 3.

**Example 1.** *Suppose that l*(*s*, *t*) = *s* + *t. Let a* = 2*, b* = 2*, p* = 3*, q* = 2*, and* ˜ *l* = 2 ∑ *t*=1 2 ∑ *s*=1 *l*(*s*, *t*) = 12 < 16+18*l*<sup>∗</sup> <sup>3</sup> <sup>=</sup> <sup>52</sup> 3 *, f and g are two functions defined as follows:*

$$f((s,t),c) = f(c) = \begin{cases} \frac{5}{4}pc^{p-1} + pc^{p-1}\sin(\frac{1}{5}\ln c^p) + \frac{1}{5}pc^{p-1}\cos(\frac{1}{5}\ln c^p), & c > 0, \\ 0, & c \le 0, \end{cases} \tag{20}$$

*and*

$$\lg((s,t),c) = \lg(c) = 2pc^{p-1}.\tag{21}$$

*Then, for each λ*<sup>1</sup> ∈ ( 49 54 , 34 <sup>27</sup> ) *and µ*<sup>1</sup> ∈ [0,( 34 <sup>216</sup> <sup>−</sup> <sup>1</sup> 8 *λ*1))*, the following problem, namely* (*Dλ*<sup>1</sup> ,*µ*<sup>1</sup> )*.*

−[∆1(*φp*(∆1*y*(*s* − 1, *t*))) + ∆2(*φp*(∆2*y*(*s*, *t* − 1)))] + *l*(*s*, *t*)*φq*(*y*(*s*, *t*)) = *λ*<sup>1</sup> *f*((*s*, *t*), *y*(*s*, *t*)) + *µ*1*g*((*s*, *t*), *y*(*s*, *t*)),(*s*, *t*) ∈ *Z*(1, 2) × *Z*(1, 2)*, with the following Dirichlet boundary conditions:*

$$y(s,0) = y(s,2) = 0, \quad s \in \mathbf{Z}(0,3),$$

$$y(0,t) = y(2,t) = 0, \quad t \in \mathbf{Z}(0,3),$$

*possesses two sequences of constant-sign solutions, which converge to zero (with one being positive and the other being negative).*

*In fact,*

$$F((s,t),c) = \int\_0^c f((s,t),\tau)d\tau = \begin{cases} \frac{5}{4}c^p + c^p \sin(\frac{1}{5}\ln c^p), & c > 0, \\ 0, & c \le 0, \end{cases} \tag{22}$$

$$G((s,t),\mathbf{c}) = \int\_0^\mathbf{c} g((s,t),\tau)d\tau = 2\mathbf{c}^p. \tag{23}$$

*c*

*Since f*((*s*, *t*), *c*) > 0, *g*((*s*, *t*), *c*) > 0 *for c* > 0*, we know that F*((*s*, *t*), *c*) *and G*((*s*, *t*), *c*) *are increasing at c* ∈ (0, +∞)*. Thus,* max 0≤*m*≤*c F*((*s*, *t*), *m*) = *F*((*s*, *t*), *c*) *and* max 0≤*m*≤*c G*((*s*, *t*), *m*) = *G*((*s*, *t*), *c*)*, for every c* > 0*. Obviously,*

$$A\_{0\*} = \liminf\_{c \to 0^+} \frac{abF((s, t), c)}{c^p} = \liminf\_{c \to 0^+} \frac{4(\frac{5}{4}c^p + c^p \sin(\frac{1}{5}\ln c^p))}{c^p} = 1,$$

$$B^{0\*} = \limsup\_{c \to 0^+} \frac{abF((s, t), c)}{c^p} = \limsup\_{c \to 0^+} \frac{4(\frac{5}{4}c^p + c^p \sin(\frac{1}{5}\ln c^p))}{c^p} = 9.$$

*<sup>c</sup>*→0<sup>+</sup> *c <sup>c</sup>*→0<sup>+</sup> *We can verify condition* (*i*5) *of Theorem 3 since*

$$\frac{p(a+b+2)^{p-1}\max\{A\_{0^+}, A\_{0^-}\}}{4^p + l\_\*(a+b+2)^{p-1}} = \frac{27}{34} < \frac{pq\min\{B^{0^+}, B^{0\_-}\}}{2bq + 2aq + p\tilde{l}} = \frac{54}{49}.$$

*Next, we can further verify condition* (*g*3) *of Theorem 3 since*

$$C^{0\*} = \limsup\_{c \to 0^+} \frac{\sum\_{t=1}^b \sum\_{s=1}^a G((s, t), c)}{c^p} = \limsup\_{c \to 0^+} \frac{ab2c^p}{c^p} = 8 < +\infty.$$

*In summary, every condition of Theorem 3 is met.*

*Therefore, for each λ*<sup>1</sup> ∈ ( 49 54 , 34 <sup>27</sup> ) *and µ*<sup>1</sup> ∈ [0,( 34 <sup>216</sup> <sup>−</sup> <sup>1</sup> 8 *λ*1))*, problem* (*Dλ*<sup>1</sup> ,*µ*<sup>1</sup> ) *possesses two sequences of constant-sign solutions, which converge to zero (with one being positive and the other being negative).*

#### **5. Conclusions**

In this paper, we studied the existence of small solutions of perturbed partial discrete Dirichlet problems with the (*p*, *q*)-Laplacian operator. Unlike the results in [25], we obtained some sufficient conditions of the existence of infinitely many small solutions, as shown in Theorems 1–3. Firstly, according to Theorem 4.3 of [28] and Lemma 1 of this paper, we obtained a sequence of positive solutions, which converges to zero in Theorem 1. Furthermore, by truncation techniques, we acquired two sequences of constant-sign solutions, which converge to zero (with one being positive and the other being negative). Secondly, the Corollaries 2–4 was acquired when *λ* = 1. Finally, as a special case of Theorem 1, we obtained a sequence of positive solutions, which converges to zero in Theorem 4. The existence of large constant-sign solutions of partial difference equations with the (*p*, *q*)-Laplacian operator will be discussed by the method used in this paper as our future research direction.

**Author Contributions:** All authors contributed equally to this paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is supported by the National Natural Science Foundation of China (grant No. 11971126), the Program for Changjiang Scholars and Innovative Research Team in University (grant No. IRT\_16R16) and the Innovation Research for the Postgraduates of Guangzhou University (grant No. 2020GDJC-D07).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interests.

#### **References**


## *Article* **Vibration Properties of a Concrete Structure with Symmetries Used in Civil Engineering**

**Sorin Vlase 1,2 , Marin Marin 3,\* and Ovidiu Deaconu <sup>4</sup>**


**Abstract:** The paper aims to study a concrete structure, currently used in civil engineering, which has certain symmetries. This type of problem is common in engineering practice, especially in civil engineering. There are many reasons why structures with identical elements or certain symmetries are used in industry, related to economic considerations, shortening the design time, for constructive, simplicity, cost or logistical reasons. There are many reasons why the presence of symmetries has benefits for designers, builders, and beneficiaries. In the end, the result of these benefits materializes through short execution times and reduced costs. The paper studies the eigenvalue and eigenmode properties of vibration for components of the constructions' structure, often encountered in current practice. The identification of such properties allows the simplification and easing of the effort necessary for the dynamic analysis of such a structure.

**Keywords:** vibrations; symmetrical structures; eigenmodes; building; concrete

#### **1. Introduction**

Frequently encountered in the design and construction of structures used in civil engineering, symmetries allow, in many cases, the simplification of calculations and dynamic analysis of such a structure. The direct consequence would be the shortening of the design and execution time and, of course, the decrease of the costs generated by these stages. The smaller information provided by a repetitive or symmetrical structure can help ease the computational effort. In the case of a static calculation, methods of approaching this problem are presented in the Strength of Materials courses. In the dynamic case, considering the elastic elements and studying the vibrations, although certain properties have long been observed [1], a systematic study of the problem has not yet been done. A case in which the symmetries introduced by two identical motors and their effect on vibrations were considered was studied [2]. The use of identical systems was applied to the design of a centrifugal pendulum vibration absorber system [3]. Circular symmetry and its induced properties have been reported previously [4,5]. Other particular cases have been studied [6] using a finite difference method and [7] for continuous systems. In the following, we will study the case of a mechanical system consisting of four trusses, two of which are identical. The transverse and torsional vibrations of such a system that are strongly coupled for the chosen case will be studied.

In the field of engineering, not only in civil engineering, but also in other fields such as the machine or machinery manufacturing industry, the automotive industry, and the aerospace industry, there are products, parts of products, machines, and components that contain identical, repetitive elements, which have, in their composition, parts that show symmetries of different types.

**Citation:** Vlase, S.; Marin, M.; Deaconu, O. Vibration Properties of a Concrete Structure with Symmetries Used in Civil Engineering. *Symmetry* **2021**, *13*, 656. https://doi.org/ 10.3390/sym13040656

Academic Editors: Jan Awrejcewicz and Calogero Vetro

Received: 22 March 2021 Accepted: 9 April 2021 Published: 12 April 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Until now, symmetries in Mechanics have been studied mainly from the point of view of the mathematics involved [8], as they have effects in writing equations of motion [9], but their applications in practice are little studied [10]. In January 2018, a special issue of the Symmetry review dedicated to applications in Structural Mechanics (Civil Engineering and Symmetry-2018, ISSN 2073-8994, [11]) was launched. The course Similarity, Symmetry and Group Theoretical Methods in Mechanics was organized at the Center for Solid Mechanics-CISM from UDINE (see ref. [12]).

The use of substructures in the design of the aeronautical industry is a relatively common procedure used to reduce working time. Finite element models of different parts are condensed into substructures. Obviously a substructure contains less degree of freedom (DOF) than the entire structure and it is easier to model it. Likewise, for identical parts that are found in the construction and design of aeronautical panels, if the substructure has been generated for one part, the generation of the whole assembly becomes easier to do. A method reduces the size of the model, useful in the design stage and in the manufacture of the entire structure. The accuracy of the finite element analysis performed is maintained. An illustration of this method is presented in a previous work [13].

The modeling of mechanical systems with repetitive or identical parts leads, finally, to systems of differential equations that describe the answer of such systems, which have, in their component, strings of identical terms. This feature leads to simpler methods of solving these systems of differential equations. Such an issue is addressed in studies [14,15].

The vast majority of buildings, works of art, halls, and in general, the constructions, have identical parts and have symmetries. It is a situation that has existed since the beginning of the first constructions made by man (antiquity) and the reasons are of several kinds as an easier, faster design, then a cheaper realization and, less important for engineers but important for beneficiaries, for aesthetic reasons. The structures have in their composition repetitive elements or present different forms of symmetry. These properties can be used successfully to facilitate static and dynamic analysis.

Group theory has been used extensively to study various phenomena in physics and chemistry, such as quantum mechanics, crystallography, and molecular structure. However, this theory can find a fertile field of application in engineering. This allows simplifying the analysis of systems that have certain symmetries or identical parts. In this way, it was efficient in the study of vibrations or the dynamic or kinematic analysis of mechanical systems. The use of group theory in engineering was analyzed [16,17]. Different aspects of the use of symmetry in engineering are presented in other papers. In a research [18], the influence of the symmetry of boundary condition in the description of the models was studied. Some theoretical basis and an attempt to classify the symmetries that occur in structures were presented [19] and application of the symmetry in engineering structures were presented [20,21].

In recent years, new and interesting methods of studying this type of problem have been studied by researchers [22,23] and new ways to deal with such problems were reported [24,25].

However, there are still many situations that can be studied and, therefore, the present paper aims to complete the cases studied and to offer proposals for the application of these properties that could help a design engineer to ease his effort.

#### **2. Model and Free Vibration Response**

We have 2 coplanar reinforced concrete beams with different properties. AD beam is a main beam, considered double clamped at both ends, with the length *L*<sup>1</sup> + *L*2. CB beam is a secondary beam, considered simply supported at both ends (nodes C and B), with length *L*<sup>3</sup> + *L*3. The beams are made monolithically, so at point O of the intersection, a rigid knot is created. The secondary beam is arranged symmetrically to the main beam. The 2 beams have the Young's moduli *E*<sup>1</sup> and *E*2. The 2 beams have different sectional properties: for beam AD, we have the moment of inertia Iz1 and the area of section *A*1, and for beam

*Symmetry* **2021**, *13*, x FOR PEER REVIEW 3 of 18

*Symmetry* **2021**, *13*, x FOR PEER REVIEW 3 of 18

**2. Model and Free Vibration Response** 

**2. Model and Free Vibration Response** 

CB, we have the moment of inertia *Iz*<sup>2</sup> and the area of section *A*2, with the property that *Iz*<sup>1</sup> > *Iz*<sup>2</sup> and *A*<sup>1</sup> > *A*2. The whole structure is presented in Figure 1. for beam AD, we have the moment of inertia Iz1 and the area of section *A*1, and for beam CB, we have the moment of inertia *Iz*2 and the area of section *A*2, with the property that *Iz*<sup>1</sup> > *Iz*2 and *A*<sup>1</sup> > *A*2. The whole structure is presented in Figure 1. CB, we have the moment of inertia *Iz*2 and the area of section *A*2, with the property that *Iz*<sup>1</sup> > *Iz*2 and *A*<sup>1</sup> > *A*2. The whole structure is presented in Figure 1.

We have 2 coplanar reinforced concrete beams with different properties. AD beam is a main beam, considered double clamped at both ends, with the length *L*1 + *L*2. CB beam is a secondary beam, considered simply supported at both ends (nodes C and B), with length *L*3 + *L*3. The beams are made monolithically, so at point O of the intersection, a rigid knot is created. The secondary beam is arranged symmetrically to the main beam. The 2 beams have the Young's moduli *E*1 and *E*2. The 2 beams have different sectional properties:

We have 2 coplanar reinforced concrete beams with different properties. AD beam is a main beam, considered double clamped at both ends, with the length *L*1 + *L*2. CB beam is a secondary beam, considered simply supported at both ends (nodes C and B), with length *L*3 + *L*3. The beams are made monolithically, so at point O of the intersection, a rigid knot is created. The secondary beam is arranged symmetrically to the main beam. The 2 beams have the Young's moduli *E*1 and *E*2. The 2 beams have different sectional properties: for beam AD, we have the moment of inertia Iz1 and the area of section *A*1, and for beam

**Figure 1.** Structure with repetitive cells. **Figure 1.** Structure with repetitive cells. **Figure 1.** Structure with repetitive cells.

Thus, the model of the mechanical system considered (Figure 2) consists of two identical trusses OB and OC rigidly fixed, perpendicular to a third bar AOD. The trusses can have transverse vibrations in a direction perpendicular to the ABDC plane and torsional vibrations. The trusses are clamped in the points B and C. In A and D, the AOD trusses are clamped, so the displacement, slope, and torsion angle at these points are zero. For point O, the transverse displacements of point O belonging to all four bars are equal. The torsion angle of the truss AO in O is equal to the torsion angle of the truss OD in O and with the slope of the bars OB and OC in O. The torsion angle of the trusses OB and OC in O is equal to the slope of the trusses AO and OD in point O. The sum of the shear forces and moments in O will be zero. Thus, the model of the mechanical system considered (Figure 2) consists of two identical trusses OB and OC rigidly fixed, perpendicular to a third bar AOD. The trusses can have transverse vibrations in a direction perpendicular to the ABDC plane and torsional vibrations. The trusses are clamped in the points B and C. In A and D, the AOD trusses are clamped, so the displacement, slope, and torsion angle at these points are zero. For point O, the transverse displacements of point O belonging to all four bars are equal. The torsion angle of the truss AO in O is equal to the torsion angle of the truss OD in O and with the slope of the bars OB and OC in O. The torsion angle of the trusses OB and OC in O is equal to the slope of the trusses AO and OD in point O. The sum of the shear forces and moments in O will be zero. Thus, the model of the mechanical system considered (Figure 2) consists of two identical trusses OB and OC rigidly fixed, perpendicular to a third bar AOD. The trusses can have transverse vibrations in a direction perpendicular to the ABDC plane and torsional vibrations. The trusses are clamped in the points B and C. In A and D, the AOD trusses are clamped, so the displacement, slope, and torsion angle at these points are zero. For point O, the transverse displacements of point O belonging to all four bars are equal. The torsion angle of the truss AO in O is equal to the torsion angle of the truss OD in O and with the slope of the bars OB and OC in O. The torsion angle of the trusses OB and OC in O is equal to the slope of the trusses AO and OD in point O. The sum of the shear forces and moments in O will be zero.

**Figure 2.** One repetitive cell. **Figure 2.** One repetitive cell. **Figure 2.** One repetitive cell.

We will study a continuous concrete truss, homogeneous, with constant section. If there are no forces distributed or concentrated along the length of the truss, the vibrations of this are described by the well-known Equations [26]: We will study a continuous concrete truss, homogeneous, with constant section. If there are no forces distributed or concentrated along the length of the truss, the vibrations of this are described by the well-known Equations [26]: We will study a continuous concrete truss, homogeneous, with constant section. If there are no forces distributed or concentrated along the length of the truss, the vibrations of this are described by the well-known Equations [26]:

$$\frac{\partial^4 v}{\partial x^4} + \frac{\rho A}{EI\_z} \frac{\partial^2 v}{\partial t^2} = 0 \tag{1}$$

The notations used in Equation (1) are the following: *v*—is the truss deflection, *A*—*s* the cross section, *ρ* is the mass density, *E*—Young's modulus, and *I<sup>z</sup>* is the second area moment of inertia with respect to the *z* axis and *x* is the ordinate of the point having the deflection *v*.

To solve Equation (1), we look for a solution of the form [27–29]:

$$v(\mathbf{x},t) = \Phi(\mathbf{x})\sin(pt+\theta). \tag{2}$$

Equation (2) must check Equation (1) at any time and, imposing this condition, we obtain:

$$\frac{\partial^4 \Phi}{\partial x^4} - p^2 \frac{\rho A}{EI\_z} \Phi = 0 \tag{3}$$

Denote:

$$
\lambda^4 = \frac{\rho A}{EI\_z} \tag{4}
$$

Then, Equation (1) becomes:

$$\frac{\partial^4 \Phi}{\partial x^4} - p^2 \lambda^4 \Phi = 0 \tag{5}$$

In Equation (5), Φ represents the function, depending on the abscissa *x*, which gives us the deformed bar (eigenmode) corresponding to its eigenpulsation *p*. The solution is:

$$\Phi(\mathbf{x}) = \mathbb{C}\_1 \sin(\lambda \sqrt{p} \mathbf{x}) + \mathbb{C}\_2 \cos(\lambda \sqrt{p} \mathbf{x}) + \mathbb{C}\_3 \text{sh}(\lambda \sqrt{p} \mathbf{x}) + \mathbb{C}\_4 \text{ch}(\lambda \sqrt{p} \mathbf{x}).\tag{6}$$

The constants *C*1, *C*2, *C*3, *C*<sup>4</sup> are determined considering the boundary conditions for this problem.

In the following, we will use Equation (5) for the domains defined by the four trusses, obtaining, in this way, four differential equations of the fourth order, corresponding to the frames AO, OD, OB, OC (see Appendix A).

The study of torsional vibrations for a straight bar, unloaded over the length, leads to the second order differential equation:

$$\frac{\partial^2 \varphi}{\partial x^2} - \frac{J}{GI\_p} \frac{\partial^2 \varphi}{\partial t^2} = 0 \tag{7}$$

where *ϕ* is the angle of torsion of the cross section being at the distance *x* from the end of the truss, *J* = *ρI<sup>p</sup>* is the polar moment of inertia, *I<sup>p</sup>* is the polar second moment of the area, and *G* is shear's modulus.

The solution Equation (7) is sought in the form:

$$
\varphi(\mathbf{x},t) = \psi(\mathbf{x})\sin(pt+\theta) \tag{8}
$$

Equation (8) must verify Equation (7), from which we obtain:

$$\frac{\partial^2 \psi}{\partial x^2} + p^2 \delta\_i^2 \Phi = 0 \tag{9}$$

where the notation was made:

$$
\delta^2 = \frac{J}{G|I\_p} \tag{10}
$$

The solution is:

$$\psi(\mathbf{x}) = D\_1 \sin(\delta p \mathbf{x}) + D\_2 \cos(\delta p \mathbf{x}) \tag{11}$$

Denoted by *M<sup>b</sup>* is the bending moment of a bar in section *x*, *T* the shear force that appears in the cross section and with *M<sup>t</sup>* the torque. For the studied system in the paper, the boundary conditions are:


(d) For the OC truss, the end C is clamped, so: *vOC*(*L*3, *t*) = 0; *M<sup>b</sup> OC*(*L*3, *t*) = 0; *M<sup>t</sup> OC*(*L*3, *t*) = 0.

Imposing these conditions, for the considered four bars, 12 boundary conditions are obtained. The two moments and the shear force are expressed by the known relationships in the mechanics of the deformable solid [30–33]:

$$M^b(\mathbf{x}) = -EI\_z \frac{\partial^2 v(\mathbf{x})}{\partial \mathbf{x}^2};\ T(\mathbf{x}) = EI\_z \frac{\partial^3 v(\mathbf{x})}{\partial \mathbf{x}^3};\ M^t(\mathbf{x}) = GI\_p \frac{\partial \rho(\mathbf{x})}{\partial \mathbf{x}}\tag{12}$$

Using the Equation (12), the solutions presented in Appendixes, and the boundary conditions (a), (b), (c), and (d) result in 12 linear equations involving the unknown constants (Appendix A).

The continuity of the elastic system at point O leads to the following conditions: we have once that the displacements in C for all trusses: *vAO*(*L*2, *t*) = *vOD*(0, *t*) = *vOB*(0, *t*) = *vOC*(0, *t*), (three conditions), the slopes in O of the trusses AO and OD are equal to the torsion angle of the trusses OB and OC in: *v* 0 *AO*(*L*2, *t*) = *v* 0 *OD*(0, *t*) = *ϕOB*(0, *t*) = −*ϕOC*(0, *t*), (three conditions), and the torsion angles of the bars AO and OD in O are equal to the slopes of the bar OB and OC in O: *ϕAO*(*L*2, *t*) = *ϕOD*(0, *t*) = *v* 0 *OB*(0, *t*) = −*v* 0 *OC*(0, *t*)) (three conditions). These mean nine conditions presented in the Appendix B, from which nine linear equations result.

Three more conditions are needed to obtain the constants in the written differential equations. They are obtained by considering the equilibrium of an infinitesimal element of the mass containing the point O.

The sum of the four shear forces in O must be zero, so:

$$T1 + T2 + T3 - T4 = \begin{array}{c} \text{0} \end{array} \tag{13}$$

A similar equilibrium relation is obtained for bending and torsional moments:

$$M^{b1} + M^{b2} + M^{t3} - M^{t4} = 0,\tag{14}$$

$$M^{t1} + M^{t2} + M^{b3} - M^{b4} = 0\tag{15}$$

Writing these three conditions results in three linear equations (Appendix C). The unknowns are:

$$\begin{Bmatrix} B^{AO} \end{Bmatrix} = \begin{bmatrix} \ \mathbb{C}\_1^{AO} & \mathbb{C}\_2^{AO} & \mathbb{C}\_3^{AO} & \mathbb{C}\_4^{AO} & D\_1^{AO} & D\_2^{AO} \end{bmatrix} \tag{16}$$

$$\left\{ \mathcal{B}^{\text{OD}} \right\} = \left[ \begin{array}{cc} \mathcal{C}\_1^{\text{OD}} & \mathcal{C}\_2^{\text{OD}} & \mathcal{C}\_3^{\text{OD}} & \mathcal{C}\_4^{\text{OD}} & D\_1^{\text{OD}} & D\_2^{\text{OD}} \end{array} \right] \tag{17}$$

$$\begin{Bmatrix} \mathcal{B}^{OB} \end{Bmatrix} = \begin{bmatrix} \mathcal{C}\_1^{OB} & \mathcal{C}\_2^{OB} & \mathcal{C}\_3^{OB} & \mathcal{C}\_4^{OB} & D\_1^{OB} & D\_2^{OB} & \cdots \end{bmatrix} \tag{18}$$

$$\left\{{B^{\rm OC}}\right\} = \left[ \begin{array}{cc} \mathbf{C}\_1^{\rm OC} & \mathbf{C}\_2^{\rm OC} & \mathbf{C}\_3^{\rm OC} & \mathbf{C}\_4^{\rm OC} & D\_1^{\rm OC} & D\_2^{\rm OC} \end{array} \right] \tag{19}$$

or:

$$\{B\} = \left\{ \begin{array}{c} \{B^{AO}\} \\ \{B^{OD}\} \\ \{B^{OB}\} \\ \{B^{OC}\} \end{array} \right\} \tag{20}$$

In such way, a homogeneous linear system with 24 equations with 24 unknowns was obtained. In order to have other solutions besides the trivial solution zero, the determinant of the system must be zero. Putting this condition, the obtained eigenfrequencies of the system can be determined from the obtained equation.

The continuous models used in our studies are excellent for a classical analysis of such systems. The boundary conditions, written for these models, ultimately lead to a linear system of homogeneous equations. For this system to have a solution other than the trivial solution, zero, it is necessary that the system determinant be equal to zero. Imposing this condition leads to writing the characteristic equation that will provide, by solving, its eigenfrequencies of vibration. Once these values are known, then, we can determine your eigenmodes for these frequencies.

The matrix of the system can be written, in a concise form:

$$A\_{24 \times 24} \left[ \begin{array}{cc} A\_{11} & 0 \\ 0 & A\_{11} & A\_{12} \\ A\_{21} & A\_{21} & A\_{22} \end{array} \right] \tag{21}$$

where the matrices - *Aij* are presented in Appendix C.

$$[A]\{B\} = \{0\} \tag{22}$$

The condition:

$$\det(A) = 0\tag{23}$$

offers the eigenvalues of the system of differential equations.

#### **3. Properties of the Eigenvalues and Eigenmodes**

Let us now consider one of the identical trusses OB and OC. The truss is clamped in O and supported in B (or C). Equations (1) and (7) are also valid for the OB bar (OC), with the boundary conditions:

For point O:

$$\text{tr} = 0; \ v(0, t) = 0; \ v'(0, t) = 0; \ q(0, t) = 0 \tag{24}$$

and for the point B(C):

$$\text{tr} = L\_{\eth \prime} \, v(L\_{\eth \prime}t) = 0 \text{; } M^b(L\_{\eth \prime}t) = 0 \text{; } M^t(L\_{\eth \prime}t) = 0 \tag{25}$$

The solution is:

$$\Phi(\mathbf{x}) = \mathbb{C}\_1 \sin(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbb{C}\_2 \cos(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbb{C}\_3 \text{sh}(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbb{C}\_4 \text{ch}(\lambda\_2 \sqrt{p} \mathbf{x})\tag{26}$$

for transversal vibrations and:

$$\psi(\mathbf{x}) = D\_1 \sin(\delta\_2 p \mathbf{x}) + D\_2 \cos(\delta\_2 p \mathbf{x}) \tag{27}$$

with the imposed boundary conditions for this case:

$$\mathcal{C}\_2^{OB} + \mathcal{C}\_4^{OB} = 0; \; \mathcal{C}\_1^{OB} + \mathcal{C}\_3^{OB} = 0; \; D\_2^{OB} = 0 \tag{28}$$

$$\mathcal{C}\_1^{\rm BO}\sin(\lambda\_2\sqrt{p}L\_3) + \mathcal{C}\_2^{\rm BO}\cos(\lambda\_2\sqrt{p}L\_3) + \mathcal{C}\_3^{\rm BO}\sin(\lambda\_2\sqrt{p}L\_3) + \mathcal{C}\_4^{\rm BO}\sin(\lambda\_2\sqrt{p}L\_3) = 0 \tag{29}$$

$$-\mathbb{C}\_1^{\rm BO}\sin(\lambda\_2\sqrt{p}\,L\_3) - \mathbb{C}\_2^{\rm BO}\cos(\lambda\_2\sqrt{p}\,L\_3) + \mathbb{C}\_3^{\rm BO}\sin(\lambda\_2\sqrt{p}\,L\_3) + \mathbb{C}\_4^{\rm BO}\mathrm{ch}(\lambda\_2\sqrt{p}\,L\_3) = 0\tag{30}$$

$$D\_1^{BO}\cos(\delta\_2 pL\_3) - D\_2^{BO}\sin(\delta\_2 pL\_3) = 0\tag{31}$$

Using conditions expressed by Equations (28)–(31), it is now possible to determine the constants *C BO* 1 , *C BO* 2 , *C BO* 3 , *C BO* 4 , *DBO* 1 , *DBO* 2 from the linear homogenous system:

$$\left\{ \left[ A\_{11} \right] \left\{ B^{OB} \right\} \right\} = 0 \tag{32}$$

where [*A*11] is the matrix determined by Equation (A76).

If the condition of the existence of non-zero solutions is set: det (*A*11) = 0 (33)

*Symmetry* **2021**, *13*, x FOR PEER REVIEW 7 of 18

$$\det\left(A\_{11}\right) = 0\tag{33}$$

It is now possible to obtain the eigenvalues of the truss OB (or OC). The following theorems will be proved in the following: **Theorem 1.** *The eigenvalues for the OB truss, clamped at one end and supported at the other, are also eigenvalues for the entire mechanical system.* 

If the condition of the existence of non-zero solutions is set:

**Theorem 1.** *The eigenvalues for the OB truss, clamped at one end and supported at the other, are also eigenvalues for the entire mechanical system.* **Proof.** We must show that det (A) = 0 implies det (S) = 0. In reference [34], this property is

It is now possible to obtain the eigenvalues of the truss OB (or OC).

**Proof.** We must show that det (A) = 0 implies det (S) = 0. In reference [34], this property is proved in a more general case. It turns out that the property is valid in our case. proved in a more general case. It turns out that the property is valid in our case. **□** It follows that the eigenvalues of a single truss in Appendix D, clamped at one end

It follows that the eigenvalues of a single truss in Appendix D, clamped at one end and supported at the other, are also eigenvalues of the composed system, clamped in A and D and with the ends B and C supported. and supported at the other, are also eigenvalues of the composed system, clamped in A and D and with the ends B and C supported. Using the matrix done by Equation (A76), and obtaining the eigenvalues for this ma-

Using the matrix done by Equation (A76), and obtaining the eigenvalues for this matrix, the eigenmodes of deformations are obtained using Equation (A85). The following two theorems will be proved: trix, the eigenmodes of deformations are obtained using Equation (A85). The following two theorems will be proved:

**Theorem 2.** *For eigenvalues that are common to the whole mechanical system (Figure 3) and to the subsystem in Figure A1 (see Theorem 1), eigenvectors are of the form:* **Theorem 2.** *For eigenvalues that are common to the whole mechanical system (Figure 3) and to the subsystem in Figure A1 (see Theorem 1), eigenvectors are of the form:* 

$$
\Phi = \left\{ \begin{array}{c} \Phi\_1 \\ - \end{array} \right\} \tag{34}
$$

*(the existence of common eigenvalues is proved by theorem T1). (the existence of common eigenvalues is proved by theorem T1).*

**Figure 3.** Geometry of a repetitive cell. **Figure 3.** Geometry of a repetitive cell.

some simple calculations:

where:

**Proof.** For the eigenvalues obtained from Equation (33), the following system must be solved: **Proof.** For the eigenvalues obtained from Equation (33), the following system must be solved:

$$
\begin{bmatrix} A\_{11} & 0 & A\_{12} \\ 0 & A\_{11} & A\_{12} \\ A\_{21} & A\_{21} & A\_{22} \end{bmatrix} \begin{Bmatrix} \Phi\_{OB} \\ \Phi\_{OC} \\ \Phi\_{AOD} \end{Bmatrix} = \begin{Bmatrix} 0 \\ 0 \\ 0 \end{Bmatrix} \tag{35}
$$

Suppose that we determined this vector. Equation (35) becomes, after performing

det ଵଵ = 0 (36)

ଵଵை = 0 (37)

where:

$$\det A\_{11} = \begin{array}{c} 0 \end{array} \tag{36}$$

Condition (36) implies that a vector Φ*OB* can be found, such that:

$$A\_{11} \Phi\_{\text{OB}} = 0 \tag{37}$$

Suppose that we determined this vector. Equation (35) becomes, after performing some simple calculations:

$$A\_{11}\Phi\_{\rm OB} + A\_{12}\Phi\_{\rm AOD} = 0\tag{38}$$

$$A\_{11}\Phi\_{\rm OC} + A\_{12}\Phi\_{\rm AOD} = 0\tag{39}$$

$$A\_{21}(\Phi\_{\rm OB} + \Phi\_{\rm OC}) + A\_{22}\Phi\_{\rm AOD} = 0\tag{40}$$

If we take into account Equation (37), the system of Equations (38)–(40) becomes:

$$A\_{12} \Phi\_{AOD} = 0\tag{41}$$

$$A\_{11}\Phi\_{\text{OC}} + A\_{12}\Phi\_{A\text{OD}} = 0\tag{42}$$

$$A\_{21}(\Phi\_{\rm OB} + \Phi\_{\rm OC}) + A\_{22}\Phi\_{\rm AOD} = 0\tag{43}$$

From Equation (41), because, in general, det*A*<sup>12</sup> 6= 0, it follows immediately:

$$
\Phi\_{A\text{OD}} = 0\tag{44}
$$

and introducing that in Equation (43), we obtain Φ*OB* = −Φ*OC*, a relation which verifies also Equation (42), if we take into account Equation (37). If Φ*OB* = Φ<sup>1</sup> is denoted, it results in Equation (34).

**Theorem 3.** *For the other eigenvalues of the system, the eigenvectors are of the form:*

$$\Phi = \left\{ \begin{array}{l} \Phi\_1 \\ \Phi\_1 \\ \Phi\_3 \end{array} \right\} \tag{45}$$

**Proof.** For the eigenvalues calculated, the system of Equation (35) must be solved, with det*A*<sup>11</sup> 6= 0 or:

$$A\_{11}\Phi\_{OB} + A\_{12}\Phi\_{AOD} = 0\tag{46}$$

$$A\_{11} \Phi\_{\rm OC} + A\_{12} \Phi\_{AOD} = 0 \tag{47}$$

$$A\_{21}(\Phi\_{\rm OB} + \Phi\_{\rm OC}) + A\_{22}\Phi\_{\rm AOD} = 0\tag{48}$$

Subtracting (47) from (46), we get:

$$A\_{11}(\Phi\_{OB} - \Phi\_{OC}) = 0\tag{49}$$

If det *A* 6= 0, it results in Φ*<sup>s</sup>* − Φ*<sup>m</sup>* = 0 and Φ*<sup>s</sup>* = Φ*<sup>m</sup>* = Φ1.

A block diagram representing the stage of the analysis is presented in Figure 4.

**Figure 4.** Block diagram of the operation. **Figure 4.** Block diagram of the operation.

For the eigenfrequencies of the system that are the same with the eigenvalues of a single clamped truss at one end and supported at the other, the eigenmodes are skew For the eigenfrequencies of the system that are the same with the eigenvalues of a single clamped truss at one end and supported at the other, the eigenmodes are skew symmetric, the two identical trusses vibrate in counterphase, and the third truss AOD rests. For the other eigenfrequencies, the identical trusses have identical eigenmodes.

ଵଵை + ଵଶை = 0 (38) ଵଵை + ଵଶை = 0 (39) ଶଵ(ை + ை) + ଶଶை = 0 (40)

ଵଶை = 0 (41) ଵଵை + ଵଶை = 0 (42) ଶଵ(ை + ை) + ଶଶை = 0 (43)

ை = 0 (44)

ଵଵை + ଵଶை = 0 (46) ଵଵை + ଵଶை = 0 (47) ଶଵ(ை + ை) + ଶଶை = 0 (48)

ଵଵ(ை − ை)=0 (49)

ൡ (45)

If we take into account Equation (37), the system of Equations (38)–(40) becomes:

From Equation (41), because, in general, det ଵଶ ≠ 0 , it follows immediately:

and introducing that in Equation (43), we obtain ை = −ை, a relation which verifies also Equation (42), if we take into account Equation (37). If ை = ଵ is denoted, it results

**Proof.** For the eigenvalues calculated, the system of Equation (35) must be solved, with

A block diagram representing the stage of the analysis is presented in Figure 4.

**Theorem 3.** *For the other eigenvalues of the system, the eigenvectors are of the form:*  =൝ଵ ଵ ଷ

If det ≠ 0, it results in ௦ − = 0 and ௦ = = ଵ. □

#### **4. Conclusions**

in Equation (34). □

det ଵଵ ≠ 0 or:

Subtracting (47) from (46), we get:

Buildings and constructions, in general, show different forms of symmetry or are made up of repetitive elements. The existence of these symmetries leads to obtaining advantages related to the calculation, design, and manufacture of the structure. One of the advantages is related to the ease of describing the system by systematizing the information used; then, the properties demonstrated in the paper allow to decrease the time required to perform calculations and all this will allow savings and simplifications in the design process. Then, the existence of identical elements or identical parts can simplify the process of making the structure, by simplifying the labor and effort required to manufacture the structure. In conclusions, the design is simpler, and the realization costs are lower. There are also aesthetic reasons that justify the realization of structures with symmetries. From the point of view of calculation and behavior in static and dynamic cases, symmetries can bring significant advantages. In the strength of materials, symmetries are widely used in the static analysis of structures. However, they can be used for dynamic analysis, so that the vibrations of such structures allow simplification of the calculation and time savings in the design process. The paper has presented several vibration properties of a symmetrical structure made of concrete, used in civil engineering. Such structures are frequently encountered in the construction of buildings and in civil engineering and the knowledge of vibration properties can prove to be an advantage that allows to reduce the time and costs related to the design. We mention that symmetries appearing in all aspects of current life and in engineering applications are common. In consequence, the results obtained can be extended to other situations that may be encountered in practice. Future research could reveal other types of symmetries that will allow the systematization of the results and the proposal of a strategy to approach the design and execution of systems with identical parts or symmetries.

**Author Contributions:** Conceptualization, S.V., M.M., and O.D.; methodology, S.V.; validation, S.V., M.M., and O.D.; formal analysis, M.M.; investigation, S.V.; resources, O.D.; writing—original draft preparation, S.V.; writing—review and editing, S.V.; visualization, M.M. and O.D.; supervision, M.M.; project administration, O.D.; funding acquisition, O.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** We want to thank the reviewers who have read the manuscript carefully and have proposed pertinent corrections that have led to an improvement in our manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

For the truss AO:

$$\frac{\partial^4 \Phi\_{AO}}{\partial x^4} - \frac{\rho\_1 A\_1}{E\_1 I\_{z1}} p^2 \Phi\_{AO} = 0 \tag{A1}$$

For the truss OD:

$$\frac{\partial^4 \Phi\_{BC}}{\partial x^4} - \frac{\rho\_1 A\_1}{E\_1 I\_{z1}} p^2 \Phi\_{OD} = 0 \tag{A2}$$

For the truss OB:

$$\frac{\partial^4 \Phi\_{OB}}{\partial x^4} - \frac{\rho\_2 A\_2}{E\_2 I\_{22}} p^2 \Phi\_{OB} = 0 \tag{A3}$$

For the truss OC:

$$\frac{\partial^4 \Phi\_{\rm OC}}{\partial x^4} - \frac{\rho\_2 A\_2}{E\_2 I\_{22}} p^2 \Phi\_{\rm OC} = 0 \tag{A4}$$

The following notations are made:

$$\frac{\rho\_1 A\_1}{E\_1 I\_{z1}} = \lambda\_1^4; \frac{\rho\_2 A\_2}{E\_2 I\_{z2}} = \lambda\_2^4 \tag{A5}$$

Using (A5), the four solutions for the four differential equations of order four (A1)–(A4) are:

$$\Phi\_{AO}(\mathbf{x}) = \mathbb{C}\_1^{AO} \sin(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathbb{C}\_2^{AO} \cos(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathbb{C}\_3^{AO} \sin(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathbb{C}\_4^{AO} \text{ch}(\lambda\_1 \sqrt{p} \mathbf{x})\tag{A6}$$

$$\boldsymbol{\Phi}\_{\rm{OD}}(\mathbf{x}) = \mathbf{C}\_1^{\rm{OD}} \sin(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathbf{C}\_2^{\rm{OD}} \cos(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathbf{C}\_3^{\rm{OD}} \mathbf{sh}(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathbf{C}\_4^{\rm{OD}} \mathbf{ch}(\lambda\_1 \sqrt{p} \mathbf{x}) \tag{A7}$$

$$\boldsymbol{\Phi}\_{\rm OB}(\mathbf{x}) = \mathbf{C}\_1^{\rm OB} \sin(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_2^{\rm OB} \cos(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_3^{\rm OB} \sin(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_4^{\rm OB} \sin(\lambda\_2 \sqrt{p} \mathbf{x})\tag{A8}$$

$$\boldsymbol{\Phi}\_{\rm OC}(\mathbf{x}) = \mathbf{C}\_1^{\rm OC} \sin(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_2^{\rm OC} \cos(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_3^{\rm OC} \sin(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_4^{\rm OC} \mathbf{ch}(\lambda\_2 \sqrt{p} \mathbf{x}) \tag{A9}$$

For torsion, the notation was made:

$$\delta\_i^2 = \frac{J\_i}{G\_i I\_{pi}} \text{ i = 1/2} \tag{A10}$$

Index 1 corresponds to trusses AO and OD and index 2 to trusses OB and OC. Applying Equation (8) for the four trusses studied, leads us to:

For torsional vibrations of the bar AO:

$$\frac{\partial^2 \psi\_{AO}}{\partial x^2} + p^2 \delta\_1^2 \psi\_{AO} = 0 \tag{A11}$$

*λ*1 √

(*λ*<sup>1</sup>

for torsional vibrations of the bar OD:

$$
\frac{\partial^2 \psi\_{\rm OD}}{\partial x^2} + p^2 \delta\_1^2 \psi\_{\rm OD} = 0 \tag{A12}
$$

for torsional vibrations of the bar OB:

$$\frac{\partial^2 \psi\_{OB}}{\partial x^2} + p^2 \delta\_2^2 \psi\_{OB} = 0 \tag{A13}$$

for torsional vibrations of the bar OC:

$$2\frac{\partial^2 \psi\_{\rm OC}}{\partial x^2} + p^2 \delta\_2^2 \psi\_{\rm OC} = 0 \tag{A14}$$

The solutions of the four equations (A11)–(A14) are:

$$\psi\_{AO}(\mathbf{x}) = D\_1^{AO}\sin(\delta\_1 p \mathbf{x}) + D\_2^{AO}\cos(\delta\_1 p \mathbf{x}) \tag{A15}$$

$$\psi\_{OD}(\mathbf{x}) = D\_1^{OD} \sin(\delta\_1 p \mathbf{x}) + D\_2^{OD} \cos(\delta\_1 p \mathbf{x}) \tag{A16}$$

$$\psi\_{\rm OB}(\mathbf{x}) = D\_1^{\rm OB} \sin(\delta\_2 p \mathbf{x}) + D\_2^{\rm OB} \cos(\delta\_2 p \mathbf{x}) \tag{A17}$$

$$\psi\_{\rm OC}(\mathbf{x}) = D\_1^{\rm OC} \sin(\delta\_2 p \mathbf{x}) + D\_2^{\rm OC} \cos(\delta\_2 p \mathbf{x}) \tag{A18}$$

The solutions contain 24 integration constants that will be determined considering the boundary conditions.

The two moments and the shear force are expressed by the known relationships in the mechanics of the deformable solid [26,27]:

$$M^b(\mathbf{x}) = -EI\_z \frac{\partial^2 v(\mathbf{x})}{\partial \mathbf{x}^2};\ T(\mathbf{x}) = EI\_z \frac{\partial^3 v(\mathbf{x})}{\partial \mathbf{x}^3};\ M^\mathbf{f}(\mathbf{x}) = GI\_p \frac{\partial \varrho(\mathbf{x})}{\partial \mathbf{x}}$$

However: Truss AO:

$$\frac{\partial v(\mathbf{x})}{\partial \mathbf{x}} = u\_0 \Phi\_{AO}'(\mathbf{x}) \sin(pt + \theta) = \tag{A19}$$

$$\overline{p} \left[ \mathsf{C}\_1^{AO} \cos(\lambda\_1 \sqrt{p} \mathbf{x}) - \mathsf{C}\_2^{AO} \sin(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathsf{C}\_3^{AO} \mathrm{ch}(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathsf{C}\_4^{AO} (\mathrm{sh} \lambda\_1 \sqrt{p} \mathbf{x}) \right] \sin(pt + \theta)$$

$$\begin{aligned} \frac{\partial^2 v(\mathbf{x})}{\partial \mathbf{x}^2} &= u\_0 \Phi\_{AO}''(\mathbf{x}) \sin(pt + \theta) = \\ \mathbf{x}(\mathbf{x}) - \mathbf{C}\_2^{AO} \cos(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathbf{C}\_3^{AO} \sin(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathbf{C}\_4^{AO} \text{ch}(\lambda\_1 \sqrt{p} \mathbf{x}) \left| \sin(pt + \theta) \right\rangle \end{aligned} \tag{A20}$$

$$\left(\sqrt{p}\right)^{2}\left[-\mathsf{C}\_{1}^{AO}\sin(\lambda\_{1}\sqrt{p}\mathbf{x})-\mathsf{C}\_{2}^{AO}\cos(\lambda\_{1}\sqrt{p}\mathbf{x})+\mathsf{C}\_{3}^{AO}\sin(\lambda\_{1}\sqrt{p}\mathbf{x})+\mathsf{C}\_{4}^{AO}\mathrm{ch}(\lambda\_{1}\sqrt{p}\mathbf{x})\right]\sin(pt+\theta);\tag{7.17}$$

$$\frac{\partial^{3}v(\mathbf{x})}{\partial\mathbf{r}^{3}}=\Phi\_{\mathrm{AO}}^{\mathrm{w}}(\mathbf{x})\sin(pt+\theta)=\tag{7.18}$$

$$\begin{aligned} \text{\(\mu}^3 \text{ -- \text{\(\mu\)}}\text{\(\mu\)\text{\(\mu\)}}\text{\(\mu\)\text{\(\(\mu\)}}\\ (\lambda\_1 \sqrt{p})^3 \left[ -\text{C}\_1^{AO} \cos(\lambda\_1 \sqrt{p} \mathbf{x}) + \text{C}\_2^{AO} \sin(\lambda\_1 \sqrt{p} \mathbf{x}) + \text{C}\_3^{AO} \text{ch}(\lambda\_1 \sqrt{p} \mathbf{x}) + \text{C}\_4^{AO} \text{sh}(\lambda\_1 \sqrt{p} \mathbf{x}) \right] \sin(pt + \theta) \end{aligned} \tag{A21}$$

$$\frac{\partial \varphi(\mathbf{x})}{\partial \mathbf{x}} = \psi\_{AO}' \sin(pt + \theta) = \delta\_1 p \left[ D\_1^{AO} \cos(\delta\_1 p \mathbf{x}) - D\_2^{AO} \sin(\delta\_1 p \mathbf{x}) \right] \sin(pt + \theta) \tag{A22}$$
 
$$\text{Truss OD:}$$

$$\frac{\partial v(\mathbf{x})}{\partial \mathbf{x}} = \lambda\_1 \sqrt{p} \left[ \mathbf{C}\_1^{\text{OD}} \cos(\lambda\_1 \sqrt{p} \mathbf{x}) - \mathbf{C}\_2^{\text{OD}} \sin(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathbf{C}\_3^{\text{OD}} \text{ch}(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathbf{C}\_4^{\text{OD}} \text{sh}(\lambda\_1 \sqrt{p} \mathbf{x}) \right] \sin(pt + \theta) \tag{A23}$$

$$\frac{\partial^2 v(\mathbf{x})}{\partial \mathbf{x}^2} = (\lambda\_1 \sqrt{p})^2 \left[ -\mathbb{C}\_1^{\text{OD}} \sin(\lambda\_1 \sqrt{p} \mathbf{x}) - \mathbb{C}\_2^{\text{OD}} \cos(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathbb{C}\_3^{\text{OD}} \sin(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathbb{C}\_4^{\text{OD}} \text{ch}(\lambda\_1 \sqrt{p} \mathbf{x}) \right] \sin(pt + \theta) \tag{A24}$$

$$\frac{\partial^3 v(\mathbf{x})}{\partial \mathbf{x}^3} = (\lambda\_1 \sqrt{p})^3 \left[ -\mathbb{C}\_1^{\text{OD}} \cos(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathbb{C}\_2^{\text{OD}} \sin(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathbb{C}\_3^{\text{OD}} \text{ch}(\lambda\_1 \sqrt{p} \mathbf{x}) + \mathbb{C}\_4^{\text{OD}} \text{sh}(\lambda\_1 \sqrt{p} \mathbf{x}) \right] \sin(pt + \theta) \tag{A25}$$

$$\frac{\partial \rho(\mathbf{x})}{\partial \mathbf{x}} = \rho\_0 \boldsymbol{\upmu}\_{AO}^{\prime} \sin(\boldsymbol{pt} + \boldsymbol{\theta}) = \rho\_0 \delta\_1 \boldsymbol{p} \left[ D\_1^{OD} \cos(\delta\_1 \mathbf{p} \mathbf{x}) - D\_2^{OD} \sin(\delta\_1 \mathbf{p} \mathbf{x}) \right] \sin(\boldsymbol{pt} + \boldsymbol{\theta}) \tag{A26}$$
 
$$\text{Truss OB:}$$

$$\frac{\partial v(\mathbf{x})}{\partial \mathbf{x}} = \lambda\_2 \sqrt{p} \left[ \mathbf{C}\_1^{\text{OB}} \cos(\lambda\_2 \sqrt{p} \mathbf{x}) - \mathbf{C}\_2^{\text{OB}} \sin(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_3^{\text{OB}} \text{ch}(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_4^{\text{OB}} \text{sh}(\lambda\_2 \sqrt{p} \mathbf{x}) \right] \sin(pt + \theta) \tag{A27}$$

$$\frac{\partial^2 v(\mathbf{x})}{\partial \mathbf{x}^2} = (\lambda\_2 \sqrt{p})^2 \left[ -\mathsf{C}\_1^{\rm OB} \sin(\lambda\_2 \sqrt{p} \mathbf{x}) - \mathsf{C}\_2^{\rm OB} \cos(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathsf{C}\_3^{\rm OB} \sin(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathsf{C}\_4^{\rm OB} \mathrm{ch}(\lambda\_2 \sqrt{p} \mathbf{x}) \right] \tag{A28}$$

$$\frac{\partial^3 \overline{v}(\mathbf{x})}{\partial \mathbf{x}^3} = (\lambda\_2 \sqrt{\overline{p}})^3 \left[ -\mathbf{C}\_1^{\rm OB} \cos(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_2^{\rm OB} \sin(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_3^{\rm OB} \mathbf{ch}(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_4^{\rm OB} \sin(\lambda\_2 \sqrt{p} \mathbf{x}) \right] \sin(pt + \theta) \tag{A29}$$

$$\frac{\partial \rho(\mathbf{x})}{\partial \mathbf{x}} = \rho\_o \delta\_2 p \left[ D\_1^{OB} \cos(\delta\_2 p \mathbf{x}) - D\_2^{OB} \sin(\delta\_2 p \mathbf{x}) \right] \sin(pt + \theta) \tag{A30}$$
  $\text{Truss OC:}$ 

$$\frac{\partial v(\mathbf{x})}{\partial \mathbf{x}} = \lambda\_2 \sqrt{p} \left[ \mathbf{C}\_1^{\mathrm{OC}} \cos(\lambda\_2 \sqrt{p} \mathbf{x}) - \mathbf{C}\_2^{\mathrm{OC}} \sin(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_3^{\mathrm{OC}} \mathrm{ch}(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_4^{\mathrm{OC}} \mathrm{sh}(\lambda\_2 \sqrt{p} \mathbf{x}) \right] \sin(pt + \theta) \tag{A31}$$

$$\frac{\partial^2 v(\mathbf{x})}{\partial \mathbf{x}^2} = (\lambda\_2 \sqrt{p})^2 \left[ -\mathbf{C}\_1^{\mathrm{OC}} \sin(\lambda\_2 \sqrt{p} \mathbf{x}) - \mathbf{C}\_2^{\mathrm{OC}} \cos(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_3^{\mathrm{OC}} \sin(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_4^{\mathrm{OC}} \sin(\lambda\_2 \sqrt{p} \mathbf{x}) \right] \sin(pt + \theta) \tag{A32}$$

$$\frac{\partial^3 \mathbf{v}(\mathbf{x})}{\partial \mathbf{x}^3} = (\lambda\_2 \sqrt{p})^3 \left[ -\mathbf{C}\_1^{\mathbf{OC}} \cos(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_2^{\mathbf{OC}} \sin(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_3^{\mathbf{OC}} \mathbf{ch}(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbf{C}\_4^{\mathbf{OC}} \mathbf{sh}(\lambda\_2 \sqrt{p} \mathbf{x}) \right] \sin(pt + \theta) \tag{A33}$$

$$\frac{\partial \rho(\mathbf{x})}{\partial \mathbf{x}} = \rho\_o \delta\_2 p \left[ D\_1^{AO} \cos(\delta\_2 p \mathbf{x}) - D\_2^{AO} \sin(\delta\_2 p \mathbf{x}) \right] \sin(pt + \theta) \tag{A34}$$

*D*

The following relations will be obtained:

$$\mathbf{C}\_2^{AO} + \mathbf{C}\_4^{AO} = \mathbf{0} \tag{A35}$$

$$\mathbf{C}\_1^{AO} + \mathbf{C}\_3^{AO} = \mathbf{0} \tag{A36}$$

$$
\epsilon\_2^{AO} = 0 \tag{A37}
$$

$$\mathbf{C}\_{1}^{\rm OD}\sin(\lambda\_{1}\sqrt{\rho}L\_{1}) + \mathbf{C}\_{2}^{\rm OD}\cos(\lambda\_{1}\sqrt{\rho}L\_{1}) + \mathbf{C}\_{3}^{\rm OD}\sin(\lambda\_{1}\sqrt{\rho}L\_{1}) + \mathbf{C}\_{4}^{\rm OD}\sin(\lambda\_{1}\sqrt{\rho}L\_{1}) = 0 \tag{A38}$$

$$\mathbf{C}\_1^{\rm OD}\cos(\lambda\_1\sqrt{\mu}L\_1) - \mathbf{C}\_2^{\rm OD}\sin(\lambda\_1\sqrt{\mu}L\_1) + \mathbf{C}\_3^{\rm OD}\mathrm{ch}(\lambda\_1\sqrt{\mu}L\_1) + \mathbf{C}\_4^{\rm OD}\mathrm{sh}(\lambda\_1\sqrt{\mu}L\_1) = 0\tag{A.39}$$

$$D\_1^{OD} \sin(\delta\_1 p L\_1) + D\_2^{OD} \cos(\delta\_1 p L\_1) = 0 \tag{A40}$$

$$\mathbf{C}\_{1}^{\rm OB}\sin(\lambda\_{2}\sqrt{\rho}\mathbf{L}\_{3}) + \mathbf{C}\_{2}^{\rm OB}\cos(\lambda\_{2}\sqrt{\rho}\mathbf{L}\_{3}) + \mathbf{C}\_{3}^{\rm OB}\sin(\lambda\_{2}\sqrt{\rho}\mathbf{L}\_{3}) + \mathbf{C}\_{4}^{\rm OB}\mathbf{ch}(\lambda\_{2}\sqrt{\rho}\mathbf{L}\_{3}) = 0\tag{A41}$$

$$-\mathbf{C}\_1^{\rm OB}\sin\left(\lambda\_2\sqrt{\rho}\mathbf{L}\_3\right) - \mathbf{C}\_2^{\rm OB}\cos\left(\lambda\_2\sqrt{\rho}\mathbf{L}\_3\right) + \mathbf{C}\_3^{\rm OB}\sin\left(\lambda\_2\sqrt{\rho}\mathbf{L}\_3\right) + \mathbf{C}\_4^{\rm OB}\mathbf{ch}\left(\lambda\_2\sqrt{\rho}\mathbf{L}\_3\right) = \mathbf{0} \tag{\text{A42}}$$

$$D\_1^{OB}\cos(\delta\_2 pL\_3) - D\_2^{OB}\sin(\delta\_2 pL\_3) = 0\tag{A43}$$

$$\mathbf{C}\_{1}^{\rm OC}\sin(\lambda\_{2}\sqrt{\rho}\mathbf{L}\_{3}) + \mathbf{C}\_{2}^{\rm OC}\cos(\lambda\_{2}\sqrt{\rho}\mathbf{L}\_{3}) + \mathbf{C}\_{3}^{\rm OC}\sin(\lambda\_{2}\sqrt{\rho}\mathbf{L}\_{3}) + \mathbf{C}\_{4}^{\rm OC}\mathbf{ch}(\lambda\_{2}\sqrt{\rho}\mathbf{L}\_{3}) = 0\tag{A44}$$

$$-\mathcal{C}\_1^{\text{OC}}\sin(\lambda\_2\sqrt{p}L\_3) - \mathcal{C}\_2^{\text{OC}}\cos(\lambda\_2\sqrt{p}L\_3) + \mathcal{C}\_3^{\text{OC}}\text{sh}(\lambda\_2\sqrt{p}L\_3) + \mathcal{C}\_4^{\text{OC}}\text{ch}(\lambda\_2\sqrt{p}L\_3) = 0\tag{A45}$$

$$D\_1^{OC} \cos(\delta\_2 pL\_3) - D\_2^{OC} \sin(\delta\_2 pL\_3) = 0\tag{A46}$$

which represent a system of 12 equations.

#### **Appendix B**

#### These nine conditions lead to nine relationships:

$$\begin{aligned} \mathbf{C}\_1^{AO} \sin(\lambda\_1 \sqrt{p} \mathbf{L}\_2) + \mathbf{C}\_2^{AO} \cos(\lambda\_1 \sqrt{p} \mathbf{L}\_2) + \mathbf{C}\_3^{AO} \sin(\lambda\_1 \sqrt{p} \mathbf{L}\_2) + \mathbf{C}\_4^{AO} \mathbf{ch}(\lambda\_1 \sqrt{p} \mathbf{L}\_2) &= \\ \mathbf{C}\_2^{OD} + \mathbf{C}\_4^{OD} = \mathbf{C}\_2^{OB} + \mathbf{C}\_4^{OB} = \mathbf{C}\_2^{OC} + \mathbf{C}\_4^{OC} \end{aligned} \tag{A47}$$

$$\begin{aligned} \lambda\_1 \sqrt{p} \left[ \mathsf{C}\_1^{AO} \cos(\lambda\_1 \sqrt{p} \mathsf{L}\_2) - \mathsf{C}\_2^{AO} \sin(\lambda\_1 \sqrt{p} \mathsf{L}\_2) + \mathsf{C}\_3^{AO} \mathrm{ch}(\lambda\_1 \sqrt{p} \mathsf{L}\_2) + \mathsf{C}\_4^{AO} \mathrm{sh}(\lambda\_1 \sqrt{p} \mathsf{L}\_2) \right] &= \\ \lambda\_1 \sqrt{p} \left[ \mathsf{C}\_1^{OD} + \mathsf{C}\_3^{OD} \right] &= D\_2^{OB} = -D\_2^{OC} \end{aligned} \tag{A48}$$

$$D\_2^{AO} = D\_2^{OD} = \lambda\_2 \sqrt{p} \left[ \mathbf{C}\_1^{OB} + \mathbf{C}\_3^{OB} \right] = \lambda\_2 \sqrt{p} \left[ \mathbf{C}\_1^{OC} + \mathbf{C}\_3^{OC} \right] \tag{A49}$$

$$\text{or: } \frac{1}{\lambda\_2 \sqrt{p}} D\_2^{AO} = \frac{1}{\lambda\_2 \sqrt{p}} D\_2^{OD} = \mathcal{C}\_1^{OB} + \mathcal{C}\_3^{OB} = \mathcal{C}\_1^{OC} + \mathcal{C}\_3^{OC} \tag{A50}$$

in which, together, (A35)–(A46) represent a system of 21 equations. From (A47), we can obtain the equations:

> *C OB* <sup>2</sup> + *C OB* <sup>4</sup> − *C OD* <sup>2</sup> − *C OD* <sup>4</sup> = 0 (A51)

$$\mathbf{C}\_2^{\rm OC} + \mathbf{C}\_4^{\rm OC} - \mathbf{C}\_2^{\rm OD} - \mathbf{C}\_4^{\rm OD} = \mathbf{0} \tag{A52}$$

$$\mathbb{C}\_{1}^{AO}\sin(\lambda\_{1}\sqrt{\mathbb{p}}L\_{2}) + \mathbb{C}\_{2}^{AO}\cos(\lambda\_{1}\sqrt{\mathbb{p}}L\_{2}) + \mathbb{C}\_{3}^{AO}\mathrm{sh}(\lambda\_{1}\sqrt{\mathbb{p}}L\_{2}) + \mathbb{C}\_{4}^{AO}\mathrm{ch}(\lambda\_{1}\sqrt{\mathbb{p}}L\_{2}) \ - \mathbb{C}\_{2}^{OD} + \mathbb{C}\_{4}^{OD} = 0. \tag{A53}$$

From (A48), we can obtain the equations:

$$\text{(2: }\mathbb{C}\_1^{OB} + \mathbb{C}\_3^{OB} - \frac{1}{\lambda\_2 \sqrt{p}} D\_2^{AO} = 0\tag{A54}$$

$$\mathcal{C}\_1^{\text{OC}} + \mathcal{C}\_3^{\text{OC}} - \frac{1}{\lambda\_2 \sqrt{p}} D\_2^{AO} = 0 \tag{A55}$$

$$D\_2^{AO} - D\_2^{OD} = 0 \tag{A56}$$

From (A49), we obtain:

$$D\_2^{OB} - \lambda\_1 \sqrt{p} \left[ \mathcal{C}\_1^{OD} + \mathcal{C}\_3^{OD} \right] = 0 \tag{A57}$$

$$\left[D\_2^{\rm OC} - \lambda\_1 \sqrt{p} \left[\mathcal{C}\_1^{\rm OD} + \mathcal{C}\_3^{\rm OD}\right] = 0;\tag{A58}$$

$$
\lambda\_1 \sqrt{p} \left[ \mathsf{C}\_1^{AO} \cos(\lambda\_1 \sqrt{p} \mathsf{L}\_2) - \mathsf{C}\_2^{AO} \sin(\lambda\_1 \sqrt{p} \mathsf{L}\_2) + \mathsf{C}\_3^{AO} \mathrm{ch}(\lambda\_1 \sqrt{p} \mathsf{L}\_2) + \mathsf{C}\_4^{AO} \mathrm{sh}(\lambda\_1 \sqrt{p} \mathsf{L}\_2) \right] - \lambda\_1 \sqrt{p} \left[ \mathsf{C}\_1^{OD} + \mathsf{C}\_3^{OD} \right] = 0 \tag{A59}
$$

#### **Appendix C**

Three more conditions are needed to obtain the constants in the written differential equations.

They are obtained by considering the equilibrium of an infinitesimal element of the mass containing the point O.

The sum of the four shear forces in O must be zero, so:

$$T1 + T2 + T3 - T4 = 0$$

Replacing the expressions of the shear force determined for the four bars in O, we obtain:

$$\begin{array}{ll} \lambda\_1^3 \operatorname{E} I\_{\overline{1}} \left( -\operatorname{C}\_1^{AO} \cos \left( \lambda\_1 \sqrt{p} \operatorname{L}\_2 \right) + \operatorname{C}\_2^{AO} \sin \left( \lambda\_1 \sqrt{p} \operatorname{L}\_2 \right) + \operatorname{C}\_3^{AO} \operatorname{ch} \left( \lambda\_1 \sqrt{p} \operatorname{L}\_2 \right) \\\ + \operatorname{C}\_4^{AO} \operatorname{sh} \left( \lambda\_1 \sqrt{p} \operatorname{L}\_2 \right) \right) + \lambda\_1^3 \operatorname{E} I\_{\overline{1}} \left( -\operatorname{C}\_1^{OD} + \operatorname{C}\_3^{OD} \right) + \lambda\_2^3 \operatorname{E} I\_{\overline{2}} \left( -\operatorname{C}\_1^{OB} + \operatorname{C}\_3^{OB} \right) \\\ - \lambda\_2^3 \operatorname{E} I\_{\overline{2}} \left( -\operatorname{C}\_1^{OC} + \operatorname{C}\_3^{OC} \right) = 0 \end{array} \tag{A60}$$

With the notation:

$$a\_1 = \frac{E\_2 I\_{p2}}{E\_1 I\_{z1}} \left(\frac{\lambda\_2}{\lambda\_1}\right)^3 \tag{A61}$$

we can write:

$$\begin{aligned} \left(-\mathsf{C}\_{1}^{AO}\cos(\lambda\_{1}\sqrt{p}\ L\_{2}) + \mathsf{C}\_{2}^{AO}\sin(\lambda\_{1}\sqrt{p}\ L\_{2}) + \mathsf{C}\_{3}^{AO}\mathrm{ch}(\lambda\_{1}\sqrt{p}\ L\_{2})\right. \\ + \left.\mathrm{C}\_{4}^{AO}\mathrm{sh}(\lambda\_{1}\sqrt{p}\ L\_{2})\right) + \left(-\mathsf{C}\_{1}^{OD} + \mathsf{C}\_{3}^{OD}\right) + a\_{1}(-\mathsf{C}\_{1}^{OB} + \mathsf{C}\_{3}^{OB}) - a\_{1}(-\mathsf{C}\_{1}^{OC} + \mathsf{C}\_{3}^{OC}) = 0 \end{aligned} \tag{A62}$$

A similar equilibrium relation is obtained for bending and torsional moments.

$$M^{b1} + M^{b2} + M^{t3} - M^{t4} = 0$$

$$M^{t1} + M^{t2} + M^{b3} - M^{b4} = 0$$

Taking into account relations (27)–(43), we obtain:

$$-E\_1 I\_{z1} \frac{\partial^2 \Phi\_{AO}(L\_2)}{\partial \mathbf{x}^2} - E\_1 I\_{z1} \frac{\partial^2 \Phi\_{OD}(\mathbf{0})}{\partial \mathbf{x}^2} + G\_2 I\_{p2} \frac{\partial \varphi\_{OB}(\mathbf{0})}{\partial \mathbf{x}} - G\_2 I\_{p2} \frac{\partial \varphi\_{OC}(\mathbf{0})}{\partial \mathbf{x}} = \mathbf{0} \tag{\text{A63}}$$

$$\mathrm{G}\_{1}\mathrm{I}\_{p1}\frac{\partial\varphi\_{AO}(\mathrm{L}\_{2})}{\partial\mathbf{x}} + \mathrm{G}\_{1}\mathrm{I}\_{p1}\frac{\partial\varphi\_{OD}(\mathrm{0})}{\partial\mathbf{x}} - \mathrm{E}\_{2}\mathrm{I}\_{\mathrm{22}}\frac{\partial^{2}\Phi\_{OB}(\mathrm{L}\_{2})}{\partial\mathbf{x}^{2}} + \mathrm{E}\_{2}\mathrm{I}\_{\mathrm{22}}\frac{\partial^{2}\Phi\_{OC}(\mathrm{0})}{\partial\mathbf{x}^{2}} = \mathbf{0} \tag{\text{A64}}$$

Using (11)–(14) and (23)–(26), the results are:

$$\begin{split} \mathcal{C}\_{1}^{AO} \sin(\lambda\_{1}\sqrt{p}\,L\_{2}) + \mathcal{C}\_{2}^{AO} \cos(\lambda\_{1}\sqrt{p}\,L\_{2}) - \mathcal{C}\_{3}^{AO} \sin(\lambda\_{1}\sqrt{p}\,L\_{2}) - \mathcal{C}\_{4}^{AO} \text{ch}(\lambda\_{1}\sqrt{p}) - \\ (-\mathcal{C}\_{2}^{OD} + \mathcal{C}\_{4}^{OD}) + \frac{\mathcal{C}\_{2}I\_{p2}}{E\_{1}I\_{21}} \frac{\delta\_{2}}{\lambda\_{1}^{2}} (D\_{1}^{OB} - D\_{1}^{OC}) = 0 \end{split} \tag{A65}$$

$$\begin{split} \frac{G\_1 I\_{p1}}{E\_2 I\_{r2}} \frac{\delta\_1}{\lambda\_2^2} (D\_1^{AO} \cos(\delta\_1 p L\_2) - D\_2^{AO} \sin(\delta\_1 p L\_2)) + \frac{G\_1 I\_{p1}}{E\_2 I\_{r2}} \frac{\delta\_1}{\lambda\_2^2} D\_1^{OD} - \\ (-\mathsf{C}\_2^{OB} + \mathsf{C}\_4^{OB}) + (-\mathsf{C}\_2^{OC} + \mathsf{C}\_4^{OC}) &= 0 \end{split} \tag{A66}$$

This denoted:

$$a\_2 = \frac{G\_2 I\_{p2}}{E\_1 I\_{z1}} \frac{\delta\_2}{\lambda\_1^{2'}}, \ a\_3 = \frac{G\_1 I\_{p1}}{E\_2 I\_{z2}} \frac{\delta\_1}{\lambda\_2^2} \tag{A67}$$

$$\begin{split} \mathbf{C}\_{1}^{AO} \sin(\lambda\_{1}\sqrt{p}\,L\_{2}) + \mathbf{C}\_{2}^{AO} \cos(\lambda\_{1}\sqrt{p}\,L\_{2}) - \mathbf{C}\_{3}^{AO} \sin(\lambda\_{1}\sqrt{p}\,L\_{2}) - \mathbf{C}\_{4}^{AO} \sin(\lambda\_{1}\sqrt{p}\,L\_{2}) - \\ \left(-\mathbf{C}\_{2}^{OD} + \mathbf{C}\_{4}^{OD}\right) + a\_{2} \left(D\_{1}^{OB} - D\_{1}^{OC}\right) = 0 \end{split} \tag{A68}$$

$$a\_3 \left( D\_1^{AO} \cos(\delta\_1 p L\_2) - D\_2^{AO} \sin(\delta\_1 p L\_2) \right) + a\_3 D\_1^{OD} - \left( -\mathcal{C}\_2^{OB} + \mathcal{C}\_4^{OB} \right) + \left( -\mathcal{C}\_2^{OC} + \mathcal{C}\_4^{OC} \right) = 0 \tag{A69}$$

This represents a system with 24 equations with 24 unknowns. From the system formed, we must determine the constants *C AO* 1 , *C AO* 2 , *C AO* 3 , *C AO* 4 , *C OD* 1 , *C OD* 2 , *C OD* 3 , *C OD* 4 , *C OB* 1 , *C OB* 2 , *C OB* 3 , *C OB* 4 , *C OC* 1 , *C OC* 2 , *C OC* 3 , *C OC* 4 , *DAO* 1 , *DAO* 2 , *DOD* 1 , *DOD* 2 , *DOB* 1 , *DOB* 2 , *DOC* 1 , *DOC* 2 . They will be denoted, in the following:

$$\{B\} = \left\{ \begin{array}{c} \{B^{AO}\} \\ \{B^{OD}\} \\ \{B^{OB}\} \\ \{B^{OC}\} \end{array} \right\} \tag{A70}$$

with:

$$\begin{Bmatrix} \mathcal{B}^{AO} \end{Bmatrix} = \begin{bmatrix} \mathcal{C}\_1^{AO} & \mathcal{C}\_2^{AO} & \mathcal{C}\_3^{AO} & \mathcal{C}\_4^{AO} & D\_1^{AO} & D\_2^{AO} \end{bmatrix} \tag{A71}$$
 
$$\begin{Bmatrix} \mathcal{B}^{OD} \end{Bmatrix} = \begin{bmatrix} \mathcal{C}\_1^{OD} & \mathcal{C}\_2^{OD} & \mathcal{C}\_2^{OD} & \mathcal{C}\_1^{OD} & D\_1^{OD} & D\_2^{OD} \end{bmatrix} \tag{A72}$$

$$\begin{Bmatrix} \mathcal{B}^{\text{OD}} \end{Bmatrix} = \begin{bmatrix} \mathcal{C}\_1^{\text{OD}} & \mathcal{C}\_2^{\text{OD}} & \mathcal{C}\_3^{\text{OD}} & \mathcal{C}\_4^{\text{OD}} & D\_1^{\text{OD}} & D\_2^{\text{OD}} & \mathbb{I} \end{bmatrix} \tag{A72}$$
 
$$\begin{Bmatrix} \mathcal{B}^{\text{OB}} \end{Bmatrix} = \begin{bmatrix} \mathcal{C}\_1^{\text{OB}} & \mathcal{C}\_2^{\text{OB}} & \mathcal{C}\_3^{\text{OB}} & \mathcal{C}\_4^{\text{OB}} & D\_4^{\text{OB}} & D\_2^{\text{OB}} & \mathbb{I} \end{bmatrix} \tag{A73}$$

$$\left\{ \begin{array}{ccccccccc} \mathcal{B}^{\text{OB}} & = & \mathbb{C}\_{1}^{\text{OB}} & \mathbb{C}\_{2}^{\text{OB}} & \mathbb{C}\_{3}^{\text{OB}} & \mathbb{C}\_{4}^{\text{OB}} & D\_{1}^{\text{OB}} & D\_{2}^{\text{OB}} & \end{array} \right\} \tag{A73}$$

$$\left\{ \mathcal{B}^{\text{OC}} \right\} = \left[ \begin{array}{cccccc} \mathcal{C}\_{1}^{\text{OC}} & \mathcal{C}\_{2}^{\text{OC}} & \mathbb{C}\_{3}^{\text{OC}} & \mathbb{C}\_{4}^{\text{OC}} & D\_{1}^{\text{OC}} & D\_{2}^{\text{OC}} & \end{array} \right] \tag{A74}$$

A homogeneous linear system was obtained. In order to have other solutions besides the trivial solution zero, the determinant of the system must be zero. Putting this condition, the obtained eigenfrequencies of the system can be determined from the obtained equation. Denote:

[*A*11] = 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 1 sin(*λ*<sup>2</sup> √*pL*3) cos(*λ*<sup>2</sup> √*pL*3) *sh*(*λ*<sup>2</sup> √*pL*3) *ch*(*λ*<sup>2</sup> √*pL*3) 0 0 − sin(*λ*<sup>2</sup> <sup>√</sup>*pL*3) <sup>−</sup> cos(*λ*<sup>2</sup> √*pL*3) sh(*λ*<sup>2</sup> √*pL*3) ch(*λ*<sup>2</sup> √*pL*3) 0 0 0 0 0 0 cos(*δ*<sup>2</sup> *pL*3) − sin(*δ*<sup>2</sup> *pL*3) (A75) [*A*12] = 0 0 0 0 0 0 0 −1 0 −1 0 0 0 0 0 0 0 <sup>−</sup> <sup>1</sup> *λ*1 √*p* 0 0 0 0 0 0 0 0 0 0 0 0 −*λ*<sup>1</sup> <sup>√</sup>*<sup>p</sup>* <sup>0</sup> <sup>−</sup>*λ*<sup>1</sup> √*p* 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (A76) [*A*21] = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −*a*<sup>1</sup> 0 −*a*<sup>1</sup> 0 0 0 *a*<sup>1</sup> 0 −*a*<sup>1</sup> 0 0 0 0 0 0 0 *a*<sup>2</sup> 0 0 0 0 0 −*a*<sup>2</sup> 0 0 1 0 −1 0 0 0 −1 0 1 0 0 (A77) In the following, we denote: [*A*22] = *A*22*a A*22*b A*22*c A*22*d* (A78) where: [*A*22*a*] = 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (A79)

$$[A\_{12}b] = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \sin(\lambda\_1 \sqrt{p} L\_1) & \cos(\lambda\_1 \sqrt{p} L\_1) & \sinh(\lambda\_1 \sqrt{p} L\_1) & \sinh(\lambda\_1 \sqrt{p} L\_1) & 0 & 0 \\ \cos(\lambda\_1 \sqrt{p} L\_1) & -\sin(\lambda\_1 \sqrt{p} L\_1) & \sinh(\lambda\_1 \sqrt{p} L\_1) & \sinh(\lambda\_1 \sqrt{p} L\_1) & 0 & 0 \\ 0 & 0 & 0 & 0 & \sin(\delta\_1 p L\_1) & \cos(\delta\_1 p L\_1) \end{bmatrix} \tag{A80}$$

$$[A\_{12}d] = \begin{bmatrix} 0 & -1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ -1 & 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & \mathbf{a}\_3 & 0 \end{bmatrix} \tag{A81}$$

$$[A\_{12}\!c] = \begin{bmatrix} \sin(\lambda\_1\sqrt{\eta}\mathcal{L}\_2) & \cos(\lambda\_1\sqrt{\eta}\mathcal{L}\_2) & \operatorname{sh}(\lambda\_1\sqrt{\eta}\mathcal{L}\_2) & \operatorname{ch}(\lambda\_1\sqrt{\eta}\mathcal{L}\_2) & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ \cos(\lambda\_1\sqrt{\eta}\mathcal{L}\_2) & -\sin(\lambda\_1\sqrt{\eta}\mathcal{L}\_2) & \operatorname{ch}(\lambda\_1\sqrt{\eta}\mathcal{L}\_2) & \operatorname{sh}(\lambda\_1\sqrt{\eta}\mathcal{L}\_2) & 0 & 0\\ -\cos(\lambda\_1\sqrt{\eta}\mathcal{L}\_2) & \sin(\lambda\_1\sqrt{\eta}\mathcal{L}\_2) & \operatorname{ch}(\lambda\_1\sqrt{\eta}\mathcal{L}\_2) & \operatorname{sh}(\lambda\_1\sqrt{\eta}\mathcal{L}\_2) & 0 & 0\\ \sin(\lambda\_1\sqrt{\eta}\mathcal{L}\_2 & \cos(\lambda\_1\sqrt{\eta}\mathcal{L}\_2) & -\operatorname{sh}(\lambda\_1\sqrt{\eta}\mathcal{L}\_2) & -\operatorname{ch}(\lambda\_1\sqrt{\eta}\mathcal{L}\_2) & 0 & 0\\ 0 & 0 & 0 & \operatorname{ag}\cos(\delta\_1\eta\mathcal{L}\_2) & -\operatorname{ag}\sin(\delta\_1\eta\mathcal{L}\_2) \end{bmatrix} \tag{A82}$$

In this case, the matrix of the system can be written, in a concise form: In this case, the matrix of the system can be written, in a concise form: ଵଵ 0 ଵଶ

$$A\_{24 \times 24} = \begin{bmatrix} A\_{11} & 0 & A\_{12} \\ 0 & A\_{11} & A\_{12} \\ A\_{21} & A\_{21} & A\_{22} \end{bmatrix} \tag{A83}$$

and the system becomes:

and the system becomes:

$$[A]\{\mathcal{B}\} = \{0\} \tag{A84}$$

The condition: The condition:

 $\text{The condition:}$ 

$$\det(A) = 0\tag{A85}$$

offers the eigenvalues of the system of differential Equations (6)–(9) and (19)–(22).

#### **Appendix D Appendix D**

Let us now consider one of the identical trusses OB and OC. The truss is clamped in O and supported in B (or C). Equation (1) is also valid for the OB bar (OC), with the boundary conditions: Let us now consider one of the identical trusses OB and OC. The truss is clamped in O and supported in B (or C). Equation (1) is also valid for the OB bar (OC), with the boundary conditions:

= 0 ; (0, ) = 0 ; <sup>ᇱ</sup>

**Figure A1.** Model of a single truss. **Figure A1.** Model of a single truss.

For point O: For point O:

$$x = 0; \ v(0, t) = 0; \ v'(0, t) = 0; \ \varphi(0, t) = 0 \tag{A86}$$

and for the point B(C):

$$\mathbf{x} = \mathbf{L}\_{\mathfrak{J}} ; \operatorname{\boldsymbol{\upsilon}}(\mathbf{L}\_{\mathfrak{J}}, t) = \mathbf{0} ; \operatorname{\boldsymbol{M}}^{\mathfrak{b}}(\mathbf{L}\_{\mathfrak{J}}, t) = \mathbf{0} ; \operatorname{\boldsymbol{M}}^{\mathfrak{t}}(\mathbf{L}\_{\mathfrak{J}}, t) = \mathbf{0} \tag{A87}$$

The solution: The solution:

$$v(x,t) = \Phi(x)\sin(pt+\theta) \tag{A88}$$

offers the new equation: offers the new equation:

$$\frac{\partial^4 \Phi}{\partial x^4} - p^2 \frac{\rho\_2 A\_2}{E\_2 I\_{z2}} \Phi = 0 \tag{A89}$$

(, ) = () sin( + ) (A93)

If noted:

If noted:

$$
\cdots \qquad \downarrow \text{2.22}
$$

$$
\lambda\_2^4 = \frac{\rho\_2 A\_2}{E\_2 I\_{22}} \tag{A90}
$$

(A91)

(0, ) = 0 ; (0, ) = 0 (A87)

the solution is:

the solution is:

$$\Phi(\mathbf{x}) = \mathbb{C}\_1 \sin(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbb{C}\_2 \cos(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbb{C}\_3 \text{sh}(\lambda\_2 \sqrt{p} \mathbf{x}) + \mathbb{C}\_4 \text{ch}(\lambda\_2 \sqrt{p} \mathbf{x})\tag{A91}$$

() = ଵsin (ଶඥ) + ଶcos (ଶඥ) + ଷsh(ଶඥ) + ସch(ଶඥ) (A92)

<sup>ସ</sup> <sup>=</sup> ଶଶ

ଶ

Torsional vibrations are described by Equation (15). By choosing as:

and introducing in Equation (15), we obtain:

Torsional vibrations are described by Equation (15). By choosing *ϕ* as:

$$
\varphi(\mathbf{x},t) = \psi(\mathbf{x})\sin(pt+\theta) \tag{A92}
$$

and introducing in Equation (15), we obtain:

$$\frac{\partial^2 \psi}{\partial x^2} + p^2 \delta\_2^2 \psi = 0 \tag{A93}$$

The solution of the differential Equation (A93) will be:

$$\psi(\mathbf{x}) = D\_1 \sin(\delta\_2 p \mathbf{x}) + D\_2 \cos(\delta\_2 p \mathbf{x}) \tag{A94}$$

#### **References**


## *Article* **Evaluating the Impact of Different Symmetrical Models of Ambient Assisted Living Systems**

**Wael Alosaimi <sup>1</sup> , Md Tarique Jamal Ansari <sup>2</sup> , Abdullah Alharbi <sup>1</sup> , Hashem Alyami <sup>3</sup> , Adil Hussain Seh <sup>4</sup> , Abhishek Kumar Pandey <sup>4</sup> , Alka Agrawal <sup>4</sup> and Raees Ahmad Khan 4,\***


**Abstract:** In recent years, numerous attempts have been made to enhance the living standard for old-aged people. Ambient Assisted Living (AAL) is an evolving interdisciplinary field aimed at the exploitation of knowledge and communication technology in health and tele-monitoring systems to combat the impact of the growing aging population. AAL systems are designed for customized, responsive, and predictive requirements, requiring high performance of functionality to ensure interoperability, accessibility, security, and consistency. Standardization, continuity, and assistance of system development have become an urgent necessity to meet the increasing needs for sustainable systems. In this article, we examine and address the methods of the different AAL systems. In addition, we analyzed the acceptance criteria of the AAL framework intending to define and evaluate different AAL-based symmetrical models, leveraging performance characteristics under the integrated fuzzy environment. The main goal is to provide an understanding of the current situation of the AAL-oriented setups. Our vision is to investigate and evaluate the potential symmetrical models of AAL systems and frameworks for the implementation of effective new installations.

**Keywords:** ambient assisted living; AAL; ambient intelligence; assisted living; user-interfaces; fuzzy logic

#### **1. Introduction**

The current digital environment, comprising smart home products, mobile devices, smart watches, and software applications, has had a significant impact on human lifestyles. These systems have provided a great deal of power to individuals, thereby significantly lowering dependence on others. These advanced devices have not only transformed lifestyles, but have also revolutionized almost every area of human existence. The idea of Ambient Assisted Living (AAL) resulted from these smart technologies, and represents the response to the task of maintaining the standard of living of elderly people. Ambient Assisted Living (AAL) offers a system consisting of smart phones, medical sensors, cellular networks, computers, and health tracking apps [1]. AAL can also be used for different reasons, such as preventing, treating, and enhancing the well-being and health of elderly people.

AAL strives to promote the protection and wellbeing of elderly people and to increase the number of years that elderly people can live comfortably in an area of their own convenience [2–5]. It also reduces the amount of anticipated costs by empowering patients to monitor their serious medical symptoms. AAL is a sub-part of Ambient Intelligence that includes the utilization of ambient intelligent strategies, processes, and technology to allow aged people to survive comfortably for as long as humanly possible despite behavioral problems.

**Citation:** Alosaimi, W.; Ansari, M.T.J.; Alharbi, A.; Alyami, H.; Seh, A.H.; Pandey, A.K.; Agrawal, A.; Khan, R.A. Evaluating the Impact of Different Symmetrical Models of Ambient Assisted Living Systems. *Symmetry* **2021**, *13*, 450. https:// doi.org/10.3390/sym13030450

Academic Editor: Jan Awrejcewicz

Received: 10 February 2021 Accepted: 6 March 2021 Published: 10 March 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

In addition, modern developments in mobile and portable sensors have contributed to realizing the vision of AAL [6,7]. Recent popular electronic applications are fitted with smart configurations, such as accelerometers, navigation systems, GPS, and many other systems that can be used to monitor user mobility. Furthermore, recent developments in digital and sensor technology are promising a new age in health sensors [8]. Scientists and researchers have created discreet sensors in the form of covers, small holter-type gadgets, mobile systems, and smart clothing for tracking health indicators. For instance, blood sugar, blood pressure and heart performance can be evaluated by means of smart technologies such as infrared or photographic sensors. Many measures, such as electroencephalography (EEG), also involve invasive devices, including electrodes. The following Figure 1 shows the graphical representation of an Ambient Assisted Living (AAL) system. allow aged people to survive comfortably for as long as humanly possible despite behav‐ ioral problems. In addition, modern developments in mobile and portable sensors have contributed to realizing the vision of AAL [6,7]. Recent popular electronic applications are fitted with smart configurations, such as accelerometers, navigation systems, GPS, and many other systems that can be used to monitor user mobility. Furthermore, recent developments in digital and sensor technology are promising a new age in health sensors [8]. Scientists and researchers have created discreet sensors in the form of covers, small holter‐type gadgets, mobile systems, and smart clothing for tracking health indicators. For instance, blood sugar, blood pressure and heart performance can be evaluated by means of smart tech‐ nologies such as infrared or photographic sensors. Many measures, such as electroenceph‐ alography (EEG), also involve invasive devices, including electrodes. The following Fig‐ ure 1 shows the graphical representation of an Ambient Assisted Living (AAL) system.

to monitor their serious medical symptoms. AAL is a sub‐part of Ambient Intelligence that includes the utilization of ambient intelligent strategies, processes, and technology to

*Symmetry* **2021**, *13*, x FOR PEER REVIEW 2 of 20

**Figure 1.** Graphical representation of an Ambient Assisted Living (AAL) system. **Figure 1.** Graphical representation of an Ambient Assisted Living (AAL) system.

This study evaluates different Ambient Assisted Living system's symmetrical mod‐ els based on the taxonomy adopted from Amina et al. [9]. The study uses integrated fuzzy Analytic Hierarchy Process‐Technique for Order Performance by Similarity to Ideal Solu‐ tion (AHP‐TOPSIS), which is a popular multiple‐criteria decision‐making (MCDM) ap‐ proach. To handle the complexity of evaluating the efficiency of various Ambient Assisted Living System Symmetrical Models on one parameter, or on the value of another high‐ precision parameter, the Analytic Hierarchy Process (AHP) method has been extensively used by numerous authors and practitioners. Ghodsypour and O'Brien [10] claim that AHP is much more reliable than other symmetrical models of scoring for analysis proce‐ dures. Conceivably, the technique is suitable because the decision‐making process has a one‐way hierarchical relationship between decision‐making stages. Interestingly, Carney and Wallnau [11] noted that the selection parameters for alternatives are not necessarily This study evaluates different Ambient Assisted Living system's symmetrical models based on the taxonomy adopted from Amina et al. [9]. The study uses integrated fuzzy Analytic Hierarchy Process-Technique for Order Performance by Similarity to Ideal Solution (AHP-TOPSIS), which is a popular multiple-criteria decision-making (MCDM) approach. To handle the complexity of evaluating the efficiency of various Ambient Assisted Living System Symmetrical Models on one parameter, or on the value of another high-precision parameter, the Analytic Hierarchy Process (AHP) method has been extensively used by numerous authors and practitioners. Ghodsypour and O'Brien [10] claim that AHP is much more reliable than other symmetrical models of scoring for analysis procedures. Conceivably, the technique is suitable because the decision-making process has a one-way hierarchical relationship between decision-making stages. Interestingly, Carney and Wallnau [11] noted that the selection parameters for alternatives are not necessarily independent of each other, but rather interconnect. In such a complicated setting, an incorrect result may be obtained. TOPSIS (Technique for Order Performance by Similarity to Idea Solution) [12] is also an appropriate approach for solving MCDM problems. TOPSIS is initiated on the principle that the optimum alternative should also have the smallest distance from the positive idea solution (PIS), and the greatest distance from the negative idea solution. The principle of TOPSIS is logical and comprehensible, and the related calculation is straightforward. Consequently, it is important to note the inherent complexity of specifying accurate subjective opinions to the parameters.

In the subsequent sections of this article, we classify the discussion into five sections. Section 2 presents the different related works. Section 3 discusses the materials and methods used in this paper. Section 4 presents the statistical results and evaluates their quality characteristics according to different metrics. Finally, Section 5 concludes the paper.

#### **2. Related Works**

Fuzzy multiple-criteria decision-making (MCDM) provides successful outcomes in the resolution of selection-based problems [13]. The approach has been chosen by several researchers because it can effectively handle the knowledge that is analyzed using a multi-resourced linguistic and quantitative decision-making challenge, and evidence that is diverse.

Lin et al. [14] suggested a fuzzy-based strategy for the selection of an effective smart technology framework for fall detection. To address this problem, a fuzzy collective intelligence methodology was used. Alpha-cut procedures were used in a fuzzy collective intelligence method to determine the fuzzy weights of the parameters with each decisionmaker. Then, the fuzzy combination was used to sum the fuzzy weights generated by each decision-maker. Consequently, a fuzzy order preference strategy comparable to the ideal solution was used to assess the appropriateness of a smart technology system for fall detection.

Samanlioglu et al. [15] proposed an approach to choosing the best employee applicant for an IT organization by combining the fuzzy analytic hierarchical process method (fuzzy AHP), including Chang's scale analysis, in addition to fuzzy TOPSIS. The decisionmakers' (DMs) verbal assessments were included in the analysis using intuitive fuzzy numbers. They first calculated the value of thirty sub-criteria weights using fuzzy AHP and then, using fuzzy TOPSIS, five IT employees' alternatives were evaluated based on fuzzy AHP weights.

Anand and Vinodh [16] presented research to evaluate Additive Manufacturing (AM) procedures for micro-manufacturing using combined fuzzy AHP-TOPSIS. Parameters weights were derived using fuzzy AHP, whereas rankings were obtained using fuzzy TOPSIS.

Nazam et al. [17] suggested hybrid fuzzy AHP analysis to measure the weight of threat parameters and sub-criteria, in addition to the order efficiency technique using the ideal solution (TOPSIS) procedure to rank and evaluate the risks resulting from the adoption of green supply chain management (GSCM) practices in a fuzzy setting. Their proposed fuzzy risk-oriented assessment theory was implemented for the realistic case of the textile automotive industry. Ultimately, the conceptual model allows academics and professionals to consider the value of performing effective risk assessments while introducing green supply chain interventions.

Ansari et al. [18] addressed the viewpoint of security professionals on the relative value of parameters for the selection of an appropriate Security Requirements Engineering (SRE) system using the multi-criteria decision-making approach. The research was designed and conducted to determine the most suitable SRE approach for quality and stable application development based on the expertise and experience of the security specialist. The hierarchical analysis was performed using a fuzzy TOPSIS method. The efficient SRE selection process was evaluated in pairs.

Kumar et al. [19] used an approach that involves the combination of fuzzy AHP with fuzzy TOPSIS processes to determine the effect of various malware detection techniques in the context of web applications. This research used different variants of a university's software system to test the effect of a variety of current malware detection techniques.

Alenezi et al. [20] used the combined approach of fuzzy AHP-TOPSIS to test security architecture strategies and their attributes. The efficacy of this method was also evaluated on the real-time software system of BBAU, Lucknow, India. In addition, various university web apps were also used to support the findings produced.

The uncertainty associated with Alzheimer's disease complicates the development of patient monitoring software. To address this problem, Coronato described a systematic approach to modelling and developing intelligent Ambient Assisted Living Systems to monitor the behavior of people with cognitive impairments. The major aspect was an automated method to detect irrational behavior, in addition to a technique for the design of secure AAL systems specifically aimed at individuals with cognitive or mental illnesses. The author also implemented their approach for modelling a monitoring system that can demonstrate feasibility through the identification of abnormal circumstances [21].

Koleva et al. highlighted the main obstacles in formulating and executing an efficient AAL system based on their expertise in the eWALL project. They also provided suggestions for solutions to these obstacles [22].

Many surveys and reviews have been conducted that address the functionality of different models of Ambient Assisted Living systems. However, the fuzzy AHP-TOPSIS approach has not been used to evaluate the impact of different symmetrical models of Ambient Assisted Living systems.

#### **3. Materials and Methods**

Recent developments in a variety of technical fields have helped to realize the potential of AAL. These innovations involve smart homes, help robots, e-textiles, and portable and implantable sensors. In the following sections, the proposed research methodology is discussed in more detail. Figure 2 shows a functional diagram of the research methodology. *Symmetry* **2021**, *13*, x FOR PEER REVIEW 5 of 20

**Figure 2.** Functional diagram of the research methodology. **Figure 2.** Functional diagram of the research methodology.

#### *3.1. Step 1: Identification of Different AAL System's Models 3.1. Step 1: Identification of Different AAL System's Models*

A variety of ventures have adopted the AAL scope as a field of research. Various references, symmetrical models, systems, and interfaces have been proposed for an ac‐ ceptable AAL scheme, but few of these have been generally accepted. In this section, we analyze the most comprehensive Ambient Assisted Living system's symmetrical models. A graphical representation of the hierarchy can be seen in Figure 3. *A. Ambient Intelligence Reference Architecture* (*AmIRA)* A variety of ventures have adopted the AAL scope as a field of research. Various references, symmetrical models, systems, and interfaces have been proposed for an acceptable AAL scheme, but few of these have been generally accepted. In this section, we analyze the most comprehensive Ambient Assisted Living system's symmetrical models. A graphical representation of the hierarchy can be seen in Figure 3.

bient intelligent systems, the intention of AmIRA is to encourage the re‐use of Ambient Intelligence (AmI) processes throughout various AmI systems [24]. The advantage of this design is that each layer is autonomous; however, there is a demand for resources from other layers. AmIRA is uncertain because there are no restrictions mentioned, however,

Continua is an attempt to provide compatibility in the area of personal telemedicine. The design identifies different steps and reference system classes [25]. Continua made an explicit option to build a framework that incorporates several domains. It is known to be the only approach that provides a uniform security system. Continua reflects the reference architecture mostly as a symmetrical layered model. Each link is composed to its counter‐ parts by a platform that has been identified. Reference system classes and platforms are the maximum representation level in the Continua design. Conversely, it depends on the

FeelGood is an initiative that introduces new architectural elements to complement existing practices by specifying the ecosystem level of service components [26]. The pri‐ mary objective was to enhance the quality of life in Finland. The suggested RA does not

the framework components do not adequately define the business processes.

execution of the design and lacks any applications.

*B. Continua*

*C. FeelGood*

MPOWER is an AAL initiative with the goal of creating and maintaining AAL frame‐ works based on trends, service‐oriented frameworks, online services, and Extensible Stylesheet Language (XSDL) transformations [35]. It emphasizes compatibility among ser‐

*Symmetry* **2021**, *13*, x FOR PEER REVIEW 7 of 20

*I. MPOWER*

vices, extensively in the areas of AAL.

**Figure 3.** Structure for evaluation of different Ambient Assisted Living system's symmetrical models. **Figure 3.** Structure for evaluation of different Ambient Assisted Living system's symmetrical models.

#### *A. Ambient Intelligence Reference Architecture (AmIRA)*

AmIRA includes an emphasis on processes, structures, and components [23]. It assimilates multi-layer application architecture. According to the given specifications of ambient intelligent systems, the intention of AmIRA is to encourage the re-use of Ambient Intelligence (AmI) processes throughout various AmI systems [24]. The advantage of this design is that each layer is autonomous; however, there is a demand for resources from other layers. AmIRA is uncertain because there are no restrictions mentioned, however, the framework components do not adequately define the business processes.

#### *B. Continua*

Continua is an attempt to provide compatibility in the area of personal telemedicine. The design identifies different steps and reference system classes [25]. Continua made an explicit option to build a framework that incorporates several domains. It is known to be the only approach that provides a uniform security system. Continua reflects the

reference architecture mostly as a symmetrical layered model. Each link is composed to its counterparts by a platform that has been identified. Reference system classes and platforms are the maximum representation level in the Continua design. Conversely, it depends on the execution of the design and lacks any applications.

*C. FeelGood*

FeelGood is an initiative that introduces new architectural elements to complement existing practices by specifying the ecosystem level of service components [26]. The primary objective was to enhance the quality of life in Finland. The suggested RA does not describe comprehensive interfaces, but offers interface specifications and refers to the applicable requirements that could be implemented. The purpose of this RA was to direct the development of the product. Stakeholders and comprehensive services were identified.

*D. RAFAALS*

RAFAALS relates to the reference design for AAL systems [27,28] as a SOA-based symmetrical model of AAL. The key architecture is based on the principle of separating the functionality and the configuration of the data flow among the layers of the process. It facilitates the transfer of events between flexible software applications wherein its modules allow use of little to no awareness of other elements. This structure distinguishes the activities within each evolved producer. The layers are analytically isolated from each other and clearly specified. This is a traditional structural platform that enables the design, formulation, and deployment of any AAL setting. It defines modules and connections in an abstract way, making them a comprehensive, functional, and systematic architecture.

*E. UniversAAL*

The UniversAAL system is an open platform designed to promote the development, delivery, and implementation of technology solutions towards assisted living environments. This system is utilized to encourage end-users (i.e., supported individuals, their parents, and communities), AAL-responsible authorities, and organizations involved in the creation and implementation of AAL services. It comprises of a wide variety of tools (some are applications and others are models/architectures) targeted at these various classes. Services are divided into three major groups: runtime assistance, production assistance, and community assistance. UniversAAL is regarded to be one of the most comprehensive RMs to date [29,30]. It represents an interpretation of AAL structures at the maximum level of abstraction [31], utilizing as few terms as possible. It also reflects the AAL domain description, the AAL space review, the types of technology included, and several other principles.

#### *F. PERSONA*

PERSONA is a service platform for AAL environments; it is designed to promote the incremental creation of AAL areas focused on a compact foundation. AAL spaces are modelled as accessible distributed systems in PERSONA. The system depends on the administrative re-configuration of the platform elements, including the Situation Reasoner, the Dialog Manager, and the Service Orchestra, to provide aggregated benefits [32].

*G. SOPRANO*

SOPRANO is a highly configurable, open AAL framework for senior citizens focused on semantic agreements. This serves as a facilitating artifact among the different modules of the process by establishing a uniform integrated terminology for various layers of abstraction. The architecture of SOPRANO offers exclusively predetermined contractbased interfaces for various stakeholders centered on structured ontology [33]. Ontology serves as a facilitating artifact between the modules of the different structure by providing a shared interconnected language for different levels of abstraction. The SOPRANO initiative has been an important guide for several other initiatives, such as OpenAAL [34], which acts as a comprehensive ontology for AAL symmetrical models.

#### *H. OpenAAL*

OpenAAL is a collaborative open-source program of the FZI Research Center for Information Technologies, Friedrich-Schiller-University of Jena, and CAS Software AG. It provides a scalable and efficient interface for AAL situations and is focused on the research findings of many German and foreign initiatives, such as the SOPRANO2 Combined Project. The OpenAAL framework allows for easy deployment, configuration, and scenariodependent provision of versatile, context-aware, and customized IT-based functions [29].

#### *I. MPOWER*

MPOWER is an AAL initiative with the goal of creating and maintaining AAL frameworks based on trends, service-oriented frameworks, online services, and Extensible Stylesheet Language (XSDL) transformations [35]. It emphasizes compatibility among services, extensively in the areas of AAL.

#### *3.2. Step 2: Fuzzy AHP*

Saaty suggested the Analytic Hierarchy Process (AHP) method in 1990. All numerical and contextual considerations are integrated into AHP in the decision-making procedure. Because the highly classified scale of 1 to 9 is often used in the AHP process, this method is commonly criticized for not integrating uncertainty into the decision-making procedure. The fuzzy AHP approach has also been used in many fields to overcome multi-criteria challenges. Haq and Kannan [36] used this approach to pick the best supplier in the supply chain. This was also used by Huang et al. [37] for the evaluation of R&D projects. Fuzzy set theory is essentially a type of classical set theory. It is based on an adjacency matrix, and assigns a rank between one and ten. If the symbol is a fuzzy package, a tilde (i.e., ∼) is placed over it. A fuzzy activity is defined by (l, m, u) whereby 'l' is the lowest number, 'm' is the most probable value, and 'u' is the maximum priority [19].

#### *3.3. Step 3: Ranking with TOPSIS*

On the basis of the results, alternatives are rated using TOPSIS. In this process, the following types of parameters or characteristics are regarded:


In this analysis, different kinds of alternatives are evaluated as follows:


TOPSIS is focused on the choice of the most appropriate alternative or initiative that is the furthest from the negative ideal solution and the nearest to the positive ideal solution. The positive and negative ideal solutions are those with the maximum and minimum benefits, respectively.

#### **4. Results**

This section addresses the various quantitative measurements of the integrated fuzzy deployment of the AHP-TOPSIS symmetrical method. To achieve this goal, in our research study, we used the combined fuzzy AHP-TOPSIS approach, a well-established and verified decision-making technique. This methodology is designed to evaluate different Ambient Assisted Living system's symmetrical models based on their impact assessment in the current information technology era. To create a more compelling result, we took recommendations from 79 experts with diverse technologies and academic abilities.

To evaluate the different Ambient Assisted Living system's symmetrical models from a user perspective, five Level-1 parameters, namely domain and context, goals, design and

infrastructure, quality attributes, and critical attributes, were defined respectively as LC1, LC2, LC3, LC4 and LC5. Further sub-parameters for domain and context were usability, independency, and purpose, defined respectively as LC11, LC12 and LC13, LC14 and LC5. Goals sub-parameters were standardization and facilitation, defined respectively as LC21 and LC22. Sub-parameters for design and infrastructure were components and connections, interfaces, policies and guidelines, details, concretization, formality, data flows, and cost, defined respectively as LC31, LC32, LC33, LC34, LC35, LC36, LC37 and LC38. Quality attributes sub-parameters were interoperability, scalability, confidentiality, maintainability, privacy, trustability, and security, defined respectively as LC41, LC42, LC43, LC44, LC45, LC46 and LC47. Critical attributes sub-parameters were reliability, integrity, performance, availability, and safety, defined respectively as LC51, LC52, LC53, LC54 and LC55. Different alternatives to the Ambient Assisted Living system's symmetrical model were AmIRA, Continua, FeelGood, RAFAALS, UniversAAL, PERSONA, SOPRANO, OpenAAL, and MPOWER denoted by AT1, AT2, AT3, AT4, AT5, AT6, AT7, AT8 and AT9 respectively. The local criteria and sub-criteria weights were calculated using pair-wise comparison matrices.

The pair-wise comparative matrix for the level 1 factor was created, as shown in Table 1. The compound pair-wise relative matrixes for the hierarchical diagram of level 2 are also specified in Tables 2–6. Table 7 shows the defuzzification matrix with alpha cut method and local weights. Tables 8–12 show aggregated pair-wise comparison matrixes at level 2 for domain and context, goals, design and infrastructure, quality attributes, and critical attributes, respectively. To be more specific, integration was executed to quantify the element weights of each point. In addition, with the support of the hierarchical structure, Table 13 and Figure 4 represent the overall weights and ranking of methods.


**Table 1.** Fuzzy pair-wise comparison matrix at level 1.

**Table 2.** Fuzzy aggregated pair-wise comparison matrix at level 2 for domain and context.


**Table 3.** Fuzzy aggregated pair-wise comparison matrix at level 2 for goals.



**Table 4.** Fuzzy aggregated pair-wise comparison matrix at level 2 for design and infrastructure.

**Table 5.** Fuzzy aggregated pair-wise comparison matrix at level 2 for quality attributes.



**Table 6.** Fuzzy aggregated pair-wise comparison matrix at level 2 for critical attributes.

**Table 7.** Defuzzification matrix with alpha cut method and local weights.


**Table 8.** Aggregated pair-wise comparison matrix at level 2 for domain and context.


**Table 9.** Aggregated pair-wise comparison matrix at level 2 for goals.



**Table 10.** Aggregated pair-wise comparison matrix at level 2 for design and infrastructure.

**Table 11.** Aggregated pair-wise comparison matrix at level 2 for quality attributes.


**Table 12.** Aggregated pair-wise comparison matrix at level 2 for critical attributes.


**Table 13.** Overall weights and ranking of methods.



**Table 13.** *Cont.*

**Figure 4.** Graphical representation of overall weights**. Figure 4.** Graphical representation of overall weights.

LC12

LC13

LC21

LC22

LC31

LC32

LC33

LC34

LC35

**Table 14.** Subjective cognition results of evaluators in linguistic terms**. AT1 AT2 AT3 AT4 AT5 AT6 AT7 AT8 AT9** LC11 5.120, 7.140, 8.720 3.150, 5.150, 6.910 2.820, 4.640, 6.640 1.550, 3.180, 5.180 5.120, 7.140, 8.720 3.150, 5.150, 6.910 2.820, 4.640, 6.640 1.550, 3.180, 5.180 Table 14 shows the subjective cognition results of evaluators in linguistic terms. Table 15 presents the normalized fuzzy decision matrix. Table 16 presents the weighted normalized fuzzy decision matrix. Finally, Table 17 and Figure 5 shows the closeness coefficients to the aspired level among the different alternatives.

> 5.120, 7.140, 8.720

> 4.280, 6.370, 8.370

> 2.820, 4.640, 6.640

> 5.120, 7.140, 8.720

> 4.280, 6.370, 8.370

> 2.820, 4.640, 6.640

> 2.910, 4.640, 6.550

> 2.820, 4.640, 6.640

> 2.820, 4.640, 6.640

3.150, 5.150, 6.910

2.820, 4.640, 6.640

2.910, 4.640, 6.550

3.150, 5.150, 6.910

2.450, 4.450, 6.450

5.360, 7.360, 9.120

4.640, 6.640, 8.550

5.360, 7.360, 9.120

5.360, 7.360, 9.120

2.450, 4.450, 6.450

1.550, 3.180, 5.180

1.450, 3.000, 4.910

3.150, 5.150, 6.910

2.450, 4.450, 6.450

2.820, 4.640, 6.640

1.820, 3.730, 5.730

1.450, 3.000, 4.910

1.450, 3.000, 4.910

2.910, 4.640, 6.550

5.120, 7.140, 8.720

4.280, 6.370, 8.370

2.820, 4.640, 6.640

2.910, 4.640, 6.550

2.820, 4.640, 6.640

2.820, 4.640, 6.640

4.280, 6.370, 8.370

4.280, 6.370, 8.370

1.450, 3.000, 4.910

3.150, 5.150, 6.910

2.450, 4.450, 6.450

5.360, 7.360, 9.120

4.640, 6.640, 8.550

5.360, 7.360, 9.120

5.360, 7.360, 9.120

2.450, 4.450, 6.450

2.450, 4.450, 6.450

1.550, 3.180, 5.180

1.450, 3.000, 4.910

3.150, 5.150, 6.910

1.550, 3.180, 5.180

1.450, 3.000, 4.910

3.150, 5.150, 6.910

2.450, 4.450, 6.450

2.820, 4.640, 6.640

1.820, 3.730, 5.730

1.450, 3.180, 5.180

1.180, 2.820, 4.820

2.450, 4.450, 6.450

2.820, 4.640, 6.640

2.450, 4.450, 6.450

2.450, 4.450, 6.450

1.180, 2.820, 4.820

1.180, 2.820, 4.820

2.450, 4.450, 6.450

1.180, 2.820, 4.820

2.820, 4.640, 6.640

2.910, 4.640, 6.550

5.120, 7.140, 8.720

2.820, 4.640, 6.640

2.910, 4.640, 6.550

5.120, 7.140, 8.720

4.280, 6.370, 8.370

2.180, 4.090, 6.140

3.550, 5.550, 7.450

4.280,

4.270, 6.270, 8.140

5.360, 7.360, 9.120

4.640, 6.640, 8.550

3.120, 5.000, 7.140

4.280, 6.370, 8.370

4.280, 6.370, 8.370

4.270, 6.270, 8.140

4.280, 6.370, 8.370


**Table 14.** Subjective cognition results of evaluators in linguistic terms.

**Table 14.** *Cont.*


**Table 15.** The normalized fuzzy decision matrix.


**Table 15.** *Cont.*


**Table 15.** *Cont.*


**Table 16.** The weighted normalized fuzzy decision matrix.


**Table 16.** *Cont.*


**Table 17.** Closeness coefficients to the aspired level among the different alternatives.


*Symmetry* **2021**, *13*, x FOR PEER REVIEW 18 of 20

0.036, 0.072, 0.162

0.044, 0.088, 0.182

**Table 17.** Closeness coefficients to the aspired level among the different alternatives**.**

0.041, 0.095, 0.198

**Alternatives d+i d−i Gap Degree of CC+i Satisfaction Degree of CC−i** AT1 1.249124 1.333548 0.5164578 0.4832657 AT2 0.699528 0.840586 0.5476598 0.4532564 AT3 0.787654 1.484745 0.6535644 0.3452657 AT4 2.168547 1.484856 0.4125471 0.5923547 AT5 2.005654 1.536954 0.4332654 0.5652547 AT6 0.445476 0.392356 0.4645687 0.5345854 AT7 0.788574 1.484657 0.4125475 0.4522564 AT8 2.160256 1.536235 0.4332654 0.3452544 AT9 2.035657 0.397596 0.4645288 0.5912556

0.061, 0.121, 0.233

0.034, 0.091, 0.200

0.032, 0.089, 0.200

0.063, 0.120, 0.233

**Figure 5.** Graphical representation of closeness coefficients to the aspired level among the different alternatives. **Figure 5.** Graphical representation of closeness coefficients to the aspired level among the different alternatives.

Consequently, RAFAALS (denoted by AT4) was found to be best among nine com‐ parative alternatives because it offers the best functionality with an efficiency score of 0.5923547 among the different Ambient Assisted Living system's symmetrical models. Al‐ ternative AT4 was followed by AT9, AT5, AT6, AT1, AT2, AT7, AT3, and AT8 with per‐ formance scores of 0.5912556, 0.5652547, 0.5345854, 0.4832657, 0.4532564, 0.4522564, 0.3452657 and 0.3452544, respectively. Consequently, RAFAALS (denoted by AT4) was found to be best among nine comparative alternatives because it offers the best functionality with an efficiency score of 0.5923547 among the different Ambient Assisted Living system's symmetrical models. Alternative AT4 was followed by AT9, AT5, AT6, AT1, AT2, AT7, AT3, and AT8 with performance scores of 0.5912556, 0.5652547, 0.5345854, 0.4832657, 0.4532564, 0.4522564, 0.3452657 and 0.3452544, respectively.

#### **5. Conclusions 5. Conclusions**

LC55

0.043, 0.096, 0.196

0.017, 0.059, 0.152

AAL symmetrical models have progressed as an outcome of the emergence of global population ageing and the change of direction of technological advances. It lies at the in‐ tersection of technological innovation and age advancement. This multidisciplinary area of scientific research sees technology as a way of enhancing the lives of older people and promoting their involvement as involved members of society. This paper explores the principles of the Ambient Assisted Living area. We tested predefined categories to illus‐ trate the key areas that need to change due to the advancement of AAL systems. We sug‐ gest that there is a lack of standardization in several of the available frameworks. This is AAL symmetrical models have progressed as an outcome of the emergence of global population ageing and the change of direction of technological advances. It lies at the intersection of technological innovation and age advancement. This multidisciplinary area of scientific research sees technology as a way of enhancing the lives of older people and promoting their involvement as involved members of society. This paper explores the principles of the Ambient Assisted Living area. We tested predefined categories to illustrate the key areas that need to change due to the advancement of AAL systems. We suggest that there is a lack of standardization in several of the available frameworks. This is the key deterrent in achieving the desired efficacy of the models. Maintenance and interoperability occur in all of the models. In some systems, there is a need for facility, interoperability, and independence. For a specific design model, the practitioners need to focus on a solid standard and develop a robust infrastructure. One of the crucial aspects of AAL systems is coping with data flow, which remains a largely unexplored domain and needs more dedicated research.

**Author Contributions:** W.A. performed formal analysis and investigation. M.T.J.A. conceived and designed the experiments, performed the computation work authored and reviewed drafts of the paper. A.A. (Abdullah Alharbi) contributed reagents/materials/analysis tools. H.A. performed the visualization, review and editing work. A.H.S. analyzed and reviewed the drafts. A.K.P. performed the review and editing work. A.A. (Alka Agrawal) analyzed the data, contributed reagents/materials/analysis tools, authored or reviewed drafts of the paper. R.A.K. conceived and designed the experiments, performed the experiments, contributed reagents/materials/analysis tools. All authors have read and agreed to the published version of the manuscript.

**Funding:** The project has been funded by Taif University, Kingdom of Saudi Arabia.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This research was supported by Taif University Researchers Supporting Project number (TURSP-2020/254), Taif University, Taif, Saudi Arabia.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Dynamical Simulation of Effective Stem Cell Transplantation for Modulation of Microglia Responses in Stroke Treatment**

**Awatif Jahman Alqarni 1,2 , Azmin Sham Rambely 2,\* and Ishak Hashim <sup>2</sup>**


**Abstract:** Stem cell transplantation therapy may inhibit inflammation during stroke and increase the presence of healthy cells in the brain. The novelty of this work, is to introduce a new mathematical model of stem cells transplanted to treat stroke. This manuscript studies the stability of the mathematical model by using the current biological information on stem cell therapy as a possible treatment for inflammation from microglia during stroke. The model is proposed to represent the dynamics of various immune brain cells (resting microglia, pro-inflammation microglia, and anti-inflammation microglia), brain tissue damage and stem cells transplanted. This model is based on a set of five ordinary differential equations and explores the beneficial effects of stem cells transplanted at early stages of inflammation during stroke. The Runge–Kutta method is used to discuss the model analytically and solve it numerically. The results of our simulations are qualitatively consistent with those observed in experiments in vivo, suggesting that the transplanted stem cells could contribute to the increase in the rate of ant-inflammatory microglia and decrease the damage from pro-inflammatory microglia. It is found from the analysis and simulation results that stem cell transplantation can help stroke patients by modulation of the immune response during a stroke and decrease the damage on the brain. In conclusion, this approach may increase the contributions of stem cells transplanted during inflammation therapy in stroke and help to study various therapeutic strategies for stem cells to reduce stroke damage at the early stages.

**Keywords:** cell transplantation; cytokines; ischemic stroke; numerical simulation; runge-kutta method; stability analysis

#### **1. Introduction**

The numbers of stroke-related deaths are increasing, and globally, stroke is now one of the topmost causes of disability and death [1–3]. During an ischemic stroke, the process by which neurons and glial cells die is known as apoptosis or necrosis [4,5]. Resident microglia are triggered by these dead cells and cause the death of other cells based on environmental toxic substances [5–8]. The components of the brain that prevent the invasion of several diseases are the microglial cells, and these cells can also help prevent stroke [7,9]. Dead cells are phagocytized by activated microglia, which also produce toxic cytokines that impact healthy cells [4,9]. The pathophysiology of ischemic stroke is evident through the inflammatory response [5,10]. Directly after arterial occlusion, inflammation begins in the vasculature and then continues throughout the brain, and systemically throughout the disease stages [10,11]. The body generates tightly regulated immune responses that confer detrimental as well as beneficial properties after stroke [12]. Some of the effects of microglia activation include inhibition of brain repair and/or considerable brain damage, including neurogenesis [11,12]. Due to variability effects on inflammatory processes, the immune response following a stroke serves as a strong determinant of brain restoration or increase

**Citation:** Alqarni, A.J.; Rambely, A.S.; Hashim, I. Dynamical Simulation of Effective Stem Cell Transplantation for Modulation of Microglia Responses in Stroke Treatment. *Symmetry* **2021**, *13*, 404. https://doi.org/10.3390/ sym13030404

Academic Editor: Sergei D. Odintsov

Received: 8 February 2021 Accepted: 25 February 2021 Published: 2 March 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the damage in brain [9,11,13]. Thus, inducing stroke recovery-directed modulation of the immune response can offer a potential therapeutic approach [11].

Recently, neurorestorative stem cell-based treatment has been the focus of many stroke studies [1,7]. The extraordinary sophistication of the pathophysiology of ischemic strokes reveals pleiotropic effects on neural stem cells (NSCs), which are potentially therapeutic for both the early (subacute) and chronic phases of stroke [4,7,14]. In addition, after an ischemic stroke for newborn neurons, various obstacles are produced by the generated pathological condition, which makes the use of endogenous repair mechanisms challenging [7,10,15]. Newly formed neurons tend to die, with just ∼ 0.2% survival rate for the remaining cells, while others live up to 5 weeks after ischemia [10].

Microglia can be polarized toward the anti-inflammatory and angiogenic phenotype (*Ma*) with stem cell transplantation [11]. In one study of ischemic rats, the dependent suppression of inflammation emerged as a classical factor secreted by *M<sup>a</sup>* microglia after the demonstration of intracerebral NSC transplantation and was associated with enhanced angiogenesis and functional recovery in terms of microglia polarization [11]. Currently, it is believed that pro-inflammatory microglia (*Mp*) can exacerbate brain injury, while anti-inflammatory *M<sup>a</sup>* microglia are neuroprotective [4,12,16,17]. Thus, these dual attributes render them appropriate for use in enhancing post-stroke brain recovery, which can be achieved by shifting their balance from the detrimental *M<sup>p</sup>* to the beneficial *M<sup>a</sup>* phenotype [18]. Evidence also shows that the polarization status of microglia can be altered by stem cells (SCs) and, as such, the observed beneficial actions of SCs can be attributed to skewing microglia toward a neuroprotective and neuroregenerative phenotype [11,14]. Thus, it becomes imperative to elucidate the mechanisms of how SCs respond to tissue damage to understand the crosstalk between inflammation and SCs [10,11]. Additionally, a possible intervention point for regenerative therapies can be met through tweaking the effects of inflammation on stem cell behavior [19].

Studies have shown that stroke SCs can offer a viable solution in the future [11,14,15] and correspondingly, scholars have proposed several models of inflammatory processes [5,20,21]. For example, Di Russo et al. [5] modeled the dynamics of inflammation from a stroke: using (i) necrosis and apoptosis, (ii) the activation and inactivation of resident microglia, and (iii) the ratio of neutrophils and macrophages in the tissue, the dynamics of dead cell densities were investigated. Many scholars have also suggested quantitative methods, such as the use of statistical and computational methods, to study adult neurogenesis [22–24]. Along the same lines, systems of a hierarchical cell-constructing model were studied by Nakata et al. [25], where the structure system was controlled by the adult cells. Ziebell et al. [23] introduced a mathematical model that portrayed the various stages of the adult hippocampus and the evolving dynamics of SCs.

The basic framework for our study is a mathematical model of the interplay between microglia and endogenous NSCs during a stroke. Alqarni et al. [4] evaluated potential mechanisms to regulate and stabilize the treatment of microglia inflammation associated with an exogenous stem cell transplantation stroke. Alharbi and Rambely used ordinary differential equations in the formulation of mathematical models to describe the effect of vitamins on strengthening the immune system and its function in delaying the development of tumour cells [26–28]. In addition, Ordinary differential equations (ODEs) have been used to explain disease behavior over time, which has improved therapeutic strategies [4,28,29].

We aim to enhance understanding of the symmetry and antisymmetry of the relationship between the functions of pro-inflammatory and anti-inflammatory microglia with the effect of transplanted stem cells on the immunomodulation of microglia functions during a stroke. We modified a mathematical model for the therapy of the inflammation process in stroke using ODEs. In this paper, we study the roles of stem cell transplantation dynamics via inhibition of inflammation from microglia and stimulate the beneficial function of microglia during a stroke. We investigate the possible mechanisms involved in the dynamics of the transplanted stem cell functions through immune activities.

This paper is organized as follows: In Section 2, a mathematical model called SCs–damage–resting microglia–pro-inflammation microglia– anti-inflammation microglia (SDRPA) is presented. In Section 3, we study the model's equilibrium points. In Section 4, we investigate the model's stability. In Section 5, numerical experiments are discussed. Finally, the study's conclusions are presented in Section 6.

#### **2. Mathematical Representation of the SDRPA Model**

In this section, we give a dynamic model for the explanation of the use of transplanted SCs to treat the inflammation produced by microglia during a stroke. The suggested model of therapy stem cell dynamics transplanted in onset stroke shows the interplay between the transplanted SCs and microglia in stroke. The mathematical model is a structure of five differential equations, which are analyzed to find the equilibrium points and to study their stability. Several previous studies have shown the possible therapeutic benefits of transplanted SCs, where the transplantation of pluripotent stem cells in stroke patients has many functions. One of the potential therapeutic of this type of treatment is the immunomodulation of microglia functions during a stroke [11,30].

There are three options for differentiating SCs [31] : (1) symmetrical self-renewal with the possibility *σ<sup>S</sup>* of two SCs, (2) asymmetric self-renewal with the probability of *σA*, where one of the cells denotes the daughter residue a stem cell, whereas another cell does not make this discrimination, and (3) symmetric involvement differentiation with probability of *σD*, where a stem cell has the ability to divide to be two involvement cells. Here we suppose that *σ<sup>S</sup>* + *σ<sup>A</sup>* + *σ<sup>D</sup>* = 1 [31]. In this manuscript, we assumed that SCs *S* reproduce at the rate *σ* and die at the rate *γ<sup>s</sup>* . Thus, transplanted SCs have considerable influence on the neuroinflammation caused by a stroke because they secrete relevant cytokines to support the anti-inflammation of microglial activation to transform the *M<sup>a</sup>* to *M<sup>p</sup>* phenotype [32]. Therefore, the terms *α*4, *α*<sup>5</sup> and *α*<sup>6</sup> were designated to describe the direct interactions between SCs and the microglia *M<sup>a</sup>* and *Mp*. The following equation describes transplanted SCs' behavior during a stroke:

$$\frac{dS}{dt} \quad = \quad [\sigma - \alpha\_5 M\_a - \alpha\_6 M\_p - \gamma\_s] S.$$

where *σ* indicates the reproduction rate of SCs, *α*<sup>5</sup> signifies the stimulating and supporting rate of SCs for *Ma*, while *α*<sup>6</sup> indicates the rate of elimination of SCs due to the *M<sup>p</sup>* and *γ<sup>s</sup>* indicates the rate of death of SCs.

During an ischemic stroke, microglia are reactivated and polarized to either a classical type, *Mp*, by pro-inflammatory cytokines that cause an immune response and lead to secondary damage in the brain; or to a substitutional type, *Ma*, an anti-inflammatory state caused through anti-inflammatory cytokines, that reduces inflammation and boosts cell repair [33–36]. For mathematical modeling purposes, we supposed that resting immune cells of microglia *M<sup>r</sup>* were produced at a constant rate of *α*0, indicating the source rate of the resting microglia cells, which are dying at a constant ratio of *γ*0.

The given differential equation describes resting microglia cells' behavior:

$$\frac{dM\_r}{dt} \quad = \quad \mathfrak{a}\_0 - (\mathfrak{a}\_1 + \mathfrak{a}\_2 + \gamma\_0)M\_r.$$

Microglia that are activated by cytokines are caused through dead cells: the *M<sup>p</sup>* phenotype is activated through signals of pro-inflammation cytokine *α*<sup>1</sup> and the *M<sup>a</sup>* phenotype is activated by signals of anti-inflammation cytokine *α*2. Moreover, a shift from *M<sup>p</sup>* to *M<sup>a</sup>* occurs at the rate of parameter *α*3, where *α*<sup>3</sup> is the beginning of transference from pro-inflammatory microglia to anti-inflammatory microglia. The modulation of the immune response is considered an important function of a possible therapeutic approach to improve brain recovery post-stroke [11,19]. Exogenous stem cell transplantation has been demonstrated to modulate the inflammatory immune microenvironment across the ischemic regions of the brain by modulating the functions of the immune cells during the

stroke [11,19,33]. Direct transplant of SCs into the brain after ischemia decreased many inflammatory and immune responses and switched the balance from a pro-inflammatory to anti-inflammatory response of microglia [37]. We present these functions by the rate of parameters *α*<sup>4</sup> and *α*5. The following equations representing microglia *M<sup>a</sup>* and *M<sup>p</sup>* thus take the form:

$$\begin{aligned} \frac{dM\_p}{dt} &= -\alpha\_1 M\_r - (\alpha\_4 S + \delta\_1 D + \alpha\_3 + \gamma\_1) M\_p, \\ \frac{dM\_d}{dt} &= -\alpha\_2 M\_r + (\alpha\_5 S - \gamma\_2) M\_d + (\alpha\_3 + \alpha\_4 S) M\_p. \end{aligned}$$

where, *δ*<sup>1</sup> signifies the rate of damage from microglia related to the production of cytokines, *α*<sup>4</sup> denotes the rate of amendment of SCs for the function of *Mp*, and *γ*<sup>1</sup> and *γ*<sup>2</sup> are the natural death rate of *M<sup>p</sup>* and *Ma*, respectively.

Ischemic stroke injury is regarded as a major factor contributing to tissue damage. We assume that the SDRPA model shows that activated microglial cells play complex functions displaying both harmful and beneficial effects, which could include from elimination of cell debris by phagocytosis process and the release of inflammatory cytokines that lead to tissue destruction and increase cell death beyond the primary ischemic region [12,34]. Therefore, the dynamics of dead brain cells after stroke can be explained via the differential equation:

$$\frac{dD}{dt} \quad = \quad [(\delta\_2 - r\_1)M\_p - r\_2M\_a]D.$$

where *δ*<sup>2</sup> indicates the death rate of brain cells due to *Mp*; and *r*<sup>1</sup> and *r*<sup>2</sup> refers to the elimination rates of the damaged cells by *M<sup>p</sup>* and *Ma*, respectively. Thus, the SDRPA model can be written in the following form:

$$\frac{dS}{dt}\Big|\_{t=0} = [\sigma - \mathfrak{a}\_5 \mathcal{M}\_a - \mathfrak{a}\_6 \mathcal{M}\_p - \gamma\_\mathfrak{s}] \mathcal{S}\_\prime \tag{1}$$

$$\frac{dD}{dt} \quad = \quad [(\delta\_2 - r\_1)M\_p - r\_2M\_a]D\_\prime \tag{2}$$

$$\frac{dM\_r}{dt} = -a\_0 - (a\_1 + a\_2 + \gamma\_0)M\_{r\prime} \tag{3}$$

$$\frac{dM\_p}{dt} = -\alpha\_1 M\_r - (\alpha\_4 S + \delta\_1 D + \alpha\_3 + \gamma\_1) M\_{p\prime} \tag{4}$$

$$\frac{dM\_a}{dt} = -a\_2 M\_r + (a\_5 S - \gamma\_2) M\_a + (a\_3 + a\_4 S) M\_p. \tag{5}$$

The initial conditions in this model are: *S*(0) = *S* ∗ , *D*(0) = *D*∗ , *Mr*(0) = *M*∗ *r* , *Mp*(0) = *M*∗ *p* , and *Ma*(0) = *M*∗ *a* , 0 ≤ *t* < ∞ where *S*(*t*), *D*(*t*), *Mr*(*t*), *Mp*(*t*), and *Ma*(*t*) represent the stem cell concentration, dead cells, resting microglia, pro-inflammation microglia, and the anti-inflammation microglia, respectively.

The proposed dynamic model of SDRPA demonstrated by (1)–(5), represents the population behavior of the transplanted SCs, immune cells, and damaged cells in a stroke. Thus, the variables *S*(*t*), *D*(*t*), *Mr*(*t*), *Mp*(*t*), and *Ma*(*t*) and all parameters are non-negative real, real, and equal to or less than one. The invariant area of SDRPA becomes:

$$\Psi = (\mathcal{S}, \mathcal{D}, \mathcal{M}\_r, \mathcal{M}\_p, \mathcal{M}\_a) \in \mathfrak{R}\_+^5 \tag{6}$$

#### **3. The SDRPA Model's Equilibrium Points**

We determine the fixed points of the SDRPA system (1)–(5) from the following: *dS dt* = 0 ⇔

$$[\sigma - \mathfrak{a}\_5 \mathcal{M}\_a - \mathfrak{a}\_6 \mathcal{M}\_p - \gamma\_\mathcal{s}] \mathcal{S} = \mathbf{0},\tag{7}$$

$$\begin{array}{ccc} \bullet & \frac{dD}{dt} = 0 \Leftrightarrow \end{array}$$

•

$$[(\delta\_2 - r\_1)M\_p - r\_2M\_a]D = 0,\tag{8}$$

$$\bullet \qquad \frac{dM\_r}{dt} = 0 \Leftrightarrow$$

$$
\alpha\_0 - (\alpha\_1 + \alpha\_2 + \gamma\_0)M\_r = 0,\tag{9}
$$

• *dMp dt* = 0 ⇔ *α*1*M<sup>r</sup>* − (*α*4*S* + *δ*1*D* + *α*<sup>3</sup> + *γ*1)*M<sup>p</sup>* = 0, (10)

\* 
$$\begin{aligned} \frac{dM\_4}{dt} &= 0 \Leftrightarrow \\\\ a\_2 M\_r + (a\_5 S - \gamma\_2) M\_a + (a\_3 + a\_4 S) M\_p &= 0. \end{aligned} \tag{11}$$

The equilibrium points of SDRPA (1)–(5) calculate by solving Equations (7)–(11), we obtain three positive real points of equilibrium by using MATHEMATICA, while the other solutions will be negative. Thus, the first positive equilibrium point is given by:

$$Q\_1(S, D, M\_{\prime \prime}, M\_{p \prime}, M\_{\mathbb{R}}) = \left(0, 0, \frac{a\_0}{z}, \frac{a\_0 a\_1}{z(a\_3 + \gamma\_1)}, \frac{a\_0(a\_1 a\_3 + a\_2(a\_3 + \gamma\_1))}{z(a\_3 + \gamma\_1)\gamma\_2}\right).$$

where

$$z = \mathfrak{a}\_1 + \mathfrak{a}\_2 + \gamma\_0. \tag{12}$$

Next, we represent the second positive equilibrium point as follows:

$$Q\_2(S, D, M\_r, M\_p, M\_a) = \left(0, \frac{q}{r\_2 \ a\_2 \delta\_1}, \frac{a\_0}{z}, -\frac{r\_2 a\_0 a\_2}{z \ p}, a\_0 \frac{a\_2 (r\_1 - \delta\_2)}{z \ p}\right).$$

where

$$\begin{array}{rcl} p &=& (r\_2\alpha\_3 + \gamma\_2(r\_1 - \delta\_2)) < 0, \\ q &=& -r\_2(\alpha\_1\alpha\_3 + \alpha\_2(\alpha\_3 + \gamma\_1)) + \alpha\_1\gamma\_2(-r\_1 + \delta\_2). \end{array}$$

The third positive equilibrium point is obtained as follows:

$$\begin{array}{rcl} \mathcal{Q}\_3(\mathcal{S}, \mathcal{D}, \mathcal{M}\_{\mathcal{V}}, \mathcal{M}\_{\mathcal{V}}, \mathcal{M}\_{\mathcal{d}}) &=& \begin{pmatrix} \mathcal{\beta}\_3 & \mathcal{\alpha}\_0 \\ \mathcal{\beta}\_1 & \mathcal{z} \end{pmatrix} \mathcal{r}\_2 \mathcal{\beta}\_{0\prime} \left(\mathcal{\delta}\_2 - r\_1\right) \mathcal{\beta}\_0 \end{pmatrix}. \end{array}$$

where

$$\begin{array}{rcl} \beta\_{0} &=& \frac{\omega}{a\_{5}(-r\_{1}+\delta\_{2})+r\_{2}a\_{6}} > 0, \beta\_{1} = (r\_{2}a\_{4}+a\_{5}(-r\_{1}+\delta\_{2}))>0, \\ \beta\_{2} &=& \frac{1}{r\_{2}z\delta\_{1}\beta\_{1}\omega} [(r\_{1}^{2}+\delta\_{2}^{2})a\_{0}a\_{1}a\_{5}^{2}+r\_{2}^{2}a\_{4}(a\_{0}(a\_{1}+a\_{2})a\_{6}-z\gamma\_{1}\omega) \\ &+& +r\_{2}(-r1+\delta\_{2})(a\_{0}a\_{5}(a\_{2}a\_{4}+a\_{1}(a\_{4}+a\_{6})))-r\_{1}(2a\_{0}a\_{1}a\_{5}^{2}\delta\_{2}), \\ &-(-r\_{1}+\delta\_{2})r\_{2}z(a\_{5}(a\_{3}+\gamma\_{1})+a\_{4}\gamma\_{2})\omega)<0, \\ \beta\_{3} &=& -p-\frac{a\_{0}a\_{2}(r\_{2}a\_{6}+a\_{5}(\delta\_{2}-r\_{1}))}{z\omega}>0, \omega=-\gamma\_{5}+\sigma>0. \end{array}$$

**Proposition 1.** *We assume that the equilibrium points for the SDRPA system, S*; *D*; *M<sup>r</sup>* ; *Mp*; *M<sup>a</sup>* > 0 *are satisfied under the conditions:*


• *α*0*α*2(*r*2*α*6+*α*5(*δ*2−*r*<sup>1</sup> )) *<sup>z</sup><sup>ω</sup>* < *γ*2(−*r*<sup>1</sup> + *δ*2) − *r*2*α*<sup>3</sup>

$$
\bullet \qquad r\_2 \alpha\_3 < \gamma\_2 (\delta\_2 - r\_1)),
$$

• *r*2(*α*1*α*<sup>3</sup> + *α*2(*α*<sup>3</sup> + *γ*1)) < *γ*2*α*1(*δ*<sup>2</sup> − *r*1)

*Non-negative real, steady states will then, and only then, exist.*

We can study the stability of equilibrium points if they continue to exist constantly over time or constantly change in equilibrium in one direction. We defined three steady states as follows:

**Definition 1.** *We define the resting microglia activation in a normal brain as the absence of a high activation for these cells when stroke occurs into pro-inflammation microglia and anti-inflammation microglia, the steady-state M<sup>r</sup>* ; *Mp*; *M<sup>a</sup>* > 0 *and D*, *S* = 0 *implies the functions of the microglia in the onset of a stroke is normal and the microglia do not cause any damage in the brain.*

**Definition 2.** *We define the starting of the damage by the increased rate of pro-inflammation cytokines where the existence of high activation of microglia will cause damage in brain, the steady-states D*; *M<sup>r</sup>* ; *Mp*; *M<sup>a</sup>* > 0*, and S* = 0 *imply the damage of the brain cells from the pro-inflammation of microglia by the damaged brain cells from inflammation by microglia due to a stroke without stem cells transplantation.*

**Definition 3.** *We define the functions of transplanted stem cells on modulation of microglia responses in the brain after stroke, the steady-state of the forms S*; *D*; *M<sup>r</sup>* ; *Mp*; *M<sup>a</sup>* > 0 *is defined by the role of stem cell transplantation in inflammation.*

#### **4. The Equilibrium Points' Stability of the SDRPA Model**

A stability study of the SDRPA model for the equilibrium points is performed. By applying the Hartman–Grobman theorem concept [38], the system's 5 × 5 Jacobian matrix for the eigenvalues associated with transplanted SCs equilibrium in the brain after a stroke (1)–(5) is given by:

$$J[\pi] = \begin{bmatrix} F\_S[\boldsymbol{\varrho}] & F\_D[\boldsymbol{\varrho}] & F\_{M\_r}[\boldsymbol{\varrho}] & F\_{M\_p}[\boldsymbol{\varrho}] & F\_{M\_s}[\boldsymbol{\varrho}] \\ G\_S[\boldsymbol{\varrho}] & G\_D[\boldsymbol{\varrho}] & G\_{M\_r}[\boldsymbol{\varrho}] & G\_{M\_p}[\boldsymbol{\varrho}] & G\_{M\_d}[\boldsymbol{\varrho}] \\ H\_S[\boldsymbol{\varrho}] & H\_D[\boldsymbol{\varrho}] & H\_{M\_r}[\boldsymbol{\varrho}] & H\_{M\_p}[\boldsymbol{\varrho}] & H\_{M\_d}[\boldsymbol{\varrho}] \\ K\_S[\boldsymbol{\varrho}] & K\_D[\boldsymbol{\varrho}] & K\_{M\_r}[\boldsymbol{\varrho}] & K\_{M\_p}[\boldsymbol{\varrho}] & K\_{M\_d}[\boldsymbol{\varrho}] \\ L\_S[\boldsymbol{\varrho}] & L\_D[\boldsymbol{\varrho}] & L\_{M\_r}[\boldsymbol{\varrho}] & L\_{M\_p}[\boldsymbol{\varrho}] & L\_{M\_d}[\boldsymbol{\varrho}] \end{bmatrix}.$$

where *ϕ* = [*S*, *D*, *M<sup>r</sup>* , *Mp*, *Ma*], *F*[*ϕ*] = *dS dt* , *<sup>G</sup>*[*ϕ*] = *dD dt* , *<sup>H</sup>*[*ϕ*] = *dM<sup>r</sup> dt* , *<sup>K</sup>*[*ϕ*] = *dM<sup>p</sup> dt* , and *L*[*ϕ*] = *dM<sup>a</sup> dt* .

**Theorem 1.** *Given that the function <sup>g</sup>* : <sup>Ψ</sup> → <<sup>5</sup> <sup>+</sup> *where* <sup>Ψ</sup> *is a domain in* <sup>&</sup>lt;<sup>5</sup> <sup>+</sup>*, and assuming that Q*<sup>1</sup> ∈ Ψ *is an equilibrium point, where one eigenvalue of the Jacobian matrix has a non-negative real part at minimum. Therefore, Q*<sup>1</sup> *indicates an unstable equilibrium point of g.*

 

**Proof.** The Jacobian matrix *J* calculated at the first equilibrium point *Q*<sup>1</sup> yields: *J*[*Q*1] =

$$
\begin{bmatrix} a\_{11} & 0 & 0 & 0 & 0 \\ 0 & a\_{22} & 0 & 0 & 0 \\ 0 & 0 & -z & 0 & 0 \\ -\frac{a\_{0}\alpha\_{1}\kappa\_{4}}{z(\alpha\_{3}+\gamma\_{1})} & -\frac{a\_{0}\alpha\_{1}\delta\_{1}}{z(\alpha\_{3}+\gamma\_{1})} & a\_{1} & -a\_{3}-\gamma\_{1} & 0 \\ a\_{15} & 0 & a\_{2} & a\_{3} & -\gamma\_{2} \end{bmatrix} \text{ where}
$$

$$
\begin{split}
a\_{11} &=& -\frac{a\_{0}(\alpha\_{2}\kappa\_{5}(\alpha\_{3}+\gamma\_{1})+\alpha\_{1}(\alpha\_{3}\kappa\_{5}+\kappa\_{6}\gamma\_{2}))}{z(\alpha\_{3}+\gamma\_{1})\gamma\_{2}} - \gamma\_{5} + \sigma\_{n}, \\
a\_{22} &=& -\frac{r\_{2}a\_{0}(\alpha\_{1}\alpha\_{3}+\alpha\_{2}(\alpha\_{3}+\gamma\_{1}))}{z(\alpha\_{3}+\gamma\_{1})\gamma\_{2}} + \frac{a\_{0}\alpha\_{1}(-r\_{1}+\delta\_{2})}{z(\alpha\_{3}+\gamma\_{1})}, \\
a\_{15} &=& \frac{a\_{0}\alpha\_{1}\kappa\_{4}}{z(\alpha\_{3}+\gamma\_{1})} + \frac{a\_{0}\alpha\_{5}(\alpha\_{1}\kappa\_{3}+\alpha\_{2}(\alpha\_{3}+\gamma\_{1}))}{z(\alpha\_{3}+\gamma\_{1})\gamma\_{2}}, \\
z &=& (a\_{1}+a\_{2}+\gamma\_{0}).
\end{split}
$$

We determine the eigenvalues of the matrix *J*[*Q*1] as follows:

$$\begin{aligned} \lambda\_1 &=& -z < 0, \\ \lambda\_2 &=& -\frac{\mathfrak{a}\_0(r\_2(\mathfrak{a}\_1 + \mathfrak{a}\_2)\mathfrak{a}\_3 + r\_2\mathfrak{a}\_2\gamma\_1 + \mathfrak{a}\_1\gamma\_2(r\_1 - \delta\_2))}{z(\mathfrak{a}\_3 + \gamma\_1)\gamma\_2} > 0, \\ \lambda\_3 &=& -\frac{\mathfrak{a}\_0(\mathfrak{a}\_2\mathfrak{a}\_5(\mathfrak{a}\_3 + \gamma\_1) + \mathfrak{a}\_1(\mathfrak{a}\_3\mathfrak{a}\_5 + \mathfrak{a}\_6\gamma\_2))}{z(\mathfrak{a}\_3 + \gamma\_1)\gamma\_2} - \gamma\_3 + \sigma < 0, \\ \lambda\_4 &=& -\gamma\_2 < 0, \lambda\_5 = -(\mathfrak{a}\_3 + \gamma\_1) < 0. \end{aligned}$$

From Proposition 1, the equilibrium points, it is obvious that *λ*<sup>2</sup> > 0. Thus, *J*(*Q*1) has at least one positive root, which denotes that the second equilibrium point *Q*<sup>1</sup> is unstable.

**Theorem 2.** *Given the function <sup>g</sup>* : <sup>Ψ</sup> → <<sup>5</sup> <sup>+</sup>*, where* <sup>Ψ</sup> *is a domain in* <sup>&</sup>lt;<sup>5</sup> <sup>+</sup>*, and assuming that Q*<sup>2</sup> ∈ Ψ *is an equilibrium point, then one eigenvalue of the Jacobian matrix has a non-negative real part at minimum. Therefore, Q*<sup>2</sup> *indicates an unstable equilibrium point of g.*

**Proof.** The Jacobian matrix *J* calculated at the second equilibrium point *Q*<sup>2</sup> yields: *J*[*Q*2] =

$$\begin{bmatrix} b\_{11} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & b\_{24} & -\frac{q}{a\_2 \delta\_1} \\ 0 & 0 & -z & 0 & 0\\ v\alpha\_4 & v\delta\_1 & a\_1 & \frac{a\_1 p}{r\_2 a\_2} & 0\\ -\frac{a\_0 a\_2 \delta\_1}{Zp} & 0 & a\_2 & a\_3 & -\gamma\_2 \end{bmatrix} \text{where}$$

$$b\_{11} = -\omega + \frac{a\_0 a\_2 (-r\_1 a\_5 + r\_2 a\_6 + a\_5 \delta\_2)}{zp}, v = \frac{r\_2 a\_0 a\_2}{zp}, b\_{24} = \frac{(-r\_1 + \delta\_2)q}{r\_2 a\_2 \delta\_1}.$$

The corresponding characteristic equation of the Jacobian *J*[*Q*2], is specified by:

$$(b\_{11} - \lambda)(-z - \lambda)(\kappa\_0 + \kappa\_1\lambda + \kappa\_2\lambda^2 + \lambda^3) = 0.$$

where

$$\kappa\_2 \quad = \quad -\frac{p \cdot \mathfrak{a}\_1}{r\_2 \mathfrak{a}\_2} + \gamma\_{2\prime} \kappa\_1 = \frac{-p \,\mathfrak{a}\_1 \,\gamma\_2 + q \,\,\upsilon \,(r\_1 - \delta\_2)}{r\_2 \mathfrak{a}\_2}, \kappa\_0 = \frac{q \,\,\upsilon \, p}{r\_2 \mathfrak{a}\_2}.$$

Thus, we can obtain the first two eigenvalues:

$$
\lambda\_1 = \omega + \frac{\mathfrak{a}\_0 \mathfrak{a}\_2 (-r\_1 \mathfrak{a}\_5 + r\_2 \mathfrak{a}\_6 + \mathfrak{a}\_5 \delta\_2)}{zp} > 0, \lambda\_2 = -z < 0.
$$

From Proposition 1, the equilibrium points, it is obvious that *λ*<sup>1</sup> > 0. Thus, *J*(*Q*2) has at least one positive root, which denotes that the second equilibrium point *Q*<sup>2</sup> is unstable.

**Theorem 3.** *Given the function <sup>g</sup>* : <sup>Ψ</sup> → <<sup>5</sup> <sup>+</sup>*, where* <sup>Ψ</sup> *denotes a domain in* <sup>&</sup>lt;<sup>5</sup> <sup>+</sup>*, and assuming that Q*<sup>3</sup> ∈ Ψ *indicates an equilibrium point, all the Jacobian matrix's eigenvalues have negative real parts at the equilibrium point Q*3*. Therefore, Q*<sup>3</sup> *is assumed as a stable equilibrium point of g.*

**Proof.** The Jacobian matrix *J* calculation at the third equilibrium point *Q*<sup>3</sup> yields: *J*[*Q*3] =

0 0 0 *c*<sup>14</sup> *c*<sup>15</sup> 0 0 0 *c*<sup>24</sup> *c*<sup>25</sup> 0 0 −*z* 0 0 *c*<sup>41</sup> *c*<sup>42</sup> *α*<sup>1</sup> *c*<sup>44</sup> 0 *c*<sup>51</sup> 0 *α*<sup>2</sup> *c*<sup>54</sup> *c*<sup>55</sup> where *c*<sup>14</sup> = − *β*3 *β*1 *α*<sup>6</sup> < 0, *c*<sup>24</sup> = *β*2(−*r*<sup>1</sup> + *δ*2) > 0 *c*<sup>15</sup> = − *β*3 *β*1 *α*<sup>5</sup> < 0, *c*<sup>25</sup> = −*β*2*r*<sup>2</sup> < 0, *c*<sup>41</sup> = −*β*0*r*2*α*<sup>4</sup> < 0, *c*<sup>42</sup> = −*β*0*r*2*δ*<sup>1</sup> < 0, *c*<sup>44</sup> = −*α*<sup>3</sup> − *β*3 *β*1 *α*<sup>4</sup> − *γ*<sup>1</sup> − *β*2*δ*<sup>1</sup> < 0, *c*<sup>51</sup> = *β*0*β*<sup>1</sup> > 0, *c*<sup>54</sup> = *α*<sup>3</sup> + *β*3 *β*1 *α*<sup>4</sup> > 0, *c*<sup>55</sup> = *β*3 *β*1 *α*<sup>5</sup> − *γ*<sup>2</sup> < 0.

The characteristic equation can be written as:

$$(z+\lambda)(\lambda^4+\eta\_3\lambda^3+\eta\_2\lambda^2+\eta\_1\lambda+\eta\_0) = 0,\tag{13}$$

Here,

 

$$\begin{aligned} \eta\_0 &= -c\_{42}c\_{51}M\_0 > 0, \eta\_1 = c\_{44}M\_6 + c\_{41}M\_4 - c\_{42}M\_5 > 0, \\ \eta\_2 &= -M\_1 + M\_2 > 0, \eta\_3 = -M\_3 > 0. \end{aligned}$$

where,

$$\begin{aligned} M\_0 &=& c\_{15}c\_{24} - c\_{14}c\_{25} < 0,\\ M\_2 &=& -c\_{15}c\_{51} + c\_{44}c\_{55} > 0,\\ M\_4 &=& c\_{15}c\_{54} - c\_{14}c\_{55} < 0,\\ M\_6 &=& c\_{15}c\_{54} - c\_{14}c\_{55} < 0,\\ M\_6 &=& c\_{15}c\_{51} < 0. \end{aligned}$$

Thus, we can determine the first eigenvalue:

$$
\lambda\_1 = -z < 0.
$$

Thus, we can use the Routh–Hurwitz theorem for *λ* <sup>4</sup> + *η*3*λ* <sup>3</sup> + *η*2*λ* <sup>2</sup> + *η*1*λ* + *η*<sup>0</sup> = 0 [39], giving


Thus, the essential and adequate conditions of all roots contain negative real parts *η<sup>i</sup>* (*i* = 1; 3; 4) > 0 and ∆ > 0. Then,

$$\begin{aligned} \Delta &=& c\_{42}c\_{51}M\_0(-c\_{42}c\_{51}M\_0M\_3^2 - (M\_1+M\_2)M\_3(c\_{41}M\_4 - c\_{42}M\_5) \\ &+& c\_{44}M\_6) - (c\_{41}M\_4 - c\_{42}M\_5 + c\_{44}M\_6)^2) > 0 \end{aligned}$$

From Proposition 1,

$$\tilde{\xi}\_1 = \frac{\eta\_3 \eta\_2 - \eta\_1}{\eta\_3} > 0, \tilde{\xi}\_2 = \frac{\tilde{\xi}\_1 \eta\_1 - \eta\_0 \eta\_3}{\tilde{\xi}\_1} > 0$$

where

$$
\zeta\_1 = \frac{(M\_1 + M\_2)M\_3 + c\_{41}M\_4 - c\_{42}M\_5 + c\_{44}M\_6}{M\_3} > 0
$$

$$
\zeta\_2 = -c\_{41}M\_4 - c\_{42}M\_5 + c\_{44}M\_6 + \frac{c\_{42}c\_{51}M\_0M\_3^2}{(M\_1 + M\_2)M\_3 + c\_{41}M\_4 - c\_{42}M\_5 + c\_{44}M\_6} > 0
$$

Equation (13) has no roots with positive real parts, and only one of its eigenvalues is negative in view of the positive signs of all the coefficients in the first column. The equilibrium point *Q*<sup>3</sup> is therefore stable.

**Remark 1.** *The impact of using SCs transplant on the functions of microglia in stroke onset, represented by the dynamic SDRPA model, can be summarized as follows:*


#### **5. Numerical Results and Analysis**

This section discusses the utilization of computational models to assess which ones affect the model's behavior and thus explore the numerical solutions of the system ((1)–(5)).

#### *5.1. Determination of Parameters*

The SDRPA model involves 18 parameters, including five parameters for the initial conditions of each compartment. Herein, the researchers list several parameters which can be assessed using the experimental studies. Furthermore, for solving the system of ordinary differential equations ((1)–(5)), other parameter values were obtained by simulations through the Software MATHEMATICA with the command NDSolve. The purpose was to assess the SCs' ability to control the functions of immune cells during a stroke. From the simulation of SDRPA model, it can be concluded that there are three parameters that directly affect the behavior of microglia and SCs after transplant, parameter *α*4, which represents the rate of amendment of SCs for the function of *Mp*, parameter *α*5, which shows the rate of stimulating and supporting SCs of anti-inflammatory immune cells and, the suppression rate of SCs as a result of damage by pro-inflammatory microglia denoted by the parameter *α*6, and the simulation results showed that the effectiveness of stem cell transplantation to modulate pro-inflammation microglia function where the highest rate of SCs will switch the balance from a pro-inflammatory to anti-inflammatory response of microglia as follows: the rate involving SCs was simulated by *α*<sup>4</sup> = 50%, *α*<sup>5</sup> = 35% and *α*<sup>6</sup> = 14%. Furthermore, it was obvious that the modulation of microglia responses occurred if the following conditions were satisfied: The rate of stem cell amendment for the function of *M<sup>p</sup>* >; and the suppression rate of SCs by *Mp*, *α*<sup>4</sup> > *α*6. Table 1 displays the reference series of values of the parameters. From Figure 1, we can observe various

behaviors depending on the transplant process. In addition, damage continually decreased, while pro-inflammation microglia had a little increase, asides from shifting to a steep curve after approximately 20 h. Furthermore, that anti-inflammation microglia had a high level after stroke onset and transplantation SCs, aside from shifting to a steep and a stable curve at approximately after 20 h to the end of simulation of the recovery stage. It can be seen that after SCs transplanted in the brain the anti-inflammation microglia had a higher level than the pro-inflammation microglia comparing with numerical results evaluating microglia effect on the brain dynamics without SCs transplantation during 72 h of strok as presented in [4]. Additionally, for the concentration of transplanted SCs, there was initial growth and subsequently a reduction over time due to the effects of the treatment on the elimination rate of pro-inflammation microglia. The system was solved numerically, and time concatenation of the solutions were plotted on the system ((1)–(5)) for the parameters to obtain the dynamics of the system. For this, the fourth-order Runge–Kutta method (RK4) was used in all simulations to obtain extra stable and approximate solutions. The time 72 h was chosen for 10−<sup>4</sup> as the step size for carrying out the simulations of the model [40,41]. Figures 2 and 3 depict the precision of the suggested numerical method presented through the residual error.

**Table 1.** Parameters values for transplanted stem cells–damaged brain cells–resting microglia– pro-inflammation microglia–anti-inflammation microglia (SDRPA) model.


**Figure 1.** Effectiveness of stem cell transplantation on immune cells' phenotype behavior and dead brain cells within 72 h. Where *α*<sup>4</sup> = 0.5314, *α*<sup>5</sup> = 0.3468 and *α*<sup>6</sup> = 0.1419.

**Figure 2.** The residual error at several steps for SDRPA model.

**Figure 3.** The residual error at time t for SDRPA model.

#### *5.2. Comparison of Experimental Results*

Parameter values of resting microglia rates, activation microglia rates, decay rates of concentration of the damage by microglia, reproduction rate of SCs, death rates, and initial conditions, were assumed on the basis of the literature. We assumed that the SCs transplantation influenced transference between the states of microglia activation and could support the anti-inflammation function of microglia, which led to faster recovery. These assumptions were in accordance with the biological understanding of SC functions. In contrast to the findings of our study regarding immune cell functions and the damage during the time of stroke with the SCs effects transplanted with the work in [16,34], we established that the studies agreed in that microglial activation had the contribution of both beneficial and harmful functions in the brain. Furthermore, stem cell transplantation is considered to attenuate ischemia-induced brain injury within hours of transplantation [1,33,40]. The simulation's findings of the model corresponded with the experimental findings for determining the effect of exogenous SCs in the brain during a stroke. We compared the simulation results of the SDRPA model with the numerical results of the effects of dynamic SCs on brain therapy after stroke by endogenous SCs and exogenous SCs. Additionally, we compared the findings in our research to those of [4], and found that stem cell transplantation has the ability to modify the cytokine environment in the brain, especially for early cell transplantation after stroke. For example, transplantation within 1 week caused a reduction in the levels of pro-inflammatory cytokines and growth in the levels of anti-inflammatory cytokines. The results of the SDRPA simulation implied that when neural endogenous SCs failed to grow and assist brain recovery, exogenous SCs were considered the best solution for brain repair processes.

#### **6. Conclusions**

Mathematics and computer science fields have worked interactively to better understand biological processes. We modified the model—SDRPA ((1)–(5)) based on the model of Alqarni et al. [4] to study the effectiveness of the role of transplanted exogenous SCs in the brain on the microglia during a stroke. The multifaceted SCs affect the tissue by inhibiting of the pro-inflammation and stimulating the function of the anti-inflammation

microglia. We discussed the results of the dynamical system and the effects of SCs transplanted and microglia during the stroke analytically and numerically. The analysis and simulation results of the system show the ability of transplanted SCs to help the brain by reducing the inflammation on the onset of stroke. Following that, the rate of damage reduced after transplantation. An evident linkage between the mathematical and biological mechanisms was observed. From the analytical results, we can deduce that the stability of the SDRPA model illustrated capacity in the exogenous stem cell implantation, which is significant for immunomodulation. In conclusion, our model may assist in conception of the effectiveness of using SCs transplanted in the brain repair processes. In the outlook, we will extend this study to model strategies that improve, stimulate, and generate the NSCs in the early stage, and where that information could contribute to understanding the effects of therapeutic strategies. We hope to conduct more experimental studies to clinically investigate the results of our mathematical model and to show more precise results. In future studies, it could be interesting to incorporate the dynamics of anti-inflammatory and pro-inflammatory cytokines from microglia and cytokines of endogenous NSCs into the SDRPA model to describe the interaction processes of the different cytokine types in ischemic stroke. Furthermore, we will develop this model by studying the effect of SCs in stimulating endogenous neural cells in a stroke and dynamically validate the ability to support the endogenous stem cells by using pharmacological drugs to improve stroke therapy. In summary, symmetry and antisymmetry are fundamental to understand the role of microglia in ischemic stroke pathobiology constituting a major challenge for the development of efficient immunomodulatory therapies by SCs.

**Author Contributions:** Conceptualization, A.J.A.; funding acquisition, A.S.R.; methodology, A.J.A.; project administration, A.S.R.; supervision, A.S.R. and I.H.; validation, A.S.R.; writing—original draft, A.J.A.; writing—review and editing, I.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research is funded by a grant from Universiti Kebangsaan Malaysia with code GUP-2017-112.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank the Universiti Kebangsaan Malaysia for the support in this study.

**Conflicts of Interest:** All authors declare that there is no conflict of interest associated with this study.

#### **References**


## *Article* **Second-Order Non-Canonical Neutral Differential Equations with Mixed Type: Oscillatory Behavior**

**Osama Moaaz 1,\* ,† , Amany Nabih 1,†, Hammad Alotaibi 2,† and Y. S. Hamed 2,†**


**Abstract:** In this paper, we establish new sufficient conditions for the oscillation of solutions of a class of second-order delay differential equations with a mixed neutral term, which are under the non-canonical condition. The results obtained complement and simplify some known results in the relevant literature. Example illustrating the results is included.

**Keywords:** non-canonical differential equations; second-order; neutral delay; mixed type; oscillation criteria

#### **1. Introduction**

This paper discusses the oscillatory behavior of solutions of second-order functional differential equation with a mixed neutral term of the form

$$\left(r(l)\left[\left(y(l) + p\_1(l)y(\rho\_1(l)) + p\_2(l)y(\rho\_2(l))\right)'\right]'\right)' + q(l)y^\gamma(\sigma(l)) = 0,\tag{1}$$

where *l* ≥ *l*0. Throughout this paper, we assume the following:

**(C1)** *γ* ∈ *Q* + *odd* := {*a*/*b* : *a*, *b* ∈ Z <sup>+</sup> are odd} and *<sup>r</sup>* <sup>∈</sup> *<sup>C</sup>*([*l*0, <sup>∞</sup>),(0, <sup>∞</sup>)); **(C2)** *ρ*1, *ρ*2, *σ* ∈ *C*([*l*0, ∞), R), *ρ*1(*l*) ≤ *l* ≤ *ρ*2(*l*), *σ*(*l*) ≤ *l* and *ρ*1, *ρ*2, *σ* → ∞ as *l* → ∞; **(C3)** *p*1, *p*2, *q* ∈ *C*([*l*0, ∞), [0, ∞)) and *q*(*l*) is not identically zero for large *l*.

Let *y* be a real-valued function defined for all *l* in a real interval [*ly*, ∞), *l<sup>y</sup>* ≥ *l*0, and having a second derivative for all *l* ∈ [*ly*, ∞). The function *y* is called a *solution* of the differential Equation (1) on [*ly*, ∞) if *y* satisfies (1) on [*ly*, ∞). A nontrivial solution *y* of any differential equation is said to be *oscillatory* if it has arbitrary large zeros; otherwise, it is said to be *nonoscillatory*. We will consider only those solutions of (1) which exist on some half-line [*l<sup>b</sup>* , ∞) for *l<sup>b</sup>* ≥ *l*<sup>0</sup> and satisfy the condition sup{|*y*(*l*)| : *l<sup>c</sup>* ≤ *l* < ∞} > 0 for any *l<sup>c</sup>* ≥ *l<sup>b</sup>* .

A delay differential equation of neutral type is an equation in which the highest order derivative of the unknown function appears both with and without delay. During the last decades, there is a great interest in studying the oscillation of solutions of neutral differential equations. This is due to the fact that such equations arise from a variety of applications including population dynamics, automatic control, mixing liquids, and vibrating masses attached to an elastic bar, biology in explaining self-balancing of the human body, and in robotics in constructing biped robots, it is easy to notice the emergence of models of the neutral delay differential equations, see [1,2].

In the following, we review some of the related works that dealt with the oscillation of the neutral differential equations of mixed-type.

**Citation:** Moaaz, O.; Nabih, A.; Alotaibi, H.; Hamed, Y.S. Second-Order Non-Canonical Neutral Differential Equations with Mixed Type: Oscillatory Behavior. *Symmetry* **2021**, *13*, 318. https://doi.org/ 10.3390/sym13020318

Academic Editors: Sergei D. Odintsov and Celina Si

Received: 12 January 2021 Accepted: 10 February 2021 Published: 14 February 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Grammatikopouls et al. [3] established oscillation criteria for the equation

$$\left(r(l)\psi'(l)\right)' + q(l)y(\sigma(l)) = 0,\tag{2}$$

where

$$z(l) = y(l) + p\_1(l)y(l - \sigma\_1) + p\_2(l)y(l + \sigma\_2)\_r$$

*r*(*l*) = 1, *p*2(*l*) = 0, 0 ≤ *p*<sup>1</sup> ≤ 1, and *q*(*l*) ≥ 0. Ruan [4] obtained some oscillation criteria for the Equation (2) by employing Riccati technique and averaging function method, when *p*2(*l*) = 0 and *σ*(*l*) = *l* − *σ*. Arul and Shobha [5] studied the oscillatory behavior of solution of (2), when 0 ≤ *p*1(*l*) ≤ *p*<sup>1</sup> < ∞ and 0 ≤ *p*2(*l*) ≤ *p*<sup>2</sup> < ∞.

Dzurina et al. [6] presented some sufficient conditions for the oscillation of the secondorder equation

$$\left(\frac{1}{r(l)}y'(l)\right)' + p(l)y(\tau(l)) + q(l)y(\sigma(l)) = 0.$$

Li [7] and Li et al. [8] studied the oscillation of solutions of the second-order equation with constant mixed arguments:

$$\left(r(l)z'(l)\right)' + q\_1(l)y(l - \sigma\_3) + q\_2(l)y(l + \sigma\_4) = 0. \tag{3}$$

Arul and Shobha [5] established some sufficient conditions for the oscillation of all solutions of Equation (3) in the canonical case, that is,

$$\int\_{l\_0}^{\infty} r^{-1}(\theta) \mathbf{d}\theta = \infty$$

Thandapani et al. [9] studied the oscillation criteria for the differential equation of the

form

$$(z^{\kappa}(l))'' + q(l)y^{\beta}(l - \tau\_1) + p(l)y^{\gamma}(l + \tau\_1) = 0.$$

Grace et al. [10] studied the oscillatory behavior of solutions of the equation

$$\left(r(l)\left(\left(y(l) + p\_1(l)y^{\beta\_1}(\sigma\_1(l)) + p\_2(l)y^{\beta\_2}(\sigma\_2(l))\right)'\right)'\right)' + q(l)y^\gamma(\tau(l)) = 0, \quad l = 1, \ldots, k$$

and considered the two cases

$$\int\_{l\_0}^{\infty} r^{-1/\gamma}(\vartheta) \mathbf{d}\vartheta = \infty,\tag{4}$$

and

$$\int\_{l\_0}^{\infty} r^{-1/\gamma}(\vartheta) \mathrm{d}\vartheta < \infty. \tag{5}$$

In [11], Tunc et al. studied the oscillatory behavior of the differential Equation (1) under the condition (4). Moreover, they considered the two following cases: *p*1(*l*) ≥ 0, *p*2(*l*) ≥ 1, and *p*2(*l*) 6= 1 eventually; *p*2(*l*) ≥ 0, *p*1(*l*) ≥ 1, and *p*2(*l*) 6= 1 eventually.

For the third-order equations, Han et al. [12] studied the oscillation and asymptotic properties of the third-order equation

$$\left(a(l)z''(l)\right)' + q\_1(l)y(l-\tau\_3) + q\_2(l)y(l+\tau\_4) = 0\_\prime$$

and established two theorems which guarantee that the above equation oscillates or tends to zero. Moaaz et al. [13] discussed the oscillation and asymptotic behavior of solutions of the third-order equation

$$\left(r(l)\left(\mathbf{x}^{\prime\prime}(l)\right)^{a}\right)' + q\_1(l)f\_1(\mathbf{y}(\sigma\_1(l))) + q\_2(l)f\_2(\mathbf{y}(\sigma\_2(l))) = 0,$$

where *x*(*l*) = *y*(*l*) + *p*1(*l*)*y*(*τ*1(*l*)) + *p*2(*l*)*y*(*τ*2(*l*)). For further results, techniques, and approaches in studying oscillation of the delay differential equations, see in [14–24].

In this paper, we study the oscillatory behavior of solutions of the second-order differential equation with a mixed neutral term (1) under condition (5). We follow a new approach based on deducing a new relationship between the solution and the corresponding function. Using this new relationship, we first obtain one condition that ensures oscillation of (1). Moreover, by introducing a generalized Riccati substitution, we get a new criterion for oscillation of (1). Often these types of equations (such as (1), (2), and (3)) are studied under condition (4). On the other hand, the works that studied these equations under the condition (5) obtained two conditions to ensure the oscillation. Therefore, our results are an extension and simplification as well as improvement of previous results in [3–5,8,11].

#### **2. Main Results**

We adopt the following notation for a compact presentation of our results:

$$\psi(l) := y(l) + p\_1(l)y(\rho\_1(l)) + p\_2(l)y(\rho\_2(l)),$$

$$\kappa(u, v) := \int\_u^v r^{-1/\gamma}(\delta) \mathbf{d}\delta,$$

$$B\_1(l) := 1 - p\_1(l) \frac{\kappa(\rho\_1(l), \infty)}{\kappa(l, \infty)} - p\_2(l)$$

and

$$B\_2(l) := 1 - p\_1(l) - p\_2(l) \frac{\kappa(l\_1, \rho\_2(l))}{\kappa(l\_1, l)}.$$

**Lemma 1.** *Assume that* Θ(*ϑ*) := *Aϑ* − *B*(*ϑ* − *C*) (*γ*+1)/*γ , where A*, *B and C are real constants; B* > 0*; and γ* ∈ *Q* + *odd. Then,*

$$\Theta(\vartheta^\*) \le \max\_{\mu \in \mathbb{R}} \Theta(\vartheta) = A\mathbb{C} + \frac{\gamma^{\gamma}}{\left(\gamma + 1\right)^{\gamma + 1}} A^{\gamma + 1} \mathcal{B}^{-\gamma}.$$

**Lemma 2.** *Assume that y is a positive solution of (1) on* [*l*0, ∞)*. If ψ is a decreasing positive function for l* ≥ *l*<sup>1</sup> *large enough, then*

$$\left(\frac{\psi(l)}{\kappa(l,\infty)}\right)' \ge 0,\text{ for } l \ge l\_1. \tag{6}$$

*While if ψ is a increasing positive function for l* ≥ *l*1*, then*

$$\left(\frac{\psi(l)}{\kappa(l\_1, l)}\right)' \le 0, \text{ for } l \ge l\_1. \tag{7}$$

**Proof.** Assume that (1) has a positive solution *y* on [*l*0, ∞). Therefore, there exists a *l*<sup>1</sup> ≥ *l*<sup>0</sup> such that, for all *<sup>l</sup>* <sup>≥</sup> *<sup>l</sup>*1, *<sup>ψ</sup>*(*l*) <sup>≥</sup> *<sup>y</sup>*(*l*) <sup>&</sup>gt; 0 and *r*(*l*)(*ψ* 0 (*l*))*<sup>γ</sup>* ≤ 0. From (1), we see that

$$\left(r(l)\left(\psi'(l)\right)^{\gamma}\right)' = -q(l)y^{\gamma}(\sigma(l)) \le 0.$$

Obviously, *ψ* is either eventually decreasing or eventually increasing. Let *<sup>ψ</sup>* be a decreasing function on [*l*1, <sup>∞</sup>). Then, lim*l*→<sup>∞</sup> *<sup>ψ</sup>*(*l*) < <sup>∞</sup>, and so

$$
\psi(l) \ge -\int\_l^\infty r^{-1/\gamma}(\theta) r^{1/\gamma}(\theta) \psi'(\theta) d\theta \ge -\kappa (l,\infty) r^{1/\gamma}(l) \psi'(l). \tag{8}
$$

Thus,

$$\left(\frac{\psi(l)}{\kappa(l,\infty)}\right)' = \frac{\kappa(l,\infty)\psi'(l) + r^{-1/\gamma}(l)\psi(l)}{\left(\kappa(l,\infty)\right)^2} \ge 0.$$

Let *ψ* be a increasing function on [*l*1, ∞). Then, we obtain

$$
\psi(l) \ge \int\_{l\_1}^{l} r^{-1/\gamma}(\vartheta) r^{1/\gamma}(\vartheta) \psi'(\vartheta) d\vartheta \ge \kappa(l\_1, l) r^{1/\gamma}(l) \psi'(l).
$$

and so

$$\left(\frac{\psi(l)}{\kappa(l\_1, l)}\right)' = \frac{\kappa(l\_1, l)\psi'(l) - r^{-1/\gamma}(l)\psi(l)}{\kappa^2(l\_1, l)} \le 0.$$

Thus, the proof is complete.

**Theorem 1.** *Assume that B*2(*l*) ≥ *B*1(*l*) > 0*. If*

$$\limsup\_{l\to\infty} \int\_{l\_1}^{l} \frac{1}{r^{1/\gamma}(\beta)} \left( \int\_{l\_1}^{\beta} q(\delta) B\_1^{\gamma}(\sigma(\delta)) \kappa^{\gamma}(\sigma(\delta), \infty) d\delta \right)^{1/\gamma} d\beta = \infty,\tag{9}$$

*then, all solutions of (1) are oscillatory.*

**Proof.** Assume the contrary that Equation (1) has a positive solution *y* on [*l*0, ∞). Then, *y*(*ρ*1(*l*)), *y*(*ρ*2(*l*)) and *y*(*σ*(*l*)) are positive for all *l* ≥ *l*1, where *l*<sup>1</sup> is large enough. Thus, from (1) and the definition of *ψ*, we note that *ψ*(*l*) ≥ *y*(*l*) > 0 and *r*(*l*)(*ψ* 0 (*l*))*<sup>γ</sup>* is nonincreasing. Therefore, *ψ* 0 is either eventually negative or eventually positive. Let *ψ* 0 (*l*) < 0 on [*l*1, ∞). By using Lemma 2, we have

$$\psi(\rho\_1(l)) \le \frac{\kappa(\rho\_1(l), \infty)}{\kappa(l, \infty)} \psi(l).$$

based on the fact that *ρ*1(*l*) ≤ *l*. Therefore,

$$\begin{aligned} \varphi(l) &= \begin{array}{rcl} \psi(l) - p\_1(l)y(\rho\_1(l)) - p\_2(l)y(\rho\_2(l)) \\ \geq & \psi(l) - p\_1(l)\psi(\rho\_1(l)) - p\_2(l)\psi(\rho\_2(l)) \\ \geq & \left(1 - p\_1(l)\frac{\kappa(\rho\_1(l),\infty)}{\kappa(l,\infty)} - p\_2(l)\right)\psi(l) \\ = & B\_1(l)\psi(l). \end{array} \end{aligned}$$

Therefore, (1) becomes

$$\left(r(l)\left(\psi'(l)\right)^{\gamma}\right)' \le -q(l)\mathcal{B}\_1^{\gamma}(\sigma(l))\psi^{\gamma}(\sigma(l)).\tag{10}$$

As *r*(*l*)(*ψ* 0 (*l*))*<sup>γ</sup>* <sup>0</sup> <sup>≤</sup> 0, we have

$$r(l) \left(\psi'(l)\right)^\gamma \le r(l\_1) \left(\psi'(l\_1)\right)^\gamma := -L < 0,\tag{11}$$

for all *l* ≥ *l*1, from (8) and (11), we have

$$
\psi^\gamma(l) \ge L\kappa^\gamma(l,\infty) \text{ for all } l \ge l\_1. \tag{12}
$$

Combining (10) with (12) yields

$$\left(r(l)\left(\psi'(l)\right)^{\gamma}\right)' \le -L\eta(l)\mathcal{B}\_1^{\gamma}(\sigma(l))\kappa^{\gamma}(\sigma(l),\infty),\tag{13}$$

for all *l* ≥ *l*1. Integrating (13) from *l*<sup>1</sup> to *l*, we obtain

$$\begin{split} r(l) \left(\boldsymbol{\upmu}^{\prime}(l)\right)^{\gamma} &\leq & r(l\_{1}) \left(\boldsymbol{\upmu}^{\prime}(l\_{1})\right)^{\gamma} - L \int\_{l\_{1}}^{l} \boldsymbol{q}(\boldsymbol{\updelta}) \boldsymbol{B}\_{1}^{\gamma}(\boldsymbol{\sigma}(\boldsymbol{\delta})) \boldsymbol{\kappa}^{\gamma}(\boldsymbol{\sigma}(\boldsymbol{\delta}), \boldsymbol{\infty}) \mathbf{d} \boldsymbol{\delta} \\ &\leq & -L \int\_{l\_{1}}^{l} \boldsymbol{q}(\boldsymbol{\delta}) \boldsymbol{B}\_{1}^{\gamma}(\boldsymbol{\sigma}(\boldsymbol{\delta})) \boldsymbol{\kappa}^{\gamma}(\boldsymbol{\sigma}(\boldsymbol{\delta}), \boldsymbol{\infty}) \mathbf{d} \boldsymbol{\delta}. \end{split}$$

Integrating the last inequality from *l*<sup>1</sup> to *l*, we get

$$\psi(l) \le \psi(l\_1) - L^{1/\gamma} \int\_{l\_1}^{l} \frac{1}{r^{1/\gamma}(\boldsymbol{\beta})} \left( \int\_{l\_1}^{\boldsymbol{\beta}} q(\boldsymbol{\delta}) B\_1^{\gamma}(\sigma(\boldsymbol{\delta})) \kappa^{\gamma}(\sigma(\boldsymbol{\delta}), \infty) d\boldsymbol{\delta} \right)^{1/\gamma} d\boldsymbol{\beta}.$$

At *l* → ∞, we get a contradiction with (9). Let *ψ* 0 (*l*) > 0 on [*l*1, ∞). From Lemma 2, we arrive at

$$
\psi(\rho\_2(l)) \le \frac{\kappa(l\_{1\prime}\rho\_2(l))}{\kappa(l\_{1\prime}l)}\psi(l). \tag{14}
$$

From the definition of *ψ*, we obtain

$$\begin{array}{rcl} y(l) &=& \psi(l) - p\_1(l)y(\rho\_1(l)) - p\_2(l)y(\rho\_2(l)) \\ &\ge & \psi(l) - p\_1(l)\psi(\rho\_1(l)) - p\_2(l)\psi(\rho\_2(l)). \end{array} \tag{15}$$

Using that (14) and *ψ*(*ρ*1(*l*)) ≤ *ψ*(*l*) where *ρ*1(*l*) < *l* in (15), we obtain

$$\begin{aligned} y(l) &\geq \quad \psi(l) \Big(1 - p\_1(l) - p\_2(l) \frac{\kappa(l\_1, \rho\_2(l))}{\kappa(l\_1, l)}\Big) \\ &\geq \quad \mathcal{B}\_2(l) \psi(l). \end{aligned} \tag{16}$$

Thus, (1) becomes

$$\left(r(l)\left(\psi'(l)\right)^{\gamma}\right)' \le -q(l)\mathcal{B}\_2^{\gamma}(\sigma(l))\psi^{\gamma}(\sigma(l)).\tag{17}$$

Now, from (9) and (C2), we have that R *<sup>l</sup> l*1 *q*(*ϑ*)*B γ* 1 (*σ*(*ϑ*))*κ γ* (*σ*(*ϑ*), ∞)d*ϑ* is unbounded. Therefore, as *κ* 0 (*l*, ∞) < 0, we obtain that

$$\int\_{l\_1}^{l} q(\theta) B\_1^{\gamma}(\sigma(\theta)) \mathrm{d}\theta \to \infty \text{ as } l \to \infty. \tag{18}$$

Integrating (17) from *l*<sup>2</sup> to *l*, we get

$$\begin{split} r(l) \left(\boldsymbol{\psi}'(l)\right)^{\gamma} &\leq \quad r(l\_{2}) \left(\boldsymbol{\psi}'(l\_{2})\right)^{\gamma} - \int\_{l\_{2}}^{l} q(\boldsymbol{\theta}) \boldsymbol{B}\_{2}^{\gamma}(\boldsymbol{\sigma}(\boldsymbol{\theta})) \boldsymbol{\psi}^{\gamma}(\boldsymbol{\sigma}(\boldsymbol{\theta})) \mathrm{d}\boldsymbol{\theta} \\ &\leq \quad r(l\_{2}) \left(\boldsymbol{\psi}'(l\_{2})\right)^{\gamma} - \boldsymbol{\psi}^{\gamma}(\boldsymbol{\sigma}(l\_{2})) \int\_{l\_{2}}^{l} q(\boldsymbol{\theta}) \boldsymbol{B}\_{2}^{\gamma}(\boldsymbol{\sigma}(\boldsymbol{\theta})) \mathrm{d}\boldsymbol{\theta}. \end{split}$$

As *B*2(*l*) > *B*1(*l*), we get

$$r(l)\left(\psi'(l)\right)^{\gamma} \le r(l\_2)\left(\psi'(l\_2)\right)^{\gamma} - \psi^{\gamma}(\sigma(l\_2))\int\_{l\_2}^{l} q(\theta)\mathcal{B}\_1^{\gamma}(\sigma(\theta))d\theta. \tag{19}$$

From (18) and (19), we get a contradiction with the positivity of *ψ* 0 (*l*). Therefore, the proof is complete.

**Theorem 2.** *Assume that B*2(*l*) ≥ *B*1(*l*) > 0*. If*

$$\limsup\_{l\to\infty} \mathbf{x}^{\gamma}(l,\infty) \int\_{l\_1}^{l} q(\theta) \mathcal{B}\_1^{\gamma}(\theta) \mathbf{d}\theta > 1,\tag{20}$$

*then, all solutions of (1) are oscillatory.*

**Proof.** Assume the contrary that Equation (1) has a positive solution *y* on [*l*0, ∞). Then, *y*(*ρ*1(*l*)), *y*(*ρ*2(*l*)) and *y*(*σ*(*l*)) are positive for all *l* ≥ *l*1, where *l*<sup>1</sup> is large enough. Thus, from (1) and the definition of *ψ*, we note that *ψ*(*l*) ≥ *y*(*l*) > 0 and *r*(*l*)(*ψ* 0 (*l*))*<sup>γ</sup>* is nonincreasing. Therefore, *ψ* 0 is either eventually negative or eventually positive. Let *ψ* 0 (*l*) < 0 on [*l*1, ∞). Integrating (10) from *l*<sup>1</sup> to *l*, we get

$$r(l)\left(\boldsymbol{\psi}'(l)\right)^{\gamma} \quad \leq \quad r(l\_1)\left(\boldsymbol{\psi}'(l\_1)\right)^{\gamma} - \int\_{l\_1}^{l} q(\boldsymbol{\theta}) \boldsymbol{\mathcal{B}}\_1^{\gamma}(\boldsymbol{\sigma}(\boldsymbol{\theta})) \boldsymbol{\uppsi}^{\gamma}(\boldsymbol{\sigma}(\boldsymbol{\theta})) \mathrm{d}\boldsymbol{\theta}$$

$$\leq \quad - \boldsymbol{\uppsi}^{\gamma}(\boldsymbol{\sigma}(l)) \int\_{l\_1}^{l} q(\boldsymbol{\theta}) \boldsymbol{\mathcal{B}}\_1^{\gamma}(\boldsymbol{\theta}) \mathrm{d}\boldsymbol{\theta}.\tag{21}$$

Using *ψ*(*σ*(*l*)) ≥ *ψ*(*l*) and (8) in (21), we obtain

$$-r(l)\left(\psi'(l)\right)^{\gamma} \ge -r(l)\left(\psi'(l)\right)^{\gamma}\kappa^{\gamma}(l,\infty)\int\_{l\_1}^{l} q(\boldsymbol{\theta})\mathcal{B}\_1^{\gamma}(\boldsymbol{\theta})d\boldsymbol{\theta}.\tag{22}$$

Divide both sides of inequality (22) by −*r*(*l*)(*ψ* 0 (*l*))*<sup>γ</sup>* and taking the limsup, we get

$$\limsup\_{l\to\infty} \kappa^{\gamma}(l,\infty) \int\_{l\_1}^{l} q(\theta) B\_1^{\gamma}(\theta) \mathbf{d}\theta \le 1.$$

Thus, we get a contradiction with (20).

Let *ψ* <sup>0</sup> > 0 on [*l*1, ∞). From (20) and the fact that *κ*(*l*, ∞) < ∞, we have that (18) holds. Then, this part of proof is similar to that of Theorem 1. Therefore, the proof is complete.

**Theorem 3.** *Assume that B*2(*l*) > 0, *B*1(*l*) > 0 *and r* 0 > 0*. If there exist positive functions µ*, *δ* ∈ *C* 1 ([*l*0, ∞)) *and l*<sup>1</sup> ∈ [*l*0, ∞) *such that*

$$\limsup\_{l\to\infty} \left\{ \frac{\kappa^{\gamma}(l,\infty)}{\delta(l)} \int\_{l\_1}^{l} \Big( \delta(\theta) q(\theta) \mathcal{B}\_1^{\gamma}(\sigma(\theta)) - \frac{r(\theta)}{(\gamma+1)^{\gamma+1}} \frac{(\delta'(\theta))^{\gamma+1}}{(\delta(\theta))^{\gamma}} \Big) \mathrm{d}\theta \right\} > 1 \tag{23}$$

*and*

$$\limsup\_{l\to\infty} \int\_{l\_1}^{l} \Big( \mu(\theta) q(\theta) B\_2^{\gamma}(\sigma(\theta)) - \frac{1}{(\gamma+1)^{\gamma+1}} \frac{r(\theta) (\mu'(\theta))^{\gamma+1}}{\mu^{\gamma}(\theta) (\sigma'(\theta))^{\gamma}} \Big) d\theta = \infty,\tag{24}$$

*then, all solutions of (1) are oscillatory.*

**Proof.** Assume the contrary that Equation (1) has a positive solution *y* on [*l*0, ∞). Then, *y*(*ρ*1(*l*)), *y*(*ρ*2(*l*)) and *y*(*σ*(*l*)) are positive for all *l* ≥ *l*1, where *l*<sup>1</sup> is large enough. Thus, from (1) and the definition of *ψ*, we note that *ψ*(*l*) ≥ *y*(*l*) > 0 and *r*(*l*)(*ψ* 0 (*l*))*<sup>γ</sup>* is nonincreasing. Therefore, *ψ* 0 is either eventually negative or eventually positive.

Let *ψ* <sup>0</sup> < 0 on [*l*1, ∞). As in proof of Theorem 1, we arrive at (10). Now, we define the function

$$
\omega(l) = \delta(l) \left( \frac{r(l)(\psi'(l))}{\psi^\gamma(l)}^\gamma + \frac{1}{\kappa^\gamma(l,\infty)} \right) \text{ on } [l\_1,\infty). \tag{25}
$$

From (8), we have that *ω* ≥ 0 on [*l*1, ∞). Differentiating (25), we get

$$\begin{split} \omega'(l) &= \quad \frac{\delta'(l)}{\delta(l)}\omega(l) + \delta(l) \frac{\left(r(l)(\Psi'(l))^{\gamma}\right)'}{\Psi^{\gamma}(l)} - \gamma\delta(l)r(l) \left(\frac{\Psi'(l)}{\Psi(l)}\right)^{\gamma+1} \\ &+ \frac{\gamma\delta(l)}{r^{1/\gamma}(l)\kappa^{\gamma+1}(l,\infty)} \\ &\leq \quad \frac{\delta'(l)}{\delta(l)}\omega(l) + \delta(l) \frac{\left(r(l)(\Psi'(l))^{\gamma}\right)'}{\Psi^{\gamma}(l)} - \frac{\gamma}{\left(\delta(l)r(l)\right)^{1/\gamma}} \left(\omega(l) - \frac{\delta(l)}{\kappa^{\gamma}(l,\infty)}\right)^{(\gamma+1)/\gamma} \\ &+ \frac{\gamma\delta(l)}{r^{1/\gamma}(l)\kappa^{\gamma+1}(l,\infty)}. \end{split} \tag{26}$$

Combining (10) and (26), we have

$$\begin{split} \omega'(l) &\leq \ -\frac{\gamma}{\left(\delta(l)r(l)\right)^{1/\gamma}} \left(\omega(l) - \frac{\delta(l)}{\kappa^{\gamma}(l,\infty)}\right)^{(\gamma+1)/\gamma} - \delta(l)q(l)\mathcal{B}\_1^{\gamma}(\sigma(l))\frac{\psi^{\gamma}(\sigma(l))}{\psi^{\gamma}(l)} \\ &+ \frac{\gamma\delta(l)}{r^{1/\gamma}(l)\kappa^{\gamma+1}(l,\infty)} + \frac{\delta'(l)}{\delta(l)}\omega(l). \end{split} \tag{27}$$

Using Lemma 1 with *A* := *δ* 0 (*l*)/*δ*(*l*), *B* := *γ*(*δ*(*l*)*r*(*l*))−1/*<sup>γ</sup>* , *C* := *δ*(*l*)/*κ γ* (*l*, ∞) and *ϑ* := *ω*, we get

$$\begin{split} \frac{\delta'(l)}{\delta(l)}\omega(l) - \frac{\gamma}{\left(\delta(l)r(l)\right)^{1/\gamma}} \Big(\omega(l) - \frac{\delta(l)}{\kappa^{\gamma}(l,\infty)}\Big)^{(\gamma+1)/\gamma} &\leq \frac{1}{\left(\gamma+1\right)^{\gamma+1}}r(l)\frac{\left(\delta'(l)\right)^{\gamma+1}}{\left(\delta(l)\right)^{\gamma}} \\ &+ \frac{\delta'(l)}{\kappa^{\gamma}(l,\infty)}. \end{split}$$

As *l* ≥ *σ*(*l*), we arrive at

$$
\psi(\sigma(l)) \ge \psi(l),
\tag{28}
$$

which, in view of (27) and (28), gives

$$\begin{split} \omega'(l) &\leq \frac{\delta'(l)}{\kappa^{\gamma}(l,\infty)} + \frac{1}{(\gamma+1)^{\gamma+1}} r(l) \frac{(\delta'(l))^{\gamma+1}}{(\delta(l))^{\gamma}} - \delta(l) q(l) B\_1^{\gamma}(\sigma(l)) \frac{\Psi^{\gamma}(\sigma(l))}{\Psi^{\gamma}(l)} \\ &+ \frac{\gamma \delta(l)}{r^{1/\gamma}(l) \kappa^{\gamma+1}(l,\infty)} \\ &\leq -\delta(l) q(l) B\_1^{\gamma}(\sigma(l)) + \left(\frac{\delta(l)}{\kappa^{\gamma}(l,\infty)}\right)' + \frac{r(l)}{(\gamma+1)^{\gamma+1}} \frac{(\delta'(l))^{\gamma+1}}{(\delta(l))^{\gamma}}. \end{split} \tag{29}$$

Integrating (29) from *l*<sup>2</sup> to *l*, we arrive at

$$\begin{split} \int\_{l\_2}^{l} \Big( \delta(\theta) q(\theta) \mathcal{B}\_1^{\gamma}(\sigma(\theta)) - \frac{r(\theta)}{\left(\gamma + 1\right)^{\gamma + 1}} \frac{\left(\delta'(\theta)\right)^{\gamma + 1}}{\left(\delta(\theta)\right)^{\gamma}} \Big) \mathrm{d}\theta & \leq \ \left(\frac{\delta(l)}{\kappa^{\gamma}(l, \infty)} - \omega(l)\right)\_{l\_2}^{l} \\ & \leq \ -\Big(\delta(l) \frac{r(l) \left(\psi'(l)\right)^{\gamma}}{\Psi^{\gamma}(l)}\Big)\_{l\_2}^{l}. \end{split} \tag{30}$$

From (8), we have

$$-\frac{r^{1/\gamma}(l)\psi'(l)}{\psi(l)} \le \frac{1}{\kappa(l,\infty)}\gamma$$

which, in view of (30), implies

$$\frac{\kappa^{\gamma}(l,\infty)}{\delta(l)} \int\_{l\_2}^{l} \Big( \delta(\theta) q(\theta) B\_1^{\gamma}(\sigma(\theta)) - \frac{r(\theta)}{(\gamma+1)^{\gamma+1}} \frac{(\delta'(\theta))^{\gamma+1}}{(\delta(\theta))^{\gamma}} \Big) \mathbf{d}\theta \le 1. \,\,\mu$$

Thus, we get a contradiction with (23).

Let *ψ* 0 (*l*) > 0 on [*l*1, ∞). As in proof of Theorem 1, we arrive at (17). Now, we define the function

$$\varphi(l) = \mu(l) \frac{r(l) (\psi'(l))^\gamma}{\psi^\gamma(\sigma(l))}.\tag{31}$$

Therefore, we have that *ω* ≥ 0 on [*l*1, ∞). Differentiating (31), we find

$$\varphi'(l) = \frac{\mu'(l)}{\mu(l)}\varphi(l) + \mu(l)\frac{\left(r(l)(\psi'(l))^\gamma\right)'}{\psi^\gamma(\sigma(l))} - \gamma\mu(l)r(l)\frac{(\psi'(l))^\gamma\psi'(\sigma(l))\sigma'(l)}{\psi^{\gamma+1}(\sigma(l))}.\tag{32}$$

Combining (17) and (32), we have

$$
\varphi'(l) \le \frac{\mu'(l)}{\mu(l)} \varphi(l) - \mu(l) q(l) \mathcal{B}\_2^\gamma(\sigma(l)) - \gamma \mu(l) r(l) \frac{(\psi'(l))^\gamma \psi'(\sigma(l)) \sigma'(l)}{\psi^{\gamma+1}(\sigma(l))}.
$$

As *r*(*l*)(*ψ* 0 (*l*))*<sup>γ</sup>* <sup>0</sup> <sup>&</sup>lt; 0 and *<sup>σ</sup>*(*l*) <sup>≤</sup> *<sup>l</sup>*, we arrive at

$$
\varrho'(l) \le \frac{\mu'(l)}{\mu(l)} \varrho(l) - \mu(l) q(l) B\_2^{\gamma}(\sigma(l)) - \gamma \mu(l) r(l) \sigma'(l) \frac{(\psi'(l))^{\gamma+1}}{\psi^{\gamma+1}(\sigma(l))}.
$$

From (31), we have

$$
\phi'(l) \le \frac{\mu'(l)}{\mu(l)} \varphi(l) - \mu(l) q(l) B\_2^{\gamma}(\sigma(l)) - \frac{\gamma \sigma'(l)}{\mu^{1/\gamma}(l) r^{1/\gamma}(l)} \phi^{(\gamma+1)/\gamma}(l).
$$

Using the inequality

$$Kv - Lv^{(\gamma+1)/\gamma} \le \frac{\gamma^{\gamma}}{(\gamma+1)^{\gamma+1}} \frac{K^{\gamma+1}}{L^{\gamma}}, L > 0,\tag{33}$$

with *K* = *µ* 0 (*l*)/*µ*(*l*), *L* = *γσ*0 (*l*)/*µ* 1/*γ* (*l*)*r* 1/*γ* (*l*) and *v* = *ϕ*, we have

$$
\varphi'(l) \le -\mu(l)q(l)\mathcal{B}\_2^{\gamma}(\sigma(l)) + \frac{1}{(\gamma+1)^{\gamma+1}} \frac{r(l)(\mu'(l))^{\gamma+1}}{\mu^{\gamma}(l)(\sigma'(l))^{\gamma}}.\tag{34}
$$

Integrating (34) from *l*<sup>2</sup> to *l*, we arrive at

$$\int\_{l\_2}^{l} \Big( \mu(\theta) q(\theta) B\_2^{\gamma}(\sigma(\theta)) - \frac{1}{(\gamma + 1)^{\gamma + 1}} \frac{r(\theta) (\mu'(\theta))^{\gamma + 1}}{\mu^{\gamma} (\theta) (\sigma'(\theta))^{\gamma}} \Big) \mathrm{d}\theta \le \varphi(l\_2).$$

Taking the lim sup on both sides of this inequality, we have a contradiction with (24). The proof of the theorem is complete.

**Example 1.** *Consider the second-order neutral differential equation*

$$\left(l^2 \left(\left(y(l) + p\_0 y\left(\frac{l}{\lambda}\right) + p\_\* y(\lambda l)\right)'\right)\right)' + q\_0 l y(\sigma\_0 l) = 0,\tag{35}$$

*where λ* > 1, *σ*<sup>0</sup> ∈ (0, 1) *and* (*λp*<sup>0</sup> + *p*∗) ∈ (0, 1). *We note that r*(*l*) = *l* 2 , *p*1(*l*) = *p*0, *p*2(*l*) = *p*∗, *ρ*1(*l*) = *l*/*λ*, *ρ*2(*l*) = *λl*, *q*(*l*) = *q*0*l and σ*(*l*) = *σ*0*l. It is easy to verify that*

$$B\_1(l) = 1 - \lambda p\_0 - p\_{\*\*}$$

*and*

$$B\_2(l) = 1 - p\_0 - p\_\* \left(\frac{l - \frac{1}{\tilde{\lambda}}}{l - 1}\right) / l$$

*and so B*<sup>2</sup> > *B*<sup>1</sup> > 0*. Now, we see that*

$$\begin{split} \limsup\_{l \to \infty} \int\_{l\_1}^{l} \frac{1}{r^{1/\gamma}(\boldsymbol{\beta})} \left( \int\_{l\_1}^{\boldsymbol{\beta}} q(\boldsymbol{\delta}) B\_1^{\gamma}(\sigma(\boldsymbol{\delta})) \kappa^{\gamma}(\sigma(\boldsymbol{\delta}), \infty) \mathbf{d} \boldsymbol{\delta} \right)^{1/\gamma} \mathbf{d} \boldsymbol{\beta} \\ = \limsup\_{l \to \infty} \int\_{l\_1}^{l} \frac{1}{\boldsymbol{\beta}^2} \left( \int\_{l\_1}^{\boldsymbol{\beta}} q\_0 \boldsymbol{\delta} (1 - \lambda p\_0 - p\_\*) \frac{1}{\sigma\_0 \boldsymbol{\delta}} \mathbf{d} \boldsymbol{\delta} \right) \mathbf{d} \boldsymbol{\beta} = \infty. \end{split}$$

*Then, by Theorem 1, we have that (35) is oscillatory.*

#### **3. Conclusions**

In this work, new criteria to test the oscillation of the solutions of second-order noncanonical neutral differential equations with mixed type were presented. These criteria are to further complement and simplify relevant results in the literature.

**Author Contributions:** Conceptualization, O.M. and Y.S.H.; methodology, H.A.; investigation, O.M. and A.N.; writing—original draft preparation, O.M., A.N. and Y.S.H.; writing—review and editing, A.N. and H.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** There is no external funding for this article.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This research was supported by Taif University Researchers Supporting Project Number (TURSP-2020/155), Taif University, Taif, Saudi Arabia.

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**


### *Article* **Generalized Attracting Horseshoe in the Rössler Attractor**

**Karthik Murthy <sup>1</sup> , Ian Jordan <sup>2</sup> , Parth Sojitra <sup>3</sup> , Aminur Rahman 4,\* and Denis Blackmore <sup>5</sup>**


**Abstract:** We show that there is a mildly nonlinear three-dimensional system of ordinary differential equations—realizable by a rather simple electronic circuit—capable of producing a generalized attracting horseshoe map. A system specifically designed to have a Poincaré section yielding the desired map is described, but not pursued due to its complexity, which makes the construction of a circuit realization exceedingly difficult. Instead, the generalized attracting horseshoe and its trapping region is obtained by using a carefully chosen Poincaré map of the Rössler attractor. Novel numerical techniques are employed to iterate the map of the trapping region to approximate the chaotic strange attractor contained in the generalized attracting horseshoe, and an electronic circuit is constructed to produce the map. Several potential applications of the idea of a generalized attracting horseshoe and a physical electronic circuit realization are proposed.

**Keywords:** generalized attracting horseshoe; strange attractors; poincaré map; electronic circuits

**Citation:** Murthy, K.; Jordan, I.; Sojitra, P.; Rahman, A.; Blackmore, D. Generalized Attracting Horseshoe in the Rössler Attracto. *Symmetry* **2021**, *13*, 30. https://dx.doi.org/10.3390/sym 13010030

Received: 8 December 2020 Accepted: 21 December 2020 Published: 27 December 2020

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/).

#### **1. Introduction**

The seminal work of Smale [1] showed that the existence of a *horseshoe* structure in the iterate space of a diffeomorphism is enough to prove it is chaotic. Often these diffeomorphisms arise from certain Poincaré maps of continuous-time *chaotic strange attractors* (CSA), which in turn are discrete-time CSAs. Some examples of such attractors are the Lorenz strange attractor [2], the Rössler attractor [3], and the double scroll attractor [4]. An example of a Poincaré map of the Lorenz equations is the Hénon map [5], which can be further simplified to the Lozi map [6]. Unsurprisingly, symmetry (and symmetry breaking) plays an important role in the analysis of these models.

In more recent years Joshi and Blackmore [7] developed an *attracting horseshoe* (AH) model for CSAs, which has two saddles and a sink. This, however, negates the possibility of the Hénon and Lozi maps, which have two saddles. Fortunately, the attracting horseshoe can be modified into a *generalized attracting horseshoe* (GAH), which can have either one or two saddles while still being an attracting horseshoe [8]. This results in a quadrilateral trapping region. While extensive analysis was done in Joshi et al. [8], a simple concrete example seemed to be illusive.

In this investigation, we implement novel numerical techniques to find the necessary Poincaré map of the Rössler attractor that would admit a quadrilateral trapping region. This trapping region represents a region of rotational symmetry as every iterate originating in the trapping region will return after a 2*π* rotation. The trapping region would also filter out any flow that does not obey this symmetry. An electronic circuit could use these properties to isolate signals of interest. Similar to the experiments of Rahman et al. [9], we design a physical realization of the dynamical system in the form of an electronic circuit.

The remainder of the paper is organized as follows. In Section 2, we give an overview of the algorithm with the MATLAB codes included in the Supplementary Materials. Once we have the tools for our numerical experiments, we first propose a carefully constructed GAH model in Section 3. Then, we give numerical examples of Poincaré maps of the Rössler attractor and the map of interest in Section 4 and real-world examples in Section 5. Finally, we end discussions in Section 6 with some concluding remarks.

#### **2. Poincaré Map Algorithm**

To produce a general Poincaré section of a flow, we break up the program into four parts: solving the ODE; computing a Poincaré section perpendicular to either *x* " 0, *y* " 0, or *z* " 0; rotating the Poincaré section; and iterating the Poincaré map. Solving the ODE is standard through ODE45 on MATLAB, which executes a modified Runge–Kutta scheme. Once we have our solution matrix, we need to approximate the values of first return maps from the discretized flow. By restricting the first return onto a Poincaré section, the iterate space of the Poincaré map can be visualized. This is easily done for a section perpendicular to the axes, but in order to locate a highly specialized object, such as the GAH, we need to be able to rotate the section. Once the desired section is found we can experiment on iterating the points of trapping region candidates.

The initial major task is approximating the first return map on a Poincaré section of a flow. Much of the ideas of our initial first return map code came from that of Gonze [10]. Once the discretized flow is found numerically a planar section for a certain value of *x*, *y*, or *z* can be defined, which in general will lie between pairs of simultaneous points. Then we may draw a line between the pair through the planar section and identify the intersecting point, which approximates a point of the first return map. This can also be done with more simultaneous points in order to get higher order approximations.

Once we can approximate a map for a section perpendicular to the axes we need to have the ability to rotate and move the map to any position. This is where our program completely diverges from that in [10]. While the first instinct might be to try to rotate the section, it is equivalent to rotate the flow in the opposite direction to the desired rotation of the section. Once the flow is rotated, the code for the first return map can be readily used. This gives us the ability to analyze the first return map of a general Poincaré section.

Finally, we would like to not only compute a first return map, but also compute the iterates of a Poincaré map of any system; that is, given an initial condition on an arbitrary Poincaré section can we find the subsequent iterates. To accomplish this, we solve the ODE for a given initial condition on the planar section to find the first return. Once we have the first return, we record its location and use that as the new initial condition. This iterates the map for as many returns as desired, thereby filling in a Poincaré map. Now we have the tools needed to run numerical experiments on GAHs.

#### **3. A Constructed GAH System**

In this section, we give a brief description of the generalized attracting horseshoe (GAH) map and devise a three-dimensional nonlinear ordinary differential equation with a Poincaré section that produces it.

#### *3.1. The GAH Map*

The GAH is a modification of the AH that can be represented as a geometric paradigm with either just one or two fixed points, both of which are saddles. Figure 1 shows a rendering of a *C* <sup>1</sup> GAH with two saddle points, which can be constructed as follows. The rectangle is first contracted vertically by a factor 0 ă *λ<sup>v</sup>* ă 1{2, then expanded horizontally by a factor 1 ă *λ<sup>h</sup>* ă 2, and finally folded back into the usual horseshoe shape in such a manner that the total height and width of the horseshoe do not exceed the height and width, respectively, of the trapping rectangle *Q*. Then, the horseshoe is translated horizontally so that it is completely contained in *Q*. Obviously, the map *f* defined by this construction is a smooth diffeomorphism. Clearly, there are also many other ways

to obtain this geometrical configuration. For example, the map *f* as described above is orientation-preserving, and an orientation-reversing variant can be obtained by composing it with a reflection in the horizontal axis of symmetry of the rectangle, or by composing it with a reflection in the vertical axis of symmetry followed by a composition with a half-turn. Another construction method is to use the standard Smale horseshoe that starts with a rectangle, followed by a horizontal composition with just the right scale factor or factors to move the image of *Q* into *Q*, while preserving the expansion and contraction of the horseshoe along its length and width, respectively.

It is important to note that subrectangle *S* with its left vertical edge through *p*, which contains the arch of the horseshoe and the keystone region *K*, plays a key role in the dynamics of the iterates of *f* . In particular, we require that the map satisfy the following additional property, which is illustrated in Figure 2:

(‹) *f maps the keystone region K* p*containing a portion of the arch of the horseshoe*q *to the left of the fixed point p and the portion of its corresponding stable manifold W<sup>s</sup>* p*p*q *containing p and contained in f*p*Q*q.

The definition above and (‹) can be shown to lead to the conclusion that

$$\mathfrak{A} = \overline{\mathcal{W}^{\mu}(p)} = \bigcap\_{n=1}^{\infty} f^n(Q)\_n$$

where *W<sup>u</sup>* p*p*q is the unstable manifold of *p*, a global chaotic strange attractor (CSA).

**Figure 1.** A planar GAH with two saddle points.

The map above can be considered to be the paradigm for a GAH, but there are many analogs. In fact, let *<sup>F</sup>* : *<sup>Q</sup>*˜ <sup>Ñ</sup> *<sup>Q</sup>*˜ be any smooth diffeomorphism of a quadrilateral trapping region *Q*˜ possessing a horseshoe-like image with a keystone region *K*˜ containing a portion of the arch of *<sup>F</sup>*p*Q*˜<sup>q</sup> analogous to that shown in Figure 1. Suppose that the map is expanding by a scale factor uniformly greater than one along the length of the horseshoe and contracting transverse to it by a scale factor uniformly less than one-half in the complement of a subset of *Q*˜ containing *K*˜. Then, if *F* satisfies an additional property analogous to (‹), it maps *K*˜ into an open subset of *Q*˜ to the left of the saddle point *p*˜, and

$$\mathfrak{A} = \overline{\mathcal{W}^{\mu}(\tilde{p})} = \bigcap\_{n=1}^{\infty} F^{n}(\tilde{Q})$$

is a global CSA.

**Figure 2.** Local (transverse) horseshoe structure of *f* <sup>2</sup> near *p*.

#### *3.2. A GAH Producing System*

We now construct an ODE in R<sup>3</sup> with a Poincaré section that is a GAH. The transversal we use is the following square in the *xz*-plane defined in Cartesian and polar coordinates

$$Q\_0 = \{(x, y, z) : 0.05 < x < 1.05, y = 0, -0.5 < z < 0.5\}$$

$$= \{(r, \theta, z) : 0.05 < r < 1.05, \theta = 0, -0.5 < z < 0.5\}.\tag{1}$$

The trick is to find a relatively simple (necessarily nonlinear) *C* <sup>1</sup> ODE having *Q*<sup>0</sup> as a transversal with an induced Poincaré first-return map *P* : *Q*<sup>0</sup> Ñ *Q*<sup>0</sup> Ą *Q*2*<sup>π</sup>* " *P*p*Q*0q such that *P*p*Q*0q Ă int*Q*<sup>0</sup> is a GAH. We chose the ODE based upon a rotation about the *z*-axis so that the square evolves into the GAH as *Q*<sup>0</sup> makes a full rotation. The first half of the metamorphosis takes care of the vertical squeezing and horizontal stretching, while the second half produces the folding. It is not difficult to show that the system (in cylindrical coordinates)

$$\dot{r} = \frac{2\log(1.2)\sin^2\theta}{\pi}r,\ \dot{\theta} = 1,\ \dot{z} = \frac{2\log(0.2)\sin^2\theta}{\pi}(z+0.2) \tag{2}$$

flows *Q*<sup>0</sup> to

$$Q\_{\pi} = \{(\mathbf{x}, y, z) : -1.26 \prec x \leqslant -0.06, y = 0, -0.26 \ll z \leqslant -0.06\} \tag{3}$$

which is the original square in the radial half-plane corresponding to *θ* " 0 stretched by a factor of 1.2 along the *x*-axis and squeezed by a factor of 1/5 with respect to *z* " ´0.2 along the *z*-axis in the radial half plane corresponding to *θ* " *π*. Consequently, (2) produces the first half of the desired result comprising the stretching and squeezing for 0 ď *θ* ď *π*.

Note that (2) can be integrated directly to obtain the following for 0 ď *θ* ď *π* and initial condition p*r*p0q, *θ*p0q, *z*p0qq " p*r*0, *θ*0, *z*0q :

$$\begin{aligned} r(t) &= r(\theta) = r\_0 \exp\left[\frac{\log(1.2)}{2\pi}(2\theta - \sin 2\theta)\right], \\ \theta(t) &= t, \\ z(t) &= z(\theta) = -0.2 + (z\_0 + 0.2) \exp\left[\frac{\log(0.2)}{2\pi}(2\theta - \sin 2\theta)\right]. \end{aligned} \tag{4}$$

Now, we have to attend to the folding for *π* ď *θ* ď 2*π*. For this we use a rotation in planes orthogonal to a fixed circle in the *xy*-plane. In these planes corresponding to a circle of radius *c*, given as *c* " 0.66, we define Euclidean coordinates with origin *r* " 0.66, *z* " 0 and corresponding polar coordinates p*ρ*, *φ*q as

$$
\rho = \sqrt{(r - 0.66)^2 + z^2} = \sqrt{\vec{r}^2 + z^2},
\tag{5}
$$

where *r*˜ " *r* ´ 0.66 " *ρ* cos *φ* and *z* " *ρ* sin *φ*. Then, when *π* ď *θ* ď 2*π*, we take the folding part for *φ* ě ´*π*{2 to be

$$\dot{\vec{r}} = \dot{r} = -2\sin^2\theta\,\rho\sin\phi,\ \dot{\theta} = 1,\ \dot{z} = 2\sin^2\theta\,\rho\cos\phi.\tag{6}$$

or equivalently

$$\dot{\vec{r}} = \dot{r} = -2\sin^2\theta \, z, \,\dot{\theta} = 1, \,\dot{z} = 2\sin^2\theta \, \vec{r}. \tag{7}$$

It is easy to verify from the above that *ρ* is constant (call it *ρ*0) for the solutions of (6) or (7) and that the solution initially (at *t* " *θ* " *π*) satisfying p*ρ*, *φ*q " p*ρ*0, *φ*0q is

$$\vec{r} = \vec{r}(t) = \rho\_0 \cos(\phi(t) + \phi\_0), \ \theta = \theta(t) = t, \ z = z(t) = \rho\_0 \sin(\phi(t) + \phi\_0), \tag{8}$$

where

$$
\phi(t) = (t - \pi) - \sin t \cos t. \tag{9}
$$

The above ((6) or (7)) describes the folding field for *π* ď *θ* ď 2*π* and ´*π*{2 ď *φ*. In order to smoothly fill in the rest of the field, we shall use the function

$$\psi(\vec{r}) = \begin{cases} 0, & \vec{r} \ll -0.6\\ \sin^2\left[\frac{\pi}{1.2}(\tilde{r} + 0.6)\right], & -0.6 \le \tilde{r} \ll 0 \end{cases} \tag{10}$$

which can be recast as

$$\zeta(r) = \begin{cases} 0, & r \ll 0.06\\ \sin^2\left[\frac{\pi}{1.2}(r - 0.06)\right], & 0.06 \ll r \ll 0.66 \end{cases}.\tag{11}$$

We have now assembled all the elements for defining an ODE that generates a GAH Poincaré section. This ODE, which incorporates (2) and (7) and is *π*-periodic in *θ*, has the following form:

$$
\dot{\tau} = \mathcal{R}(r, \theta, z), \; \dot{\theta} = 1, \; \dot{z} = Z(r, \theta, z), \tag{12}
$$

subject to the initial condition

$$(r(0), \theta(0), z(0)) = (r\_0, 0, z\_0) \in Q\_{0\prime} \tag{13}$$

where

$$\mathbf{R} = \begin{cases} \frac{\log(12)\sigma(\theta)\tau}{\pi}, & 0 \leqslant \theta \leqslant \pi \\ -\sigma(\theta)z, & (\pi \leqslant \theta \leqslant 2\pi) \text{ and } (((r > 0.66)\,\text{or}(z \geqslant 0)) = (-\pi/2 \leqslant \phi \leqslant \pi)) \\ -\mathfrak{z}(r)\sigma(\theta)z, & (\pi \leqslant \theta \leqslant 2\pi) \text{ and } ((r < 0.66)\,\text{and}(z \in [-0.26, -0.06])) \\ 0, & (\pi \leqslant \theta \leqslant 2\pi) \text{ and } ((r < 0.66)\,\text{and}(z < 0)\,\text{and}(z \notin [-0.26, -0.06])) \end{cases}$$

$$\mathbf{Z} = \begin{cases} \frac{(\log(2)\sigma(\theta)\lceil z \leqslant 0.2\pi)}{\pi}, & 0 \leqslant \theta \leqslant \pi \\ \sigma(\theta)\overline{\mathbf{r}}, & (\pi \leqslant \theta \leqslant 2\pi) \text{ and } (((r > 0.66)\,\text{or}(z \geqslant 0)) = (-\pi/2 \leqslant \phi \leqslant \pi)) \\ 0, & (\pi \leqslant \theta \leqslant 2\pi) \text{ and } (((r < 0.66)\,\text{and}(z < 0)) = (-\pi \leqslant \phi \leqslant -\pi/2)) \end{cases}$$

and

$$
\sigma(\theta) = 2\sin^2\theta = 1 - \cos 2\theta.
$$

Finally, it is not difficult to show that the Poincaré section of the transversal (and trapping region) *Q*<sup>0</sup> under the system (12) is a GAH with an image that is simply a symmetric reflection about the vertical axis of the horseshoe in Figure 1. However, it appears that the construction of an electronic circuit simulating (12) would be a rather formidable undertaking, so we selected a simpler system, namely, the Rössler attractor model, which is a mildly nonlinear three-dimensional ODE that has a straightforward circuit realization.

#### **4. Poincaré Maps and Circuit Realization of the Rössler Attractor**

We consider the Rössler attractor

$$\begin{aligned} \dot{\mathbf{x}} &= \mathbf{y} - z \\ \dot{\mathbf{y}} &= \mathbf{x} + a\mathbf{y} \\ \dot{z} &= b + z(\mathbf{x} - c); \end{aligned} \tag{14}$$

,

where we use the parameters *a* " 0.2, *b* " 0.1, and *c* " 10. This produces the chaotic strange attractor in Figure 3, and it can also be realized by a rather simple electronic circuit.

**Figure 3.** The Rössler attractor with parameters *a* " 0.2, *b* " 0.1, and *c* " 10, and a rotation (represented by *x*ˆ and *y*ˆ) of *θ* " 2*π*{5 in spherical coordinates.

#### *4.1. The Poincaré Map*

One can use the algorithm in Section 2 to compute any Poincaré section of the attractor; however, what we are particularly interested in is identifying a trapping region for a generalized attracting horseshoe. Assuming the system contains a GAH, we first look for a Poincaré section with a horseshoe-like structure as shown in Figure 4.

**Figure 4.** Poincaré section (*r* " 5, *θ* " 2*π*{5) of the Rössler attractor containing a horseshoe-like structure. Plot is shown in the rotated frame.

Now, if we can find a rotationally symmetric trapping region around this horseshoe, we shall have shown evidence for the existence of a GAH. First, we identify vertices of a quadrilateral that fully encompasses the horseshoe-like structure. Then, using a recursive algorithm (described in Section 2) we compute the first return map of those vertices on that particular Poincaré section, i.e., the first iteration of the Poincaré map of those points. If the iterates are contained within that quadrilateral, the points on the quadrilateral itself can be tested. In Figure 5, four-thousand points on the quadrilateral are iterated, and it is illustrated that this first return is completely contained in the quadrilateral. While this is not a proof, the grid spacing on the quadrilateral provides compelling evidence that this is a trapping region for the GAH.

In order to provide more compelling evidence, we compute higher-order iterations of the Poincaré map in Figure 6.

**Figure 5.** The first return (blue markers) of the quadrilateral trapping region (red markers) with vertices located at p*x*ˆ, *y*ˆq " p´3.55, ´27q, p11.91, ´6.6q, p12, 0q, p´8.5, 3.5q. While the quadrilateral edges look "continuous", it should be noted that it is in fact discretized using four thousand points, which are then mapped back to the Poincaré section (*r* " 5, *θ* " 2*π*{5). Plot is shown in the rotated frame with *x*ˆ and *y*ˆ denoting rotated axes.

**Figure 6.** First five iterations of the Poincaré map (blue markers) of the quadrilateral trapping region (red markers) with vertices located at p*x*ˆ, *y*ˆq " p´3.55, ´27q, p11.91, ´6.6q, p12, 0q, p´8.5, 3.5q. While the quadrilateral edges look "continuous", it should be noted that it is in fact discretized using four thousand points, which are then mapped back to the Poincaré section (*r* " 5, *θ* " 2*π*{5). Plot is shown in the rotated frame with *x*ˆ and *y*ˆ denoting rotated axes.

#### *4.2. Circuit Realization of the Rössler System*

It happens that there are several known examples of electronic circuits realizing the Rössler attractor system. We chose the one, obtained from [11], shown in Figure 7 with a list of components in Table 1.

**Figure 7.** Multisim circuit diagram for Röossler attractor.

The physical realization of the Rössler attractor circuit was constructed using summing amplifiers, integrators, and a multiplier. Due to the nature of this system, the operational amplifier must operate within ˘15 volts in order to avoid clipping of the Rössler Attractor output waveform. In this circuit, resistors were used to represent constant values for parameters *a* and *b* in (14). A potentiometer was used to vary the parameter value of b in order to observe the bifurcations of the physical system. We first test the circuit on *Multisim* and observe the aforementioned bifurcations in Figure 8.

**Figure 8.** Multisim outputs of the Rössler attractor showing a period doubling Hopf bifurcation leading to chaos.


**Table 1.** List of components for the Röossler attractor circuit.

Next, we built the circuit and observed oscilloscope outputs as shown in Figure 9. The Poincaré section that we chose was a particular vertical plane through the top arch of the output shown (see also Figure 3). The acceptable planes were obtained by trial and

error via varying the system parameters and rotation of the plane about a vertical axis through the apex of the arc.

**Figure 9.** Oscilloscope output from Rössler attractor circuit.

#### **5. Potential Applications**

One can imagine several practical applications of devices containing electronic circuit realizations of a GAH. Two, which are related to communications and intelligence gathering, immediately come to mind: First, the circuit could be embedded in a communication receiving device, and tuned to certain "static" frequencies different from those in the expected incoming messages. The strong global attracting characteristics of the circuit would separate the static from the incoming messages, thereby enhancing the receiving capabilities of system. In effect, the GAH circuit would filter out the static.

Second, a stationary or compact mobile device incorporating the GAH circuit could be used to penetrate and analyze various communication systems. Either by connecting remotely in the case of a stationary device or directly for a mobile version, the global attracting properties could be employed to extract crucial characteristics of the system to which it is connected. Moreover, the same attracting features of the GAH circuit device could be used to absorb various parts of sent messages that would render them useless, false, or simply misleading.

The two rather basic applications mentioned provide just a glimpse of the possible applications of GAH circuits, most of which would probably be related to information systems, data collection, and filtering. Moreover, there are more applications that could exploit the chaotic strange attractor associated with a GAH circuit. For, example a GAH circuit device could be used either to control chaos, introduce chaos or adjust the fractal dimension of outputs of a variety of applicable processes based on dynamical systems.

Such mechanisms may aid in a variety of fields including cryptography and cyber security. While encryption techniques reliant on the iterate by iterate behavior of chaotic maps have had their short comings due to irreversibly in analog form and a lack of proper security when implemented in software, the global properties of a GAH circuit may provide a useful intermediate stage during various forms of symmetric encryption by allowing a signal of importance to pass both through and around such a subsystem in parallel. Long-term global properties from the signal sent through a GAH circuit can be extracted and used to manipulate the unaltered signal before reaching the recipient, thereby substantially increasing the difficulty of decryption. As the global properties of such a system (i.e., factors taken from its geometry and macro-scale structure) are less influenced by minor imperfections in the circuit, reversing the process becomes far more tangible.

Furthermore, adding to our second practical application claim, a GAH device could prove to have various applications in machine learning, specifically regarding the creation and prevention of adversarial attacks on deep neural networks. By altering carefully chosen properties of the data meant to be received by the network intelligible yet incorrect results can be forced. In the same light, the proper extraction of dominant incoming signals may help prevent the same sort of issue in special cases.

#### **6. Conclusions**

We constructed a rather complicated nonlinear three-dimensional ordinary differential equation (ODE) having a Poincaré section that is a GAH map, but it is not particularly amenable to electronic circuit realization, which was a goal of the investigation. Therefore, instead of the initial ODE, we selected the Rössler attractor; a mildly nonlinear threedimensional ODE that has a reasonably simple circuit realization and can actually produce GAH maps for carefully chosen Poincaré sections. We constructed the corresponding GAH circuit and used a novel iteration procedure to generate good approximations of the chaotic strange attractors associated to the GAH maps. Finally, in addition to the experimental and analytic aspects of our investigation, we discussed a number of potential practical applications of the GAH circuit. Most of the envisioned applications were in the realms of communication and information gathering.

**Supplementary Materials:** The following are available at https://www.mdpi.com/2073-8994/13/1 /30/s1.

**Author Contributions:** Conceptualization, A.R. and D.B.; Formal analysis, K.M.; Investigation, K.M., I.J., P.S. and A.R.; Methodology, A.R.; Project administration, A.R.; Supervision, A.R. and D.B.; Validation, I.J.; Visualization, K.M.; Writing—original draft, A.R. and D.B.; Writing—review & editing, I.J., A.R. and D.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** All data and computer programs that were developed are included as Supplementary materials.

**Acknowledgments:** The authors would like to thank the NJIT Provost Research Grants for funding this research. K.M. was funded by the Provost high school internship, and P.S. was funded by Phase-1 Provost Undergraduate Research Grant, with D.B. as faculty mentor and A.R. as graduate student mentor. K.M. appreciates the support of Bridgewater-Raritan High School, P.S. and I.J. appreciate the support of the Electrical and Computer Engineering Department at NJIT, and A.R. and D.B. appreciate the support of the Department of Mathematical Sciences and the Center for Applied Mathematics and Statistics (CAMS) at NJIT. In addition, K.M., I.J., and A.R. would like to acknowledge their current institutions: K.M. appreciates the support of the Department of Computer Science at UI-UC, I.J. appreciates the support of the Department of Applied Mathematics and Statistics at SU, and A.R. appreciates the support of the Department of Applied Mathematics at UW. Finally, the authors would like to give their sincere thanks to the reviewers for their insightful comments that helped improve the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Multiple Solutions for a Class of Nonlinear Fourth-Order Boundary Value Problems**

#### **Longfei Lin , Yansheng Liu \* and Daliang Zhao**

School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China; 2019020438@stu.sdnu.edu.cn (L.F.); dlzhao928@sdnu.edu.cn (D.Z.)

**\*** Correspondence: ysliu@sdnu.edu.cn

Received: 14 November 2020; Accepted: 29 November 2020; Published: 2 December 2020

**Abstract:** This paper is concerned with multiple solutions for a class of nonlinear fourth-order boundary value problems with parameters. By constructing a special cone and applying fixed point index theory, the multiple solutions for the considered systems are obtained under some suitable assumptions. The main feature of obtained solutions (*u*(*t*), *v*(*t*)) is that the solution *u*(*t*) is positive, and the other solution *v*(*t*) may change sign. Finally, two examples with continuous function *f*<sup>1</sup> being positive and *f*<sup>2</sup> being semipositone are worked out to illustrate the main results.

**Keywords:** multiple solutions; fixed point theory; boundary value problems

#### **1. Introduction**

It is well known that the subject of the existence of solutions to numerous boundary value problems (BVP) for differential equations such as second-order [1–3], fourth-order [4–6], even fractional order BVP [7–11] has gained considerable attention and popularity. A growing number of outstanding progress has been made in the theory of such BVP in the last decades due mainly to their extensive applications in the fields of hydrodynamics, nuclear physics, biomathematics, chemistry, and control theory. For further details, please see References [12–29] and references therein.

It is noted that fourth-order boundary value problems have an important application in practical problems, that is, they can be used to describe the deformation of elastic beam, see References [30–33] and references therein. For example, in Reference [32], by means of the theory of fixed point index on cone, Y. Li investigated the following boundary value problems of fourth-order ordinary differential equation

$$\begin{cases} \ u^{(4)}(t) + \beta u^{\prime\prime}(t) - \omega u(t) = f(t, u), & 0 < t < 1; \\ u(0) = u(1) = u^{\prime\prime}(0) = u^{\prime\prime}(1) = 0, \end{cases}$$

where *f* ∈ *C*([0, 1] × R <sup>+</sup>, R <sup>+</sup>), *<sup>α</sup>*, *<sup>β</sup>* <sup>∈</sup> <sup>R</sup> and satisfy *<sup>β</sup>* <sup>&</sup>lt; <sup>2</sup>*<sup>π</sup>* 2 , *α* ≥ −*β* <sup>4</sup>/4, *α*/*π* <sup>4</sup> + *β*/*π* <sup>2</sup> < 1. By constructing a special cone, the existence of at least one positive solution was obtained under some suitable assumptions.

Recently, in Reference [33], Q. Wang and L. Yang studied the following boundary value problems

$$\begin{cases} \boldsymbol{u}^{(4)}(t) + \beta\_1 \boldsymbol{u}^{\prime\prime}(t) - \boldsymbol{u}\_1 \boldsymbol{u}(t) = f\_1(t, \boldsymbol{u}(t), \boldsymbol{v}(t)), & 0 < t < 1; \\ \boldsymbol{v}^{(4)}(t) + \beta\_2 \boldsymbol{v}^{\prime\prime}(t) - \boldsymbol{u}\_2 \boldsymbol{v}(t) = f\_2(t, \boldsymbol{u}(t), \boldsymbol{v}(t)), & 0 < t < 1; \\ \boldsymbol{u}(0) = \boldsymbol{u}(1) = \boldsymbol{u}^{\prime\prime}(0) = \boldsymbol{u}^{\prime\prime}(1) = 0; \\ \boldsymbol{v}(0) = \boldsymbol{v}(1) = \boldsymbol{v}^{\prime\prime}(0) = \boldsymbol{v}^{\prime\prime}(1) = 0, \end{cases} \tag{1}$$

where *f*1, *f*<sup>2</sup> ∈ *C*([0, 1] × R <sup>+</sup> <sup>×</sup> <sup>R</sup> <sup>+</sup>, R <sup>+</sup>), and *β<sup>i</sup>* , *α<sup>i</sup>* ∈ R(*i* = 1, 2) satisfy the following conditions:

$$
\beta\_{\rm i} < 2\pi^2, \ -\beta\_{\rm i}/4 \le a\_{\rm i}, a\_{\rm i}/\pi^4 + \beta\_{\rm i}/\pi^2 < 1. \tag{2}
$$

These conditions involve a two-parameter non-resonance condition. By constructing two classes of cones and using the fixed point theory, the existence of at least one positive solution was obtained. It is remarkable that the premise of this establishment of the result in Reference [33] is that the nonlinear term *f*<sup>2</sup> must be positive.

We point out that there are some limitations in those existing results of fourth-order boundary value problems. All solutions obtained in the above references are positive, and moreover, the corresponding conclusions in them are not valid when the nonlinear term is allowed to be non-positive. Considering that two variables *u* and *v* in the nonlinear term usually have some connections in many practical problems, there is no description of the relationship between them in the aforementioned papers. It is an interesting problem to seek such solutions for BVP (1) that one variable is positive and the other may be non-positive under the assumptions that nonlinearity may be semipositoned, and some connection will be added between these two variables. As far as we know, there is no paper considering such problem for BVP (1). The purpose of the present paper is to fill this gap.

This paper, motivated by all the above mentioned discussions, investigates the multiple solutions for BVP (1) under the more different conditions compared with Reference [33]. By constructing a very special cone and using the fixed point index theory, the existence and multiplicity results of solutions to (1) are obtained when *β<sup>i</sup>* , *α<sup>i</sup>* ∈ R (*i* = 1, 2) satisfy the conditions (2), *f*<sup>1</sup> ∈ *C*([0, 1] × R <sup>+</sup> <sup>×</sup> <sup>R</sup>, <sup>R</sup> <sup>+</sup>), and *f*<sup>2</sup> ∈ *C*([0, 1] × R <sup>+</sup> <sup>×</sup> <sup>R</sup>, <sup>R</sup>).

The nonlinear term *f*<sup>2</sup> is allowed to change sign by contrast, *f*<sup>2</sup> ∈ *C*([0, 1] × R <sup>+</sup> <sup>×</sup> <sup>R</sup>, <sup>R</sup>). A relationship is imposed between two variables *u*, *v* in nonlinear terms, which is that the variable *v* is controlled by *u*. In obtained solution (*u*, *v*), the component *u* is positive, but the component *v* is allowed to be negative in comparison with Reference [33].

The rest of this paper is organized as follows—Section 2 contains some background materials and preliminaries. The main results will be given and proved in Section 3. Finally, in Section 4, two examples are given to support our results.

#### **2. Background Materials and Preliminaries**

The basic space used in this paper is *E* := *C*[0, 1] × *C*[0, 1]. It is a Banach space endowed with the norm k(*u*, *v*)k = max{k*u*k, k*v*k} for (*u*, *v*) ∈ *E*, where k*u*k = max *t*∈[0,1] |*u*(*t*)|, k*v*k = max *t*∈[0,1] |*v*(*t*)|. Under the condition (2), as in Reference [32], let

$$\mathfrak{f}\_{i,1} = \frac{-\beta\_i + \sqrt{\beta\_i^2 + 4\alpha\_i}}{2}, \\ \mathfrak{f}\_{i,2} = \frac{-\beta\_i - \sqrt{\beta\_i^2 + 4\alpha\_i}}{2}, \\ (i = 1, 2), $$

and let *Gi*,*j*(*t*,*s*)(*i*, *j* = 1, 2) be the Green's function of the linear boundary value problem

$$\begin{cases} -\mathfrak{u}\_{i}^{\prime\prime}(t) + \mathfrak{f}\_{i,j}\mathfrak{u}\_{i}(t) = 0, & 0 < t < 1; \\\ \mathfrak{u}\_{i}(0) = \mathfrak{u}\_{i}(1) = 0, & i, j = 1, 2. \end{cases}$$

Then for *h<sup>i</sup>* ∈ *C*[0, 1], the solution of the following nonlinear boundary value problem

$$\begin{cases} \boldsymbol{u}\_{i}^{(4)}(t) + \beta\_{i} \boldsymbol{u}\_{i}^{\prime\prime}(t) - \boldsymbol{a}\_{i} \boldsymbol{u}\_{i} = \boldsymbol{h}\_{i}(t), & 0 < t < 1; \\\ \boldsymbol{u}\_{i}(0) = \boldsymbol{u}\_{i}(1) = \boldsymbol{u}\_{i}^{\prime\prime}(0) = \boldsymbol{u}\_{i}^{\prime\prime}(1) = 0, & \boldsymbol{i}\_{i} \boldsymbol{j} = 1, 2 \end{cases}$$

can be expressed as

$$u\_i(t) = \int\_0^1 \int\_0^1 G\_{i,1}(t,\tau) G\_{i,2}(\tau,s) h\_i(s) ds d\tau, \ t \in [0,1].$$

**Lemma 1.** *The function Gi*,*j*(*t*,*s*)(*i* = 1, 2) *has the following properties: (1) Gi*,*j*(*t*,*s*) > 0 *for t*,*s* ∈ (0, 1)*; (2) Gi*,*j*(*t*,*s*) ≤ *Ci*,*jGi*,*j*(*s*,*s*) *for t*,*s* ∈ [0, 1]*, where Ci*,*<sup>j</sup>* > 0 *is a constant; (3) Gi*,*j*(*t*,*s*) ≥ *δi*,*jGi*,*j*(*t*, *t*)*Gi*,*j*(*s*,*s*) *for t*,*s* ∈ [0, 1]*, where δi*,*<sup>j</sup>* > 0 *is a constant; (4) G*2,*j*(*t*,*s*) ≤ *NjG*1,*j*(*t*,*s*) *for t*,*s* ∈ [0, 1]*, where N<sup>j</sup>* > 0 *is a constant.*

**Proof of Lemma 1.** (1)–(3) can be seen from Reference [32]. In addition, by careful calculation and Lemma 2.1 in Reference [32], it is not difficult to prove that *N<sup>j</sup>* := sup 0<*t*,*s*<1 *G*2,*<sup>j</sup>* (*t*,*s*) *G*1,*<sup>j</sup>* (*t*,*s*) < +∞. Immediately,

(4) is derived.

The main tool used here is the following fixed-point index theory.

**Lemma 2** ([34])**.** *Let E*<sup>1</sup> *be a Banach space and P be a cone in E*1*. Denote P<sup>r</sup>* = {*u* ∈ *P* : k*u*k < *r*} *and ∂P<sup>r</sup>* = {*u* ∈ *P* : k*u*k = *r*} (∀*r* > 0)*. Let T* : *P* → *P be a complete continuous mapping, then the following conclusions are valid.*


#### **3. Main Results**

In this section, we shall establish the existence and multiplicity results, which is based on the fixed point index theory. For this matter, first we define the mappings *T*1, *T*<sup>2</sup> : *E* → *C*[0, 1], and *T* : *E* → *E* by

$$T\_1(u,v)(t) = \int\_0^1 \int\_0^1 \mathcal{G}\_{1,1}(t,\tau) \mathcal{G}\_{1,2}(\tau,s) f\_1(s,u(s),v(s)) ds d\tau,$$

$$T\_2(u,v)(t) = \int\_0^1 \int\_0^1 \mathcal{G}\_{2,1}(t,\tau) \mathcal{G}\_{2,2}(\tau,s) f\_2(s,u(s),v(s)) ds d\tau,$$

$$T(u,v)(t) = (T\_1(u,v)(t), T\_2(u,v)(t)), \quad \forall (u,v) \in E.$$

Then, BVP (1) in operator forms becomes

$$T(u,v) = T(u,v). \tag{3}$$

By (3), one can easily see that the existence of solutions for BVP (1) is equivalent to the existence of nontrivial fixed point of *T*. Therefore, we need to find only the nontrivial fixed point of *T* in the following work.

Subsequently, for simplicity and convenience, set

$$M\_{\boldsymbol{i},\boldsymbol{j}} = \max\_{\boldsymbol{t} \in [0,1]} \mathbb{G}\_{\boldsymbol{i},\boldsymbol{j}}(\boldsymbol{t},\boldsymbol{t}), \quad \mathbb{C}\_{\boldsymbol{i}} = \int\_{0}^{1} \mathbb{G}\_{\boldsymbol{i},1}(\boldsymbol{\tau},\boldsymbol{\tau}) \mathbb{G}\_{\boldsymbol{i},2}(\boldsymbol{\tau},\boldsymbol{\tau}) d\boldsymbol{\tau}, \quad \text{and } \lambda\_{\boldsymbol{i}} = \boldsymbol{\pi}^{4} - \beta\_{\boldsymbol{i}} \boldsymbol{\pi}^{2} - \mathfrak{a}\_{\boldsymbol{i}}.$$

Then, *Mi*,*<sup>j</sup>* , *C<sup>i</sup>* , and *λi*(*i*, *j* = 1, 2) are positive numbers.

Now let us list the following assumptions satisfied throughout the paper.

(H1) *f*<sup>1</sup> ∈ *C*([0, 1] × R <sup>+</sup> <sup>×</sup> <sup>R</sup>, <sup>R</sup> <sup>+</sup>), *<sup>f</sup>*<sup>2</sup> <sup>∈</sup> *<sup>C</sup>*([0, 1] <sup>×</sup> <sup>R</sup> <sup>+</sup> <sup>×</sup> <sup>R</sup>, <sup>R</sup>), and there exists *<sup>N</sup>*<sup>3</sup> <sup>&</sup>gt; <sup>0</sup> such that | *f*2(*t*, *u*, *v*)| ≤ *N*<sup>3</sup> *f*1(*t*, *u*, *v*) for (*t*, *u*, *v*) ∈ [0, 1] × R <sup>+</sup> <sup>×</sup> <sup>R</sup>.


In addition, for the sake of obtaining the nontrivial fixed point of operator *T*, let

$$P = \{(u, v) \in E : u(t) \ge \sigma(t) \|u\| \text{ and } |v(t)| \le N u(t), \forall t \in [0, 1] \},$$

where *<sup>σ</sup>*(*t*) = *<sup>δ</sup>*1,1*δ*1,2*C*<sup>1</sup> *C*1,1*C*1,2*M*1,1 *G*1,1(*t*, *t*) and *N* = *N*1*N*2*N*3. *N*1, *N*2, and *N*<sup>3</sup> are defined in Lemma 1 and (H1), respectively.

Obviously, *P* is a nonempty, convex, and closed subset of *E*. Furthermore, one can prove that *P* is a cone of Banach space *E*.

For convenience, set

$$\Lambda\_{\mathbf{Y}} = \{(u, v) \in \mathbb{R}^+ \times \mathbb{R} : u \in \mathbf{Y} \subset \mathbb{R}^+, |v| \le Nu\}\_{\prime \prime}$$

$$P\_r = \{(u, v) \in P : ||u|| < r\},$$

$$\partial P\_r = \{(u, v) \in P : ||u|| = r\},$$

$$\bar{P}\_r = \{(u, v) \in P : ||u|| \le r\}.$$

It is not difficult to see that *P<sup>r</sup>* is a relatively open and bounded set of *P* for each *r* > 0.

**Lemma 3.** *To calculate the fixed point index of T in P<sup>r</sup> , we first need to prove the following result. Assume that* (*H*1) *hold. Then T* : *P* → *P is completely continuous, and T*(*P*) ⊂ *P.*

**Proof of Lemma 3.** For (*u*, *v*) ∈ *P*, by virtue of Lemma 1, one can easily obtain that

$$\begin{aligned} T\_1(\boldsymbol{u}, \boldsymbol{v})(t) &= \int\_0^1 \int\_0^1 \mathcal{G}\_{1,1}(t, \tau) \mathcal{G}\_{1,2}(\tau, s) f\_1(s, \boldsymbol{u}(s), \boldsymbol{v}(s)) ds d\tau \\ &\geq \frac{\delta\_{1,1} \delta\_{1,2} \mathcal{C}\_1}{\mathcal{C}\_{1,1} \mathcal{C}\_{1,2} M\_{1,1}} G\_{1,1}(t, t) \| \left| T\_1(\boldsymbol{u}, \boldsymbol{v}) \right| \| = \sigma(t) \| \left| T\_1(\boldsymbol{u}, \boldsymbol{v}) \right| \|, \quad \forall t \in [0, 1]. \end{aligned}$$

Moreover, (H1) together with Lemma 1 guarantees that

$$\begin{split} |T\_2(\mu, v)(t)| &= |\int\_0^1 \int\_0^1 \mathcal{G}\_{2,1}(t, \tau) \mathcal{G}\_{2,2}(\tau, s) f\_2(s, \mu(s), v(s)) ds d\tau| \\ &\leq N\_3 \int\_0^1 \int\_0^1 \mathcal{G}\_{2,1}(t, \tau) \mathcal{G}\_{2,2}(\tau, s) f\_1(s, \mu(s), v(s)) ds d\tau \\ &\leq N\_1 N\_2 N\_3 \int\_0^1 \int\_0^1 \mathcal{G}\_{1,1}(t, \tau) \mathcal{G}\_{1,2}(\tau, s) f\_1(s, \mu(s), v(s)) ds d\tau \\ &= N |T\_1(\mu, v)(t)|. \end{split}$$

Therefore, *T*(*u*, *v*) ∈ *P*, namely, *T*(*P*) ⊂ *P*. In addition, since *f*1, *f*2, and *Gi*,*<sup>j</sup>* are continuous, one can deduce that *T* is completely continuous by using normal methods such as Arscoli-Arzela theorem, and so forth.

Now we are in a position to prove our main results in the following.

**Theorem 1.** *Under the assumptions (H1) and (H2), the BVP (1) admits at least one nontrivial solution.*

**Proof of Theorem 1.** To obtain the nontrivial solution for BVP (1), we will choose a bounded open set *PR*<sup>1</sup> \ *P*¯ *r*1 in cone *P* and calculate the fixed point index *i*(*T*, *PR*<sup>1</sup> \ *Pr*<sup>1</sup> , *P*). For this, the proof of Theorem 1 will be carried out in three steps.

First, notice that by (H2), there exist *ε* ∈ (0, 1) and *r*<sup>1</sup> > 0 such that

$$f\_1(t, u, v) \le (1 - \varepsilon)\lambda\_1 u \quad \forall t \in [0, 1], (u, v) \in \Lambda\_{[0, r\_1]}.\tag{4}$$

We claim that

$$
\mu T(u, v) \neq (u, v), \; \forall \mu \in (0, 1], \; (u, v) \in \partial P\_{r\_1}.\tag{5}
$$

To this end, suppose on the contrary that there exist *µ*<sup>0</sup> ∈ (0, 1] and (*u*0, *v*0) ∈ *∂Pr*<sup>1</sup> such that

$$
\mu\_0 T(\mathfrak{u}\_{0\prime} v\_0) = (\mathfrak{u}\_{0\prime} v\_0) .
$$

Therefore, (*u*0, *v*0) satisfies the following differential equation

$$\begin{cases} \ u\_0^{(4)}(t) + \beta\_1 u\_0''(t) - \mathfrak{a}\_1 u\_0(t) = f\_1(t, u(t), v(t)), & 0 < t < 1; \\\ u\_0(0) = \mathfrak{u}\_0(1) = \mathfrak{u}\_0''(0) = \mathfrak{u}\_0''(1) = 0; \end{cases} \tag{6}$$

It follows from (4) and (6) that

$$
\mu\_0^{(4)}(t) + \beta\_1 \mu\_0^{\prime\prime}(t) - \mathfrak{a}\_1 \mathfrak{u}\_0(t) \le f\_1(t, \mathfrak{u}\_0(t), \mathfrak{v}\_0(t)) \le (1 - \varepsilon)\lambda\_1 \mathfrak{u}\_0(t).
$$

Multiplying the above inequality by sin(*πt*) and then integrating from 0 to 1, one can easily get

$$\int\_0^1 \lambda\_1 u\_0(t) \sin(\pi t) dt \le (1 - \varepsilon) \int\_0^1 \lambda\_1 u\_0(t) \sin(\pi t) dt.$$

Noticing that <sup>Z</sup> <sup>1</sup> 0 *λ*1*u*0(*t*) sin(*πt*)*dt* > 0, we obtain a contradiction. Second, from (H2), there exist *ε* > 0 and *m* > 0 such that

$$f\_1(t, u, v) \ge (1 + \varepsilon)\lambda\_1 u \quad \forall t \in [0, 1], (u, v) \in \Lambda\_{[m, +\infty)}.\tag{7}$$

Set *C* := max *t*∈[0,1] (*u*,*v*)∈Λ[0,*m*] | *f*1(*t*, *u*, *v*) − (1 + *ε*)*λ*1*u*| + 1. Then one can easily find that

$$f\_1(t, \mu, v) \ge (1 + \varepsilon)\lambda\_1 \mu - \mathcal{C}\_\prime \,\,\forall t \in [0, 1], \,\,(\mu, v) \in \Lambda\_{\mathbb{R}^+}.\tag{8}$$

Now, we will show that there exists *R*<sup>1</sup> > *r*<sup>1</sup> such that

$$\inf\_{(u,v)\in\partial P\_{\mathbb{R}\_1}} \|T(u,v)\| > 0 \text{ and } \mu T(u,v) \neq (u,v), \,\forall \mu \ge 1, \,(u,v) \in \partial P\_{\mathbb{R}\_1}.\tag{9}$$

Suppose, on the contrary, that there exist *µ*<sup>0</sup> ≥ 1 and (*u*0, *v*0) ∈ *∂PR*<sup>1</sup> such that *µ*0*T*(*u*0, *v*0) = (*u*0, *v*0). Combining (6) with (8), we immediately get

$$
\mu\_0^{(4)}(t) + \beta\_1 \mu\_0''(t) - \mathfrak{a}\_1 \mathfrak{u}\_0(t) \ge f\_1(t, \mathfrak{u}\_0(t), \mathfrak{v}\_0(t)) \ge (1 + \varepsilon)\lambda\_1 \mathfrak{u}\_0(t) - \mathcal{C}.
$$

Hence,

$$\int\_0^1 \lambda\_1 u\_0(t) \sin(\pi t) dt \ge (1+\varepsilon) \int\_0^1 \lambda\_1 u\_0(t) \sin(\pi t) dt - \frac{2\mathcal{C}}{\pi} \lambda\_1$$

which yields

$$\int\_0^1 \lambda\_1 u\_0(t) \sin(\pi t) dt \le \frac{2C}{\pi \varepsilon \lambda\_1}.$$

On the other hand, in view of the definition of cone *P*, one can easily obtain that

$$\|\|u\_0\|\int\_0^1 \sigma(t)\sin(\pi t)dt \le \int\_0^1 u\_0(t)\sin(\pi t)dt \le \frac{2\mathcal{C}}{\pi \varepsilon \lambda\_1}.$$

which means

4

4

$$\|\|\mu\_0\|\| \le \frac{2\mathcal{C}}{\pi \varepsilon \lambda\_1 \int\_0^1 \sigma(t) \sin(\pi t) dt} := R\_1^\*. \tag{10}$$

Therefore, if *R*<sup>1</sup> > *R* ∗ 1 , immediately, one can get *µT*(*u*, *v*) 6= (*u*, *v*) for *µ* ≥ 1 and (*u*, *v*) ∈ *∂PR*<sup>1</sup> . In addition, if *R*<sup>1</sup> > *m* min *t*∈[ 1 4 , 3 4 ] *σ*(*t*) := *m σ* ∗ , then by the definition of cone *P*, one can get that for any *t* ∈ [ 1 , 3 ] and (*u*, *v*) ∈ *∂PR*<sup>1</sup> ,

$$
\mu(t) \ge \min\_{t \in \left[\frac{1}{4}, \frac{3}{4}\right]} \mu(t) \ge \sigma^\* R\_1 > m. \tag{11}
$$

So, by (7), (11), and Lemma 1, one can get that for all (*u*, *v*) ∈ *∂PR*<sup>1</sup> ,

<sup>k</sup>*T*(*u*, *<sup>v</sup>*)k ≥ *<sup>T</sup>*1(*u*, *<sup>v</sup>*)( <sup>1</sup> 2 ) = Z 1 0 Z 1 0 *G*1,1( 1 2 , *τ*)*G*1,2(*τ*,*s*)*f*1(*s*, *u*(*s*), *v*(*s*))*dsdτ* ≥ *δ*1,1*δ*1,2*G*1,2( 1 2 , 1 2 )*C*<sup>1</sup> Z 1 0 *G*1,2(*s*,*s*)*f*1(*s*, *u*(*s*), *v*(*s*))*ds* ≥ *δ*1,1*δ*1,2*G*1,2( 1 2 , 1 2 )*C*1(1 + *ε*)*λ*<sup>1</sup> Z <sup>3</sup> 4 1 4 *G*1,2(*s*,*s*)*u*(*s*)*ds* ≥ *δ*1,1*δ*1,2*G*1,2( 1 2 , 1 2 )*C*1(1 + *ε*)*λ*1*R*1*σ* ∗ Z <sup>3</sup> 4 1 4 *G*1,2(*s*,*s*)*ds* ≥ *δ*1,1*δ*1,2*G*1,2( 1 2 , 1 2 )*C*1(1 + *ε*)*λ*1*m* Z <sup>3</sup> 4 1 4 *G*1,2(*s*,*s*)*ds* > 0.

That is, inf (*u*,*v*)∈*∂PR*<sup>1</sup> k*T*(*u*, *v*)k > 0. So, we can ultimately choose *R*<sup>1</sup> > max{*R* ∗ 1 ,*r*1, *m σ* <sup>∗</sup> } such that

(9) holds.

Based on (5), (9), Lemma 2, and Lemma 3, we have

$$i(T\_\prime P\_{\overline{R}\_1} \backslash \overline{P}\_{r\_1 \prime} P) = i(T\_\prime P\_{\overline{R}\_1 \prime} P) - i(T\_\prime P\_{r\_1 \prime} P) = 0 - 1 = -1.$$

As a result, the conclusion of this theorem follows.

**Theorem 2.** *Assume that (H1) and (H3) hold. Then the BVP (1) has at least one nontrivial solution.*

**Proof of Theorem 2.** In the following, we divide the proof of Theorem 2 into three steps. *Step* 1. From condition (H3), there exist *ε* > 0 and *r*<sup>2</sup> > 0 such that

$$f\_1(t, u, v) \ge (1 + \varepsilon)\lambda\_1 u\_\prime \,\,\forall t \in [0, 1], \,\,(u, v) \in \Lambda\_{[0, r\_2]}.\tag{12}$$

Subsequently, we claim that

$$\inf\_{(u,v)\in\partial P\_{r\_2}} \|T(u,v)\| > 0 \text{ and } \mu T(u,v) \neq (u,v), \; \forall \mu \ge 1, \; (u,v) \in \partial P\_{r\_2}.\tag{13}$$

In fact, if there exist *µ*<sup>0</sup> ≥ 1 and (*u*0, *v*0) ∈ *∂Pr*<sup>2</sup> such that *µ*0*T*(*u*0, *v*0) = (*u*0, *v*0), then by (6) and (12), one can obtain immediately

$$
\mu\_0^{(4)}(t) + \beta\_1 \mu\_0^{\prime\prime}(t) - \mu\_1 \mu\_0(t) \ge f\_1(t, \mu\_0(t), \nu\_0(t)) \ge (1 + \varepsilon)\lambda\_1 \mu\_0(t).
$$

Hence,

$$\int\_0^1 \lambda\_1 u\_0(t) \sin(\pi t) dt \ge (1+\varepsilon) \int\_0^1 \lambda\_1 u\_0(t) \sin(\pi t) dt.$$

Noticing that <sup>Z</sup> <sup>1</sup> 0 *λ*1*u*0(*t*) sin(*πt*)*dt* > 0, we get a contradiction. In addition, it follows from Lemma 1 and (12) that for (*u*, *v*) ∈ *∂Pr*<sup>2</sup> ,

$$\begin{split} \|T(u,v)\| &\geq T\_{1}(u,v)(\frac{1}{2}) \\ &= \int\_{0}^{1} \int\_{0}^{1} G\_{1,1}(\frac{1}{2},\tau) G\_{1,2}(\tau,s) f\_{1}(s,u(s),v(s)) ds d\tau \\ &\geq \delta\_{1,1} \delta\_{1,2} G\_{1,2}(\frac{1}{2},\frac{1}{2}) \mathbb{C}\_{1} \int\_{0}^{1} G\_{1,2}(s,s) f\_{1}(s,u(s),v(s)) ds \\ &\geq \delta\_{1,1} \delta\_{1,2} G\_{1,2}(\frac{1}{2},\frac{1}{2}) \mathbb{C}\_{1} (1+\varepsilon) \lambda\_{1} \int\_{0}^{1} G\_{1,2}(s,s) u(s) ds \\ &\geq \delta\_{1,1} \delta\_{1,2} G\_{1,2}(\frac{1}{2},\frac{1}{2}) \mathbb{C}\_{1} (1+\varepsilon) \lambda\_{1} r\_{2} \int\_{0}^{1} G\_{1,2}(s,s) \sigma(s) ds > 0, \end{split}$$

which yields inf (*u*,*v*)∈*∂Pr*<sup>2</sup> k*T*(*u*, *v*)k > 0.

*Step* 2. The assumption (H3) implies that there exist *ε* ∈ (0, 1) and *m* > 0 such that

$$f\_1(t, \mu, v) \le (1 - \varepsilon)\lambda\_1 \mu\_\prime \,\,\forall t \in [0, 1], \,\,(\mu, v) \in \Lambda\_{[m, +\infty)}.\tag{14}$$

Moreover, by the continuity of *f*<sup>1</sup> and *f*2, there exists *C* ∗ > 0 such that

$$f\_1(t, \mu, v) \le (1 - \varepsilon)\lambda\_1 \mu + \mathbb{C}^\* \text{ } \forall t \in [0, 1] \text{ } (\mu, v) \in \Lambda\_{\mathbb{R}^+}.\tag{15}$$

We claim that there exists a large enough *R*<sup>2</sup> > *r*<sup>2</sup> such that

$$
\mu T(\mu, \upsilon) \neq (\mu, \upsilon), \ \forall \mu \in (0, 1], \ (\mu, \upsilon) \in \partial P\_{\mathbb{R}\_2}.\tag{16}
$$

Suppose, on the contrary, there exist *µ*<sup>0</sup> ∈ (0, 1] and (*u*0, *v*0) ∈ *∂PR*<sup>2</sup> such that *µ*0*T*(*u*0, *v*0) = (*u*0, *v*0). Then (6) together with (15) guarantees

$$
\mu\_0^{(4)}(t) + \beta\_1 \mu\_0''(t) - \mathfrak{a}\_1 \mathfrak{u}\_0(t) \le f\_1(t, \mathfrak{u}\_0(t), \mathfrak{v}\_0(t)) \le (1 - \varepsilon)\lambda\_1 \mathfrak{u}\_0(t) + \mathcal{C}^\*.
$$

Consequently,

$$\int\_0^1 \lambda\_1 u\_0(t) \sin(\pi t) dt \le (1 - \varepsilon) \int\_0^1 \lambda\_1 u\_0(t) \sin(\pi t) dt + \frac{2\mathcal{C}^\*}{\pi} \lambda$$

namely,

$$\int\_0^1 u\_0(t) \sin(\pi t) dt \le \frac{2C^\*}{\pi e \lambda\_1}.$$

Moreover, based on the definition of cone *P*, we can immediately get

$$\|\|u\_0\|\int\_0^1 \sigma(t)\sin(\pi t)dt \le \int\_0^1 u\_0(t)\sin(\pi t)dt \le \frac{2C^\*}{\pi \varepsilon \lambda\_1}.$$

*Symmetry* **2020**, *12*, 1989

which means

$$\|\mathfrak{u}\_0\| \le \frac{2\mathcal{C}^\*}{\pi \varepsilon \lambda\_1 \int\_0^1 \sigma(t) \sin(\pi t) dt} \coloneqq R\_2^\*. \tag{17}$$

So, one can choose *R*<sup>2</sup> > max{*R* ∗ 2 ,*r*2} such that (16) holds. *Step* 3. From (13), (16), Lemma 2, and Lemma 3, we deduce that

$$i(T\_\prime P\_{R\_2} \backslash \tilde{P}\_{r\_{2\prime}}P) = i(T\_\prime P\_{R\_{2\prime}}P) - i(T\_\prime P\_{r\_{2\prime}}P) = 1 - 0 = 1.1$$

As a result, BVP(1) has at least one nontrivial solution.

Up to now, some existence results of BVP(1) have been obtained by applying the fixed point index theory. In the following, the multiple solutions will be considered for BVP (1).

**Theorem 3.** *Assume that (H1) holds. In addition, suppose that*

*(1)* lim |*v*|≤*Nu <sup>u</sup>*→0<sup>+</sup> sup max *t*∈[0,1] *f*1 (*t*,*u*,*v*) *<sup>u</sup>* < *λ*1*,* lim |*v*|≤*Nu u*→+∞ sup max *t*∈[0,1] *f*1 (*t*,*u*,*v*) *<sup>u</sup>* < *λ*1*;*

*(2) There exists r* > 0 *and a continuous nonnegative function* Φ*<sup>r</sup> such that*

$$f\_1(t, \mu, v) \ge \Phi\_r(t), \forall (t, \mu, v) \in [0, 1] \times (\sigma(t)r, r) \times [-Nr, Nr]^\perp$$

*and*

$$\max\_{t \in [0,1]} \int\_0^1 \int\_0^1 G\_{1,1}(t,\tau) G\_{1,2}(\tau,s) \Phi\_r(s) ds d\tau > r.c.$$

*Then the BVP (1) has at least two nontrivial solutions.*

**Proof of Theorem 3.** In order to obtain this conclusion, we firstly claim that

$$\inf\_{(u,v)\in\partial P\_r} \|T(u,v)\| > 0 \text{ and } \mu T(u,v) \neq (u,v), \forall \,\mu \ge 1, \ (u,v) \in \partial P\_r. \tag{18}$$

Suppose, on the contrary, there exist *µ*<sup>0</sup> ≥ 1 and (*u*0, *v*0) ∈ *∂P<sup>r</sup>* such that *µ*0*T*(*u*0, *v*0) = (*u*0, *v*0). Then,

$$\begin{split} \|u\_{0}\| &\quad \geq \|T(u\_{0},v\_{0})\| \geq T\_{1}(u\_{0},v\_{0})(t) \\ &= \int\_{0}^{1} \int\_{0}^{1} \mathbf{G}\_{1,1}(t,\tau) \mathbf{G}\_{1,2}(\tau,s) f\_{1}(s,u(s),v(s)) ds d\tau \\ &\geq \int\_{0}^{1} \int\_{0}^{1} \mathbf{G}\_{1,1}(t,\tau) \mathbf{G}\_{1,2}(\tau,s) \Phi\_{\tau}(s) ds d\tau. \end{split} \tag{19}$$

Taking the maximum for both sides of the above inequality in *t* ∈ [0, 1], we get that

$$\|\mu\_0\| \ge \max\_{t \in [0,1]} \int\_0^1 \int\_0^1 \mathbb{G}\_{1,1}(t,\tau) \mathbb{G}\_{1,2}(\tau,s) \Phi\_r(s) ds d\tau > r. \tag{20}$$

This means (*u*0, *v*0)∈ *∂P<sup>r</sup>* , which is a contradiction. Moreover, one can easily see that inf (*u*,*v*)∈*∂P<sup>r</sup>* k*T*(*u*, *v*)k > 0 holds from (19) and (20).

Next, similar to the process of proving (5) and (16), there exist *r*<sup>1</sup> ∈ (0,*r*) and *R*<sup>2</sup> ≥ max{*R* ∗ 2 ,*r*2,*r*} such that

$$
\mu T(u, v) \neq (u, v), \forall \mu \in (0, 1], \; \forall (u, v) \in \partial P\_{r\_1 \nu} \tag{21}
$$

$$
\mu T(u, v) \neq (u, v), \; \forall \mu \in (0, 1], \; \forall (u, v) \in \partial P\_{R\_2}.\tag{22}
$$

Thus, by (18), (21), (22), Lemma 2, and Lemma 3, one can immediately obtain that

$$i(T\_\prime P\_{R\_2} \backslash \tilde{P}\_{r\prime} P) = i(T\_\prime P\_{R\_2\prime} P) - i(T\_\prime P\_{r\prime} P) = 1 - 0 = 1\,,$$

$$i(T\_\prime P\_{r\_\perp} \backslash P\_{r\_{1'}}P) = i(T\_\prime P\_{r\_\prime}P) - i(T\_\prime P\_{r\_{1'}}P) = 0 - 1 = -1.$$

Namely, there exist (*u*1, *<sup>v</sup>*1) <sup>∈</sup> *<sup>P</sup><sup>r</sup>* \ *<sup>P</sup>*¯ *r*1 and (*u*2, *v*2) ∈ *PR*<sup>2</sup> \ *P*¯ *<sup>r</sup>* satisfying *T*(*u<sup>i</sup>* , *vi*) = (*u<sup>i</sup>* , *vi*)(*i* = 1, 2), that is, (*u<sup>i</sup>* , *vi*)(*i* = 1, 2) is the solution of BVP(1).

Finally, we show (*u*1, *v*1) 6= (*u*2, *v*2). To see this we need only to prove BVP(1) has no solution on *∂P<sup>r</sup>* . Suppose on the contrary, there exists (*u* ∗ , *v* ∗ ) ∈ *∂P<sup>r</sup>* being a solution of BVP(1). Then *T*(*u* ∗ , *v* ∗ ) = (*u* ∗ , *v* ∗ ). By a similar process of obtaining (20), one can get k*u* <sup>∗</sup>k = k*T*1(*u* ∗ , *v* ∗ )k > *r*, which is a contradiction. To sum up, Theorem 3 is proved.

From a process similar to the above, the following conclusion can be obtained.

**Theorem 4.** *Suppose that (H1) holds. In addition, suppose that*

*(1)* lim |*v*|≤*Nu <sup>u</sup>*→0<sup>+</sup> inf min *t*∈[0,1] *f*1 (*t*,*u*,*v*) *<sup>u</sup>* > *λ*1*,* lim |*v*|≤*Nu u*→+∞ inf min *t*∈[0,1] *f*1 (*t*,*u*,*v*) *<sup>λ</sup>*1*<sup>u</sup>* > *λ*1*;*

*(2) There exists R* > 0*, and a continuous nonnegative function* Ψ*<sup>R</sup> such that*

$$f\_1(t, \iota, v) \le \Psi\_R(t), \forall (t, \iota, v) \in [0, 1] \times [\sigma(t)\mathbb{R}, \mathbb{R}] \times [-N\mathbb{R}, N\mathbb{R}]$$

and

$$\max\_{t \in [0,1]} \int\_0^1 \int\_0^1 G\_{1,1}(t,\tau) G\_{1,2}(\tau,s) \Psi\_R(s) ds d\tau < R.$$

*Then the BVP (1) has at least two nontrivial solutions.*

#### **Proof of Theorem 4.** We firstly prove that

$$
\mu T(\mu, v) \neq (\mu, v), \,\forall \mu \in (0, 1], \,(\mu, v) \in \partial P\_{\mathbb{R}}.\tag{23}
$$

To this end, suppose on the contrary that there exist *µ*<sup>0</sup> ∈ (0, 1] and (*u*0, *v*0) ∈ *∂P<sup>R</sup>* such that *µ*0*T*(*u*0, *v*0) = (*u*0, *v*0). Hence, we get *u*<sup>0</sup> = *µ*0*T*1(*u*0, *v*0), that is

$$u\_0(t) \le T\_1(u\_0, v\_0)(t) \quad \le \int\_0^1 \int\_0^1 G\_{1,1}(t, \tau) G\_{1,2}(\tau, s) \Psi\_R(s) ds d\tau < \mathcal{R}.\tag{24}$$

Noticing that (*u*0, *v*0) ∈ *∂PR*, this is a contradiction.

Next, from a process similar to (9) and (13), there exist *R*<sup>1</sup> > max{*R*, *R* ∗ 1 ,*r*1, *m σ* <sup>∗</sup> } and *r*<sup>2</sup> ∈ (0, *R*) such that

$$\inf\_{(u,v)\in\partial P\_{\mathbb{R}\_1}} \|T(u,v)\| > 0 \text{ and } \mu T(u,v) \neq (u,v), \,\forall \mu \ge 1, \,(u,v) \in \partial P\_{\mathbb{R}\_1}.\tag{25}$$

$$\inf\_{(u,v)\in\partial P\_{\mathcal{I}\_2}} \|T(u,v)\| > 0 \text{ and } \mu T(u,v) \neq (u,v), \; \forall \mu \ge 1, \; (u,v) \in \partial P\_{\mathcal{I}\_2}.\tag{26}$$

So, by (23)–(26), Lemma 2, and Lemma 3, one can get

$$i(T\_\prime P\_{R\_1} \backslash \tilde{P}\_{R\prime} P) = i(T\_\prime P\_{R\_1\prime} P) - i(T\_\prime P\_{R\prime} P) = 0 - 1 = -1,$$

$$i(T\_\prime P\_R \backslash \tilde{P}\_{r\_{2\prime}} P) = i(T\_\prime P\_{R\prime} P) - i(T\_\prime P\_{r\_{2\prime}} P) = 1 - 0 = 1.$$

Finally, from a process similar to the end of proof of Theorem 3, BVP(1) has at least two nontrivial solutions. As a result, the conclusion of this theorem follows.

#### **4. Examples**

In this section, two illustrative examples are worked out to show the effectiveness of the obtained results.

**Example 1.** *Consider the following BVP of fourth-order ordinary differential systems*

$$\begin{cases} u^{(4)}(t) + u''(t) - \pi^2 u(t) = f\_1(t, u, v), \ 0 < t < 1; \\ v^{(4)}(t) + \frac{1}{2} v''(t) - \frac{\pi^2}{2} v(t) = f\_2(t, u, v), \ 0 < t < 1; \\ u(0) = u(1) = u''(0) = u''(1) = 0; \\ v(0) = v(1) = v''(0) = v''(1) = 0, \end{cases} \tag{27}$$

where

$$f\_1(t,u,v) = \begin{cases} \begin{array}{llll} (\pi^4 - 2\pi^2)(1+\sin(\pi t))uv^{\frac{1}{4}} & \text{if } 0 < u < 1, \ |v| < u; \\ (\pi^4 - 2\pi^2)(1+\sin(\pi t))v^{\frac{1}{4}} & \text{if } u = 1, \ |v| < u; \\ (\pi^4 - 2\pi^2)(1+\sin(\pi t))u^{\frac{1}{4}}v^{\frac{1}{4}} & \text{if } u > 1, \ |v| < u; \end{cases} \\\\ f\_2(t,u,v) = \begin{cases} (\pi^4 - 2\pi^2)(1+\cos(\pi t))uv^{\frac{1}{4}} & \text{if } 0 < u < 1, \ |v| < u; \\ (\pi^4 - 2\pi^2)(1+\cos(\pi t))v^{\frac{1}{4}} & \text{if } u = 1, \ |v| < u; \\ (\pi^4 - 2\pi^2)(1+\cos(\pi t))u^{\frac{1}{4}}v^{\frac{1}{4}} & \text{if } u > 1, \ |v| < u. \end{cases}$$

Then, BVP (27) has at least two nontrivial solutions.

**Proof of Example 1 .** BVP (27) can be regarded as a BVP of the form (1). Choosing *α*<sup>1</sup> = *π* 2 , *β*<sup>1</sup> = 1, and *λ*<sup>1</sup> = *π* <sup>4</sup> <sup>−</sup> <sup>2</sup>*<sup>π</sup>* <sup>2</sup> > 0, then we have

$$\xi\_{1,1}^{\star} = \frac{-\beta\_1 + \sqrt{\beta\_1^2 + 4a\_1}}{2} = \frac{-1 + \sqrt{1 + 4\pi^2}}{2}, \\ \xi\_{1,2}^{\star} = \frac{-\beta\_1 - \sqrt{\beta\_1^2 + 4a\_1}}{2} = \frac{-1 - \sqrt{1 + 4\pi^2}}{2}.$$

Clearly, *α*<sup>1</sup> and *β*<sup>1</sup> satisfy the condition (2). Moreover, by careful calculation and Lemma 2.1 in Reference [32], one can obtain that

$$G\_{1,1}(t,s) = \begin{cases} \frac{\sinh \, w\_{1,1}t \sinh \, w\_{1,1}(1-s)}{w\_{1,1} \sinh \, w\_{1,1}} & 0 \le t \le s \le 1;\\ \frac{\sinh \, w\_{1,1}s \sinh \, w\_{1,1}(1-t)}{w\_{1,1} \sinh \, w\_{1,1}} & 0 \le s \le t \le 1; \end{cases}$$

$$G\_{1,2}(t,s) = \begin{cases} \frac{\sin \, w\_{1,2}t \sin \, w\_{1,2}(1-s)}{w\_{1,2} \sin \, w\_{1,2}} & 0 \le t \le s \le 1;\\ \frac{\sin \, w\_{1,2}s \sin \, w\_{1,2}(1-t)}{w\_{1,2} \sin \, w\_{1,2}} & 0 \le s \le t \le 1; \end{cases}$$

where *w*1,*<sup>i</sup>* = p |*ξ*1,*<sup>i</sup>* |(*i* = 1, 2).

Now, |*v*| ≤ 2*u*, | *f*2(*t*, *u*, *v*)| ≤ 2| *f*1(*t*, *u*, *v*)|, and *N* = *N*1*N*2*N*3. Thus, one can easily get that (H1) holds by choosing *<sup>N</sup>*<sup>3</sup> <sup>≥</sup> max{2, <sup>2</sup> *N*1*N*<sup>2</sup> }, where *N<sup>j</sup>* = sup 0<*t*,*s*<1 *G*2,*<sup>j</sup>* (*t*,*s*) *G*1,*<sup>j</sup>* (*t*,*s*) , *j* = 1, 2.

In addition, by calculation, we get that

$$\lim\_{\substack{|v| \le Nu \\ u \to 0^+}} \sup\_{t \in [0,1]} \frac{f\_1(t, u, v)}{u} = \lim\_{\substack{|v| \le Nu \\ u \to 0^+}} \sup\_{t \in [0,1]} \frac{(\pi^4 - 2\pi^2)(1 + \sin(\pi t)) u v^{\frac{1}{4}}}{u} = 0 < \lambda\_1 \lambda\_2$$

$$\lim\_{\substack{|v| \le \mathbb{N}u \\ u \to +\infty}} \sup\_{t \in [0,1]} \frac{f\_1(t, u, v)}{u} = \lim\_{\substack{|v| \le \mathbb{N}u \\ u \to +\infty}} \sup\_{t \in [0,1]} \frac{(\pi^4 - 2\pi^2)(1 + \sin(\pi t))u^{\frac{1}{4}}v^{\frac{1}{4}}}{u} = 0 < \lambda\_1.$$

Choose

$$\tau = \min\{1, \left[\delta\_{1,1}\delta\_{1,2}\int\_0^1 \sqrt{\sigma(t)}\sin(\pi t)dt \min\_{t \in \left[\frac{1}{2}, \frac{3}{4}\right]} \left(G\_{1,1}(t,t)G\_{1,2}(t,t)\right)^2\right] > 0\}$$

and

$$
\Phi\_r(t) = \sqrt{r\sigma(t)}\sin(\pi t).
$$

Then, it is not difficult to obtain that the condition (2) in Theorem 3 holds. Hence, our conclusion follows from Theorem 3.

**Example 2.** *Consider the following BVP of fourth-order ordinary differential systems*

$$\begin{cases} u^{(4)}(t) + u''(t) - \pi^2 u(t) = f\_1(t, u, v), \ 0 < t < 1; \\ v^{(4)}(t) + \frac{1}{2} v''(t) - \frac{\pi^2}{2} v(t) = f\_2(t, u, v), \ 0 < t < 1; \\ u(0) = u(1) = u''(0) = u''(1) = 0; \\ v(0) = v(1) = v''(0) = v''(1) = 0. \end{cases} \tag{28}$$

where

$$f\_1(t,u,v) = \begin{cases} (\pi^4 - 2\pi^2)(2+t)u^{\frac{1}{2}}v^{\frac{1}{3}} & \text{if } 0 < u < 1, \ 0 < v < 1; \\ (\pi^4 - 2\pi^2)(2+t)v^{\frac{1}{3}} & \text{if } u = 1, \ 0 < v < 1; \\ (\pi^4 - 2\pi^2)(2+t)u^2v^{\frac{1}{3}} & \text{if } u > 1, \ 0 < v < 1, \end{cases}$$

$$f\_2(t,u,v) = \begin{cases} (\pi^4 - 2\pi^2)(1+\cos(\pi t))u^{\frac{1}{2}}v^{\frac{1}{3}} & \text{if } 0 < u < 1, \ 0 < v < 1; \\ (\pi^4 - 2\pi^2)(1+\cos(\pi t))v^{\frac{1}{3}} & \text{if } u = 1, \ 0 < v < 1; \\ (\pi^4 - 2\pi^2)(1+\cos(\pi t))u^2v^{\frac{1}{3}} & \text{if } u > 1, \ 0 < v < 1. \end{cases}$$

Then, BVP (28) has at least two nontrivial solutions.

**Proof of Example 2 .** BVP (28) can be regarded as a BVP of the form (1). Using a similar process of the proof of Example 1, one can easily obtain that

$$\lim\_{\substack{|v| \le Nu \\ u \to 0^{+}}} \inf \min\_{t \in [0,1]} \frac{f\_1(t, u, v)}{u} = \lim\_{\substack{|v| \le Nu \\ u \to 0^{+}}} \inf \min\_{t \in [0,1]} \frac{(\pi^4 - 2\pi^2)(2+t)u^{\frac{1}{2}}v^{\frac{1}{2}}}{u} = +\infty > \pi^4 - 2\pi^2 = \lambda\_1 > 0$$

$$\lim\_{\substack{|v| \le Nu \\ u \to +\infty}} \inf \min\_{t \in [0,1]} \frac{f\_1(t, u, v)}{u} = \lim\_{\substack{|v| \le Nu \\ u \to +\infty}} \inf \min\_{t \in [0,1]} \frac{(\pi^4 - 2\pi^2)(2+t)u^{2}v^{\frac{1}{2}}}{u} + \infty > \pi^4 - 2\pi^2 = \lambda\_1 > 0$$

In addition, it is obvious that (H1) holds by choosing *N*<sup>3</sup> = 2. In the following, set

$$R = \max\{1, \frac{2}{5\pi^2 \mathcal{C}\_{1,1} \mathcal{C}\_{1,2}} \max\_{t \in [0,1]} [\mathcal{G}\_{1,1}(t,t)\mathcal{G}\_{1,2}(t,t)]\} > 0$$

and

$$
\Psi\_R(t) = \pi^4 R^2 (2+t).
$$

Then, it is trivial to verify that assumption (2) of Theorem 3 is true. As a result, by Theorem 4, system (28) has at least two nontrivial solutions.

#### **5. Conclusions**

In this paper, we have obtained some appropriate results corresponding to multiple solutions for a class of nonlinear fourth-order boundary value problems with parameters. The multiple solutions for the considered systems are obtained under some suitable assumptions via fixed point index theory. The whole theoretical results has been demonstrated by providing two interesting examples. Hence, we claim that fixed point index theory can be used as a strong technique to study nonlinear fourth-order boundary value problems with parameters.

**Author Contributions:** Conceptualization and Visualization, Y.L.; Formal analysis and Investigation, D.Z.; Writing original draft and Investigation, L.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by NNSF of P.R.China (62073202 and 11671237), and a project of Shandong Province Higher Educational Science and Technology Program of China under the grant J18KA233.

**Acknowledgments:** The authors are thankful to the editor and anonymous referees for their valuable comments and suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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## *Article* **Existence of Three Solutions for a Nonlinear Discrete Boundary Value Problem with** *φc***-Laplacian**

**Yanshan Chen 1,2 and Zhan Zhou 1,2,**<sup>∗</sup>


Received: 14 October 2020; Accepted: 3 November 2020; Published: 6 November 2020

**Abstract:** In this paper, based on critical point theory, we mainly focus on the multiplicity of nontrivial solutions for a nonlinear discrete Dirichlet boundary value problem involving the mean curvature operator. Without imposing the symmetry or oscillating behavior at infinity on the nonlinear term *f* , we respectively obtain the sufficient conditions for the existence of at least three non-trivial solutions and the existence of at least two non-trivial solutions under different assumptions on *f* . In addition, by using the maximum principle, we also deduce the existence of at least three positive solutions from our conclusion. As far as we know, our results are supplements to some well-known ones.

**Keywords:** *φc*-Laplacian; boundary value problem; critical point theory; three solutions

#### **1. Introduction**

Let Z and R denote all integers and real numbers, respectively. Let *N* be a fixed positive integer. Define Z(*a*, *b*) = {*a*, *a* + 1, · · · , *b*} with *a* ≤ *b* for any *a*, *b* ∈ Z.

Difference equations are widely used in various research fields, such as computer science, discrete optimization, economics and biological neural networks [1–4]. On the existence and multiplicity of solutions for the boundary value problems of difference equations, many authors have come to important conclusions by exploiting various methods, including the method of upper and lower solutions, Brouwer degree and invariant sets of descending flow [5–7]. Critical point theory was used largely to explore differential equations much earlier on in history. In 2003, Guo and Yu in [8] used critical point theory for the first time to obtain sufficient conditions for the existence of periodic solutions and subharmonic solutions of difference equations. This crucial breakthrough inspired many scholars to use critical point theory to study the dynamics of difference equations and many meaningful and interesting results have been obtained, especially in periodic solutions [9–11], homoclinic solutions [12–16] and boundary value problems [17–23].

In [24], Agarwal, Perera and O'Regan employed the critical point theory to establish the existence of at least two positive solutions of the following second order discrete boundary value problem

$$\begin{cases} -\Delta^2 \mathfrak{x}(k-1) = f(k, \mathfrak{x}(k)), \; k \in \mathbb{Z}(1, N),\\ \mathfrak{x}(0) = \mathfrak{x}(N+1) = 0, \end{cases}$$

where ∆*x*(*k*) = *x*(*k* + 1) − *x*(*k*), ∆ <sup>2</sup>*x*(*k*) = <sup>∆</sup>(∆*x*(*k*)) and *<sup>f</sup>* : <sup>Z</sup>(1, *<sup>N</sup>*) <sup>×</sup> <sup>R</sup> <sup>→</sup> <sup>R</sup> is a continuous function. In [18], by using the three critical point theorems proposed by Bonanno [25], Jiang and Zhou obtained

sufficient conditions for the existence of at least three solutions of the following Dirichlet boundary value problem with *φp*-Laplacian

$$\begin{cases} -\Delta(\phi\_p(\Delta u(k-1))) = \lambda f(k, u(k)), \; k \in \mathbb{Z}(1, N), \\ u(0) = u(N+1) = 0, \end{cases}$$

where *φ<sup>p</sup>* is the *p*-Laplacian defined by *φp*(*u*) = |*u*| *<sup>p</sup>*−2*<sup>u</sup>* (*<sup>p</sup>* <sup>&</sup>gt; <sup>1</sup>) with *<sup>u</sup>* <sup>∈</sup> <sup>R</sup> and *<sup>λ</sup>* <sup>&</sup>gt; <sup>0</sup> is a positive parameter. Different from the conclusion of [18], Bonanno in [26] obtained the existence of three positive solutions without the asymptotic condition of the nonlinear function *f* . In particular, Bonanno obtained the sufficient conditions for the existence of at least four nontrivial solutions when *f* satisfies the growth condition at zero and infinity [26], which improved the result in [18].

In [27], by using critical point theory, Nastasi and Vetro obtained the existence of at least two positive solutions to the following Dirichlet boundary value problem with (*p*, *q*)-Laplacian

$$\begin{cases} -\Delta(\phi\_p(\Delta u(z-1))) - \Delta(\phi\_\emptyset(\Delta u(z-1))) + a(z)\phi\_p(u(z)) + \beta(z)\phi\_\emptyset(u(z)) = \lambda g(z, u(z)), \; z \in \mathbb{Z}(1, N), \\ u(0) = u(N+1) = 0, \end{cases}$$

where 1 < *q* < *p* < +∞, *α*, *β* : Z(1, *N*) → R and *g* : [1, *N* + 1] × R → R are a continuous function with *g*(*N* + 1, *t*) = 0 for all *t* ∈ R.

It is well known that the differential equation with *φc*-Laplacian (*φ<sup>c</sup>* is the mean curvature operator defined by *φc*(*s*) = *s*/ √ 1 + *s* 2 for *s* ∈ R) was studied by many scholars in the past decades [28–31]. It is usually regarded as a variant of the Liouville–Bratu–Gelfand problem, which is used to study the dynamic model of combustible gases. As mentioned above, we find that the research on the difference equation largely focuses on the case with *φp*-Laplacian. However, there are only a few results on the boundary value problems involving *φc*-Laplacian [32]. Recently, Zhou and Ling in [33] considered the existence of multiple solutions of the following discrete Dirichlet boundary value problem with *φc*-Laplacian.

Problem 1

$$\begin{cases} -\Delta \left( \frac{\Delta u(k-1)}{\sqrt{1+(\Delta u(k-1))^2}} \right) = \lambda f(k, u(k)), \; k \in \mathbb{Z}(1, N), \\ u(0) = u(N+1) = 0. \end{cases}$$

The authors in [33] found that the properties of the nonlinear term *F*(*k*, *u*) = R *<sup>u</sup>* 0 *f*(*k*, *t*)*dt* plays an important role on the existence of multiple solutions. When *F*(*k*, *u*) has oscillating behavior at infinity, there are infinite solutions to the boundary value (Problem 1) [33]. Naturally, we would like to ask: What will happen if *F*(*k*, *u*) does not oscillate at infinity?

To address this problem, in this paper, we will study the existence of solutions for the boundary value (Problem 1) without oscillating nonlinear terms. In fact, based on the theorems of G. Bonanno (Theorem 4.1 in [22] and Theorem 2.1 in [23]), we will give the conditions of the existence of at least three nontrivial solutions for (Problem 1), when *F*(*k*, *u*) does not have oscillation property at infinity. In addition, when *f*(*k*, 0) > 0, we obtain the sufficient conditions for the existence of at least three positive solutions of (Problem 1). Moveover, we give two examples to illustrate our main results.

For convenience, we end this section by recalling some classical definitions and two well-known lemmas, which are the main tools of this paper.

Let (*X*, k · k) be a real Banach space. We say that *I* : *X* → R is coercive on *X* if limk*u*k→+<sup>∞</sup> *<sup>I</sup>*(*u*) = +∞. If *<sup>I</sup>* is a continuously Gâteaux differentiable functional, we say that *<sup>I</sup>* satisfies the Palais–Smale condition ((PS)-condition in short), if any sequence *u<sup>n</sup>* ⊂ *X* such that {*I*(*un*)} is bounded and {*I* 0 (*un*)} is convergent to 0 in *X* ∗ , has a convergent subsequence in *X*.

(*H*) *Let* (*X*, k · k) *be a real finite dimensional Banach space and let* Φ*,* Ψ : *X* → R *be two continuously Gâteaux differentiable functionals with* Φ *coercive and such that*

$$\inf\_{X} \Phi = \Phi(0) = \Psi(0) = 0.$$

**Lemma 1** ([22])**.** *Assume that (H) holds and there exist r* > 0 *and x* ∈ *X, with* 0 < *r* < Φ(*x*)*, such that:*

$$\begin{aligned} (a\_1) \quad &\frac{\sup\_{\Phi(\mathbf{x}) \le r} \Psi(\mathbf{x})}{r} < \frac{\Psi(\overline{\mathbf{x}})}{\Phi(\overline{\mathbf{x}})};\\ (a\_2) \quad &\text{for each } \lambda \in \Lambda\_r := \left(\frac{\Phi(\overline{\mathbf{x}})}{\Psi(\overline{\mathbf{x}})}, \frac{r}{\sup\_{\Phi(\mathbf{x}) \le r} \Psi(\mathbf{x})}\right), \text{the functional } \Phi - \lambda \Psi \text{ is coercive.} \end{aligned}$$

*Then, for each λ* ∈ Λ*<sup>r</sup> , the functional I<sup>λ</sup>* = Φ − *λ*Ψ *has at least three distinct critical points in X.*

**Lemma 2** ([23])**.** *Let X be a real Banach space and let* Φ*,* Ψ *: X* → R *be two continuously Gâteaux differentiable functionals such that* inf *X* Φ = Φ(0) = Ψ(0) = 0. *Assume that there are r* > 0 *and u*˜ ∈ *X, with* 0 < Φ(*u*˜) < *r, such that*

$$\frac{\sup\_{\Phi(u)\le r} \Psi(u)}{r} < \frac{\Psi(\mathfrak{u})}{\Phi(\tilde{u})} \tag{1}$$

*for each λ* ∈ Φ(*u*˜) Ψ(*u*˜) , *r* sup Φ(*u*)≤*r* Ψ(*u*) *, and the functional I<sup>λ</sup>* = Φ − *λ*Ψ *satisfies (PS)-condition and it is unbounded from below. Then, for each λ* ∈ Φ(*u*˜) Ψ(*u*˜) , *r* sup Φ(*u*)≤*r* Ψ(*u*) *, the functional I<sup>λ</sup> admits at least two non-zero critical points uλ*,1, *uλ*,2 *such that Iλ*(*uλ*,1) < 0 < *Iλ*(*uλ*,2).

#### **2. Preliminaries**

In this section, we recall some definitions, notations and properties. Consider the *N*-dimensional Banach space

$$S = \{ \mu : [0, N+1] \to \mathbb{R} : \mu(0) = \mathfrak{u}(N+1) = 0 \}$$

endowed with the norm

$$\|u\| = \left(\sum\_{k=1}^{N+1} |\Delta u(k-1)|^2\right)^{\frac{1}{2}}.$$

We define the functional Φ, Ψ : *S* → R in the following way

$$\Phi(u) = \sum\_{k=1}^{N+1} \left( \sqrt{1 + (\Delta u(k-1))^2} - 1 \right), \quad \Psi(u) = \sum\_{k=1}^{N} F(k, u(k)) \tag{2}$$

for each *u* ∈ *S*, where

$$F(k, \xi) := \int\_0^{\xi} f(k, t)dt, \quad \xi \in \mathbb{R}.$$

It is easy to check that Φ, Ψ ∈ *C* 1 (*S*, *R*) and we have

$$\Phi'(u)(v) = -\sum\_{k=1}^{N} \Delta(\phi\_c \left(\Delta u(k-1)\right)) v(k)$$

and

$$\Psi'(\boldsymbol{\mu})(v) = -\sum\_{k=1}^{N} f(k, \boldsymbol{\mu}(k)) v(k).$$

By a standard argument, it can be shown that the critical points of the functional

$$I\_{\lambda} = \Phi - \lambda \Psi$$

are the solutions of (Problem 1).

Let

$$\|\mu\|\_{\infty} = \max\left\{ |\mu(k)| : k \in \mathbb{Z}(1, N) \right\}.$$

We see that k · k<sup>∞</sup> is another norm in *S*. From Lemma 2.2 of [18], we have

**Lemma 3** ([18])**.** *For any u* ∈ *S, the following relation holds*

$$\|u\|\_{\infty} \le \frac{\sqrt{N+1}}{2} \|u\|.$$

From (2.1) and (2.3) in [22], we have

**Lemma 4** ([22])**.** *For any u* ∈ *S, one has*

$$\frac{1}{\sqrt{N\lambda\_N}}||u|| \le ||u||\_{\infty} \le \frac{1}{\sqrt{\lambda\_1}}||u||\_{\prime\prime}$$

*where λ*<sup>1</sup> = 4 sin<sup>2</sup> *<sup>π</sup>* 2(*N*+1) *and λ<sup>N</sup>* = 4 sin<sup>2</sup> *<sup>N</sup><sup>π</sup>* 2(*N*+1) *.*

Finally, in order to obtain the positive solutions of (Problem 1), we need the following strong maximum principle, which can be found in Theorem 2.1 of [33].

**Lemma 5** ([33])**.** *Assume u* ∈ *S such that either*

$$
\mu(k) > 0 \quad \text{or} \quad \Delta(\phi\_\mathfrak{c}(\Delta \mu(k-1))) \le 0,
$$

*for all k* ∈ Z(1, *N*). *Then, either u* > 0 *in* Z(1, *N*) *or u* ≡ 0*.*

#### **3. Main Results**

For convenience, set

$$F\_t := \sum\_{k=1}^{N} F(k, t)\_t \text{ for all } t > 0.$$

Our first result is the following theorem.

**Theorem 1.** *Assume that there exist two positive constants c and d with*

$$2\sqrt{1+d^2} > 1 + \sqrt{1 + \frac{4c^2}{N+1}}\tag{3}$$

*such that*

$$(i) \quad f(k, \xi) > 0 \text{ for each } k \in \mathbb{Z} \\ (1, N) \text{ and } \xi \in [-c, c]\varphi$$

$$\left(\left(\mathrm{ii}\right)\_{-}\right)\_{\frac{F\_d}{2\left(\sqrt{1+d^2}-1\right)}} > \frac{F\_c}{-1+\sqrt{1+\frac{4c^2}{N+1}}};$$

*(iii)* lim sup |*ξ*|→+∞ *F*(*k*,*ξ*) <sup>|</sup>*ξ*<sup>|</sup> <sup>&</sup>lt; 2*Fc N*( √ 4*c* <sup>2</sup>+*N*+1− √ *N*+1) *.*

$$\text{Then, for every } \lambda \in \Lambda := \left(\frac{2(\sqrt{1+d^2}-1)}{\mathbb{F}\_d}, \frac{-1+\sqrt{1+\frac{4r^2}{N+1}}}{\mathbb{F}\_d}\right), \text{(Problem 1) has at least three nontrivial solutions.}$$

**Proof.** We take *X* = *S*, Φ and Ψ as in (2). Clearly, Φ and Ψ are two continuously Gâteaux differentiable functionals. Now, we prove the coercivity of Φ. In fact, one has

$$\begin{aligned} \Phi(u) &= \sum\_{k=1}^{N+1} \left( \sqrt{1 + (\Delta u(k-1))^2} - 1 \right) \\ &\ge \left( \sum\_{k=1}^{N+1} \left[ 1 + (\Delta u(k-1))^2 \right] \right)^{\frac{1}{2}} - \sum\_{k=1}^{N+1} 1 \\ &\ge ||u|| - N - 1. \end{aligned}$$

This means lim k*u*k→+∞ Φ(*u*) = +∞ and verifies the coercivity of Φ. Moreover, by the definition of Φ and Ψ, we can obtain

$$\inf\_{X} \Phi = \Phi(0) = \Psi(0) = 0.$$

To summarize, condition (*H*) in Lemma 1 holds.

According to Lemma 1, it is clear that Theorem 1 holds if we can verify (*a*1) and (*a*2) of Lemma 1. Put

$$r = -1 + \sqrt{1 + \frac{4c^2}{N+1}}.$$

If Φ(*u*) ≤ *r*, let

$$v(k) = \sqrt{1 + (\Delta u(k))^2} - 1 \text{ for } k \in \mathbb{Z}(0, N).$$

Then, we have *N* ∑ *k*=0 *v*(*k*) ≤ *r* and

$$\sum\_{k=0}^{N} (\Delta u(k))^2 = \sum\_{k=0}^{N} \left( v(k)^2 + 2v(k) \right) \le \left( \sum\_{k=0}^{N} v(k) \right)^2 + 2\sum\_{k=0}^{N} v(k) \le r^2 + 2r = \frac{4c^2}{N+1}.$$

Thus, k*u*k<sup>∞</sup> ≤ √ *N*+1 2 k*u*k ≤ *c* by Lemma 3.

By (i), we see that *F*(*k*, *ξ*) is increasing in *ξ* ∈ [−*c*, *c*]. Thus

$$\sup\_{\substack{\Phi(u)\le r\\r}} \frac{\Psi(u)}{r} \le \frac{\sup\_{\|u\|\_{\infty}\le c} \sum\_{k=1}^{N} F(k, u(k))}{-1 + \sqrt{1 + \frac{4c^2}{N+1}}} \le \frac{\sum\_{k=1}^{N} \max\_{|\tilde{\xi}| \le c} F(k, \tilde{\xi})}{-1 + \sqrt{1 + \frac{4c^2}{N+1}}} = \frac{F\_c}{-1 + \sqrt{1 + \frac{4c^2}{N+1}}}.\tag{4}$$

Then, it is easy to get

$$\sup\_{\Phi(u)\le r} \Psi(u) \\ \qquad \frac{1}{\lambda} < \frac{1}{\lambda}. \tag{5}$$

Now, let *u* ∈ *S* be defined by

$$
\overline{\mu}(k) = \begin{cases} d, & \text{if } k \in \mathbb{Z}(1, N), \\ 0, & \text{if } k = 0 \text{ or } k = N + 1. \end{cases}
$$

Then, we see from (3) that

$$\Phi(\overline{\pi}) = 2(\sqrt{1+d^2}-1) > \sqrt{1+\frac{4c^2}{N+1}} - 1 = r.s.$$

*Symmetry* **2020**, *12*, 1839

Moreover, it holds that

$$\frac{\Psi(\overline{u})}{\Phi(\overline{u})} = \frac{\sum\_{k=1}^{N} F(k, \overline{u}(k))}{2(\sqrt{1+d^2}-1)} = \frac{F\_d}{2(\sqrt{1+d^2}-1)}.\tag{6}$$

Therefore, we have

$$\frac{\Psi(\overline{u})}{\Phi(\overline{u})} > \frac{1}{\lambda}. \tag{7}$$

Hence, condition (*a*1) of Lemma 1 follows by combining (5) with (7). Next, we prove the coercivity of the functional *Iλ*. From (*iii*), there is an *ε* > 0 such that

$$\limsup\_{|\xi| \to +\infty} \frac{F(k, \xi)}{|\xi|} < \varepsilon < \frac{2F\_{\varepsilon}}{N(\sqrt{4c^2 + N + 1} - \sqrt{N + 1})}.$$

Then, there is a positive constant *h* such that

$$F(k, \xi) \le \varepsilon |\xi| + h$$

for each *ξ* ∈ R and *k* ∈ Z(1, *N*). By Lemma 3 and *λ* < −1+ q 1+ <sup>4</sup>*<sup>c</sup>* 2 *N*+1 *Fc* , one has

$$\begin{aligned} \lambda \sum\_{k=1}^{N} F(k, u(k)) &\leq \lambda \sum\_{k=1}^{N} \left[ \varepsilon |u(k)| + h \right] \\ &\leq \lambda \frac{\varepsilon N \sqrt{N+1}}{2} ||u|| + \lambda Nh \\ &\leq \frac{\varepsilon N \left( \sqrt{N+1+4c^2} - \sqrt{N+1} \right)}{2F\_c} ||u|| + \frac{\left( -1 + \sqrt{1 + \frac{4c^2}{N+1}} \right) Nh}{F\_c} \end{aligned}$$

for each *u* ∈ *S*. Therefore, it is true that

$$\begin{split} I\_{\lambda}(u) &= \Phi(u) - \lambda \Psi(u) \\ &= \sum\_{k=1}^{N+1} \left[ \sqrt{1 + (\Delta u(k-1))^2} - 1 \right] - \lambda \sum\_{k=1}^{N} F(k, u(k)) \\ &\geq \left[ \sum\_{k=1}^{N+1} \left[ 1 + (\Delta u(k-1))^2 \right] \right]^{\frac{1}{2}} - N - 1 - \lambda \sum\_{k=1}^{N} F(k, u(k)) \\ &\geq \left[ 1 - \frac{\varepsilon N \left( \sqrt{N+1+4c^2} - \sqrt{N+1} \right)}{2F\_{\mathbb{C}}} \right] \|u\| - N - 1 - \frac{\left( -1 + \sqrt{1 + \frac{4c^2}{N+1}} \right) Nh}{F\_{\mathbb{C}}}. \end{split}$$

Thus, we get that *I<sup>λ</sup>* is coercive, and condition (*a*2) of Lemma 1 is verified. In summary, all assumptions of Lemma 1 are proved, and so the functional *I<sup>λ</sup>* has at least three distinct critical points in *X* for each *λ* ∈ Λ. Since *u* ≡ 0 is not a solution to the (Problem 1), the proof is completed.

**Remark 1.** *It is obvious that the mean curvature operator φ<sup>c</sup> is odd symmetric (φc*(−*s*) = −*φc*(*s*)*, s* ∈ R*). When the nonlinear term f is also odd symmetric (f*(·, −*s*) = −*f*(·,*s*)*), the variational functional I<sup>λ</sup> is even symmetric (Iλ*(−*u*) = *Iλ*(*u*)*, u* ∈ *S). In this case, it is easy to obtain multiple solutions to (Problem 1) by using the critical point theory with symmetries. However, in this paper, we obtain multiple solutions to (Problem 1) without the symmetry on f .*

Now, let

$$F^+(k, \mathfrak{f}) = \int\_0^\mathfrak{f} f(k, t^+) dt, \quad (k, \mathfrak{f}) \in \mathbb{Z}(1, N) \times \mathbb{R}\_+$$

where *t* <sup>+</sup> <sup>=</sup> max{0, *<sup>t</sup>*} and define *<sup>I</sup>* + *<sup>λ</sup>* <sup>=</sup> <sup>Φ</sup> <sup>−</sup> *<sup>λ</sup>*Ψ+, with <sup>Φ</sup> defined in (2) and

$$\Psi^+(\mathfrak{u}) := \sum\_{k=1}^N F^+(k, \mathfrak{u}(k)).$$

It is well known that *I* + *λ* ∈ *C* 1 (*S*, R) and the critical points of *I* + *λ* are precisely the solutions of the following problem.

Problem 2

$$\begin{cases} -\Delta(\phi\_c\left(\Delta u(k-1)\right)) = \lambda f(k, u^+(k)), \; k \in \mathbb{Z}(1, N),\\ u(0) = u(N+1) = 0. \end{cases}$$

We have the following corollary.

**Corollary 1.** *Assume that there exist two positive constants c and d with*

$$2\sqrt{1+d^2} > 1 + \sqrt{1 + \frac{4c^2}{N+1}}$$

*such that*

$$f(i) \quad f(k, \xi) > 0 \text{ for each } k \in \mathbb{Z}(1, N) \text{ and } \xi \in [0, c];$$

$$\left(\left(\mathrm{ii}\right)\_{-\frac{F\_d}{2\left(\sqrt{1+d^2}-1\right)}} > \frac{F\_c}{-1 + \sqrt{1 + \frac{4c^2}{N+1}}}\right;$$

$$(iii)\ \limsup\_{\xi \to +\infty} \frac{F(k,\xi)}{\xi} < \frac{2F\_0}{N(\sqrt{4c^2 + N + 1} - \sqrt{N + 1})}.$$

*Then, for every λ* ∈ Λ*, (Problem 1) has at least three positive solutions.*

**Proof.** For each *k* ∈ Z(1, *N*), consider (Problem 2) with

$$f^+(k,\xi) = \begin{cases} f(k,\xi), & \text{if } \xi > 0, \\ f(k,0), & \text{if } \xi \le 0. \end{cases}$$

One has that condition (*i*) of Theorem 1 holds. Besides, we have

$$\limsup\_{\mathfrak{f}\to-\infty} \frac{F^+(k,\mathfrak{f})}{|\mathfrak{f}|} = \limsup\_{\mathfrak{f}\to-\infty} \frac{\mathfrak{f}f(k,0)}{|\mathfrak{f}|} = -f(k,0) < \frac{2\mathcal{F}\_{\mathfrak{c}}}{N(\sqrt{4c^2+N+1}-\sqrt{N+1})} \cdot \frac{1}{2}$$

So, all conditions of Theorem 1 are true. Moreover, since *u* ≡ 0 is not a solution of problem (*P* + *λ*, *f* ), we can get that (Problem 2) has at least three nontrivial solutions. Assume *u* = {*u*(*k*)} is one of the nontrivial solution, for any *k* ∈ Z(1, *N*), one has either *u*(*k*) > 0 or

$$-\Delta(\phi\_{\varepsilon}\left(\Delta u(k-1)\right)) = \lambda f(k, u^+(k)) = \lambda f(k, 0) > 0.$$

Then, we have *u* > 0 for all *k* ∈ Z(1, *N*) by Lemma 5, i.e., *u* is a positive solution. In addition, if *u* is a positive solution of (Problem 2), then *u* is a positive solution of (Problem 1) obviously and Corollary 1 is proved.

Next, we will use Lemma 2 to obtain another conclusion of this paper.

**Theorem 2.** *Assume that there exist two positive constants c and d with*

$$2\sqrt{1+d^2} < 1 + \sqrt{1 + \frac{4c^2}{N+1}}\tag{8}$$

*such that*

$$\begin{aligned} (T\_1) \quad f(k, \xi) &> 0 \text{ for each } k \in \mathbb{Z}(1, N) \text{ and } \xi \in [-c, c];\\ (T\_2) \quad \max \left\{ \frac{2(\sqrt{1+d^2}-1)}{F\_d}, \frac{2\sqrt{N(N+1)}\sin\frac{N\pi}{2(N+1)}}{\beta} \right\} &< \frac{-1 + \sqrt{1 + \frac{4c^2}{N+1}}}{F\_c}; \end{aligned}$$

*(T*3*) there exist a positive constant β such that*

$$\liminf\_{|t| \to +\infty} \frac{F(k, t)}{|t|} > \beta$$

*for each k* ∈ Z(1, *N*).

$$\begin{aligned} \text{Then, for every } \lambda \in \left( \max\left\{ \frac{2(\sqrt{1+d^2}-1)}{\mathbb{F}\_d}, \frac{2\sqrt{N(N+1)}\sin\frac{N\pi}{2(N+1)}}{\mathbb{F}} \right\}, \frac{-1+\sqrt{1+\frac{4t^2}{N+1}}}{\mathbb{F}\_\mathbb{C}} \right), \text{ (Problem 1) has at } \lambda \in \left( \mathbb{F}\_\lambda(u\_{\lambda,1}), \begin{cases} \frac{2}{\sqrt{1+d^2}-1} \le \frac{2}{\sqrt{N(N+1)}}, \frac{2}{\sqrt{N(N+1)}} \le \frac{2}{\sqrt{N(N+1)}}, \\ 1-\frac{2}{\sqrt{1+d^2}-1} \le \frac{2}{\sqrt{N(N+1)}}, \end{cases} \right). \end{aligned}$$

**Proof.** Clearly, Φ, Ψ are two continuously Gâteaux differentiable functionals and

$$\inf\_{X} \Phi = \Phi(0) = \Psi(0) = 0.$$

Let *u* and *r* be the same as the ones defined in the proof of Theorem 1. Then we have

$$\frac{\sup\_{\Psi(u)\le r} \Psi(u)}{r} \le \frac{F\_c}{-1 + \sqrt{1 + \frac{4c^2}{N+1}}} < \frac{1}{\lambda}$$

and

$$\frac{\Psi(\overline{\mu})}{\Phi(\overline{\mu})} = \frac{F\_d}{2(\sqrt{1+d^2}-1)} > \frac{1}{\lambda}.$$

So inequality (1) in Lemma 2 holds. Besides, form (*T*3) and Lemma 4, there is a constant *h* such that, for any *u* ∈ *S* and *k* ∈ Z(1, *N*),

$$\begin{aligned} \lambda \sum\_{k=1}^N F(k, u(k)) &\geq \lambda \sum\_{k=1}^N (\beta |u(k)| - h) \\ &\geq \lambda \beta ||u||\_{\infty} - \lambda Nh \\ &\geq \frac{\lambda \beta}{\sqrt{N\lambda \lambda\_N}} ||u|| - \lambda Nh. \end{aligned}$$

#### Therefore, we have

$$\begin{split} \Lambda\_{\lambda}(u) &= \Phi(u) - \Psi(u) \\ &= \sum\_{k=1}^{N+1} \left[ \sqrt{1 + (\Delta u(k-1))^2} - 1 \right] - \lambda \sum\_{k=1}^{N} F(k, u(k)) \\ &\leq \sum\_{k=1}^{N+1} \left[ 1 + \sqrt{(\Delta u(k-1))^2} - 1 \right] - \frac{\lambda \beta}{\sqrt{N\lambda\lambda\_N}} ||u|| + \lambda Nh \\ &\leq \left( \sum\_{k=1}^{N+1} 1^2 \right)^{\frac{1}{2}} \left( \sum\_{k=1}^{N+1} (\Delta u(k-1))^2 \right)^{\frac{1}{2}} - \frac{\lambda \beta}{\sqrt{N\lambda\_N}} ||u|| + \lambda Nh \\ &\leq \left( \sqrt{N+1} - \frac{\lambda \beta}{\sqrt{N\lambda\_N}} \right) ||u|| + \lambda Nh \\ &= \left( \sqrt{N+1} - \frac{\lambda \beta}{2\sqrt{N}\sin\frac{N\pi}{2(N+1)}} \right) ||u|| + \lambda Nh. \end{split}$$

Thus, we see that lim k*u*k→+∞ *Iλ*(*u*) = −∞, which means that the functional *I<sup>λ</sup>* is unbounded from below. Moreover, we can get that −*I<sup>λ</sup>* is coercive. Therefore, *I<sup>λ</sup>* satisfies the (PS)-condition and the proof is completed.

#### **4. Examples**

**Example 1.** *Fix N* ∈ Z(1, 20) *and consider the boundary value (Problem 1) with*

$$f(k,u) = f(u) = \begin{cases} 3u^2 + \frac{1}{4}\cos u, & \text{if } u \le \frac{3\pi}{2},\\ \frac{27\pi^2}{4}\cos(u + \frac{\pi}{2}), & \text{if } u > \frac{3\pi}{2}. \end{cases}$$

*for k* ∈ Z(1, *N*)*. Then, we have*

$$F(k,\xi) = F(\xi) = \begin{cases} \xi^3 + \frac{1}{4}\sin\xi, & \text{if } \xi \le \frac{3\pi}{2},\\ \frac{27\pi^3}{8} - \frac{1}{4} + \frac{27\pi^2}{4}\sin(\xi + \frac{\pi}{2}), & \text{if } \xi > \frac{3\pi}{2}. \end{cases}$$

*Let c* = <sup>1</sup> 2 *and d* = 4*. Then one has*

$$2\sqrt{1+d^2} = 2\sqrt{17} > 1 + \sqrt{1+\frac{1}{N+1}} = 1 + \sqrt{1+\frac{4c^2}{N+1}}.$$

*In addition, as shown in Figure 1, <sup>f</sup>*(*k*, *<sup>ξ</sup>*) <sup>&</sup>gt; <sup>0</sup> *for each <sup>ξ</sup>* <sup>∈</sup> [0, <sup>1</sup> 2 ]*. Thus condition* (*i*) *of Corollary 1 follows. Moreover, we have*

$$\frac{F\_c}{-1 + \sqrt{1 + \frac{4e^2}{N+1}}} = \frac{N(1 + 2\sin\frac{1}{2})}{8\left(-1 + \sqrt{1 + \frac{1}{N+1}}\right)} \le \frac{N(1 + 2\sin\frac{1}{2})}{8\left(-1 + \sqrt{\frac{21}{20}}\right)} \approx 9.915N\tag{9}$$

*and*

$$\frac{F\_d}{2(\sqrt{1+d^2}-1)} = \frac{N(256+\sin 4)}{8(\sqrt{17}-1)} \approx 10.216N. \tag{10}$$

*Since*

$$\frac{256 + \sin 4}{8(\sqrt{17} - 1)} - \frac{1 + 2\sin\frac{1}{2}}{8\left(-1 + \sqrt{\frac{21}{20}}\right)} > 0,$$

*by combining (9) and (10), condition* (*ii*) *of Corollary 1 holds.*

**Figure 1.** The images of *f*(*u*) and *F*(*u*) in Example 1.

*We can further verify condition* (*iii*) *of Corollary 1, since*

$$\limsup\_{\substack{\xi \to +\infty \\ \xi \to +\infty}} \frac{F(k,\xi)}{\xi} = \limsup\_{\substack{\xi \to +\infty \\ \xi \to +\infty}} \frac{\frac{27\pi^3}{8} - \frac{1}{4} + \frac{27\pi^2}{4}\sin(\xi + \frac{\pi}{2})}{\xi} = 0 < \frac{2F\_{\mathbb{C}}}{N(\sqrt{4c^2 + N + 1} - \sqrt{N + 1})},$$

*To sum up, all the conditions of Corollary 1 are satisfied. Hence, for*

$$
\lambda \in \Lambda = \left( \frac{8(\sqrt{17}-1)}{N(256+\sin 4)}, \frac{8\left(-1+\sqrt{1+\frac{1}{N+1}}\right)}{N(1+2\sin\frac{1}{2})} \right) \sqrt{2}
$$

*the boundary value problem admits at least three positive solutions.*

**Example 2.** *Let N* = 3 *and β* = 10*, consider the boundary value (Problem 1) with*

$$f(k,u) = f(u) = \begin{cases} \frac{1}{6} \left[ (3 - 2u) \cos \frac{u}{2} + e^{u-2} \right], & \text{if } u \le 2\pi, \\\frac{1}{6} \left( 4\pi - 3 + e^{2\pi - 2} \right), & \text{if } u > 2\pi, \end{cases}$$

*for each k* ∈ Z(1, *N*). *Then, we have*

$$F(k,\xi) = F(\xi) = \begin{cases} \frac{1}{3}\left[\left(3 - 2\xi\right)\sin\frac{\xi}{2} + 4\left(1 - \cos\frac{\xi}{2}\right)\right] + \frac{\varepsilon^{-2}}{6}\left(e^{\xi} - 1\right), & \text{if } \xi \le 2\pi, \\\frac{1}{6}\left(4\pi - 3 + e^{2\pi - 2}\right)\xi - \frac{\pi}{3}\left(4\pi - 3 + e^{2\pi - 2}\right) + \frac{\varepsilon^{-2}}{6}\left(e^{2\pi - 1}\right) + \frac{8}{3}, & \text{if } \xi > 2\pi. \end{cases}$$

*Moreover, the images of f*(*u*) *and F*(*u*) *are shown in Figure 2. Letting c* = √ *N* + 1 = 2 *and d* = <sup>1</sup> 2 *, we have*

$$12\sqrt{1+d^2} = \sqrt{5} < 1 + \sqrt{5} = 1 + \sqrt{1 + \frac{4c^2}{N+1}}.$$

*Now, we prove f*(*u*) > 0 *for each u* ∈ [−2, 2]*. Let*

$$f\_1(\mu) = (3 - 2\mu)\cos\frac{\mu}{2} \quad \text{and} \quad f\_2(\mu) = e^{\mu - 2}.$$

**Figure 2.** The images of *f*(*u*) and *F*(*u*) in Example 2.

*Then f*(*u*) = <sup>1</sup> 6 (*f*1(*u*) + *f*2(*u*)) *for u* ≤ 2*π. Besides, it is obvious that f*2(*u*) > 0 *on* [−2, 2] *and <sup>f</sup>*1(*u*) <sup>≥</sup> <sup>0</sup> *on* - <sup>−</sup>2, <sup>3</sup> 2 *. When u* ∈ - 3 2 , 2 *, we have*

$$f\_1'(u) = \frac{1}{2} \left[ (2u - 3) \sin \frac{u}{2} - 4 \cos \frac{u}{2} \right] \le \frac{1}{2} (\sin 1 - 4 \cos 1) < 0.$$

*Thus f*1(*u*) *is decreasing on* [ 3 2 , 2]*. Since f*2(*u*) *is an increasing function and f*1(2) + *f*2( 3 2 ) = − cos 1 + *e* − 1 <sup>2</sup> > 0*, we can get f*(*u*) > 0 *on* [−2, 2]*. In addition, we have*

$$F\_d = \sum\_{k=1}^{N} F(k, d) = N \int\_0^d \frac{1}{6} \left[ (3 - 2u) \cos \frac{u}{2} + e^{u - 2} \right] du \approx 0.6634$$

*and*

$$F\_{\mathcal{C}} = \sum\_{k=1}^{N} F(k, \mathcal{c}) = N \int\_{0}^{\mathcal{C}} \frac{1}{6} \left[ (3 - 2u) \cos \frac{u}{2} + e^{u - 2} \right] du \approx 1.430...$$

*It follows that*

$$\frac{2(\sqrt{1+d^2}-1)}{F\_d} = \frac{\sqrt{5}-2}{F\_d} \approx 0.356\sqrt{5}$$

*and*

$$\frac{-1 + \sqrt{1 + \frac{4c^2}{N+1}}}{F\_c} = \frac{\sqrt{5} - 1}{F\_c} \approx 0.865.$$

*The above together with*

$$\frac{2\sqrt{N(N+1)}\sin\frac{N\pi}{2(N+1)}}{\beta} = \frac{4\sqrt{3}\sin\frac{3\pi}{8}}{10} \approx 0.640$$

*confirms condition* (*T*2) *of Theorem 2.*

*Besides,*

$$\begin{split} \liminf\_{|\tilde{\xi}| \to +\infty} \frac{F(k, \tilde{\xi})}{\tilde{\xi}} &= \liminf\_{\tilde{\xi} \to +\infty} \frac{\frac{1}{6} \left( 4\pi - 3 + e^{2\pi - 2} \right) \xi - \frac{\pi}{3} (4\pi - 3 + e^{2\pi - 2}) + \frac{e^{-2}}{6} (e^{2\pi - 1}) + \frac{8}{3} \xi} \\ &= \frac{1}{6} \left( 4\pi - 3 + e^{2\pi - 2} \right) \\ &\approx 13.673 > \beta \end{split}$$

*for each k* ∈ Z(1, *N*)*. We can see from Theorem 2 that the boundary value problem has at least two non-zero solutions for each λ* ∈ 4 √ 3 sin <sup>3</sup>*<sup>π</sup>* 8 <sup>10</sup> , √ 5−1 3 R 2 0 1 6 [(3−2*u*) cos *<sup>u</sup>* <sup>2</sup> +*e u*−2 ]*du* ≈ (0.640, 0.865)*.*

#### **5. Conclusions**

A discrete Dirichlet boundary value problem involving the mean curvature operator is studied in this paper. Unlike the existing result in [33], we obtained different sufficient conditions of the existence of multiple solutions without assuming that the nonlinear term oscillates at infinity, as shown in Theorems 1 and 2. First, according to the research results of Bonanno in [22], we obtain at least three non-trivial solutions in Theorem 1. In addition, as a supplement to Theorem 1, we prove the existence of at least three positive solutions through the maximum principle. Note that inequality (3) plays an important role in the proof of Theorem 1. For the situation that inequality (3) is not satisfied, under another suitable assumption on the nonlinear term, we still can obtain the existence of at least two non-trivial solutions based on Theorem 2.1 in [23]. It seems that the method used in this paper can be adapted to discuss the existence of homoclinic solutions or periodic solutions of difference equations with *φc*-Laplacian. This will be left as our future work.

**Author Contributions:** All authors contributed equally and significantly in writing this paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is supported by the National Natural Science Foundation of China (Grant No. 11971126) and the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT 16R16).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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*Article*

## **On the Absolute Stable Difference Scheme for Third Order Delay Partial Differential Equations**

#### **Allaberen Ashyralyev 1,2,3,**<sup>∗</sup> **, Evren Hincal <sup>1</sup> and Suleiman Ibrahim <sup>1</sup>**


Received: 16 April 2020; Accepted: 15 June 2020; Published: 19 June 2020

**Abstract:** The initial value problem for the third order delay differential equation in a Hilbert space with an unbounded operator is investigated. The absolute stable three-step difference scheme of a first order of accuracy is constructed and analyzed. This difference scheme is built on the Taylor's decomposition method on three and two points. The theorem on the stability of the presented difference scheme is proven. In practice, stability estimates for the solutions of three-step difference schemes for different types of delay partial differential equations are obtained. Finally, in order to ensure the coincidence between experimental and theoretical results and to clarify how efficient the proposed scheme is, some numerical experiments are tested.

**Keywords:** time delay; third order differential equations; difference scheme; stability

#### **1. Introduction**

Various problems in elasticity theory such as the problems of the longitudinal oscillations of a non-uniform viscoelastic rod, the problem of the longitudinal impact of a perfectly rigid body on a non-uniform finite-length viscoelastic rod with a variable cross-section, problems of wave propagation in a visco-elastic body, etc., lead to third order differential equations without the time delay term ([1–3]). Over the years, nonlocal and local boundary value problems have been of great interest due to their importance in the fields of engineering and science, especially in applied mathematics. Such problems have formed various research fields. Several nonlocal and local boundary value problems for differential equations have been investigated extensively in various works (for example, see [4–12] and the references given therein).

Differential equations having a delay term are used to model sociological, biological, as well as physical processes. They are used to model naturally occurring oscillation systems. A typical example of the occurrence of time delay can be seen in a sampled data control in control theory (see, for example, [13–17]). The presence of delay term in differential equations usually leads to difficulties in analyzing the differential equation. The boundedness and stability and the oscillation property of solutions for a third order delay ordinary differential and difference problems were widely studied (for example, see [18–25] and the references given therein).

Delay partial differential equations (DPDEs) arise in many applications such as control theory, climate models, medicine, biology, and much more (for example, see [26] and the references therein). The independent variables of partial differential equations having delay terms are time *t* together with one or more dimensional variable *x*, representing the position in space. It can also stand for the size of cells, relative DNA content, their level of mutation, as well as other parameters. The

solutions of partial differential equations having delay terms may stand for voltage, temperature, or densities or concentrations of various particles, for instance chemicals, cells, animals, bacteria, and so on. Numerical methods for partial differential equations with delay terms usually lead to specific difficulties, which are usually not present in equations without delay terms. The theory and applications of parabolic and hyperbolic partial differential equations having a time delay term were studied by numerous authors (for example, see [13,27–34] and the references given therein). Recent publications on third order DPDEs are not many.

Several physical models lead to initial-boundary value problems for third order DPDEs (see, e.g., [1,3,8]). It is known that such types of problems can be replaced with the initial value problem for a third order delay differential equation:

$$\begin{cases} \left. u\_{ttt}(t) + Au\_l(t) = bAu(t - w) + f(t), \, 0 < t < \infty, \\\left. u(t) = g(t), \, -w \le t \le 0 \end{cases} \right. \tag{1}$$

in a Hilbert space *H* with unbounded operator *A*. Here, *b* ∈ *R* 1 . Assume that *f*(*t*) is a continuous function on [0, ∞) and *f*(*t*) ∈ *D*(*A* 1/2), *<sup>g</sup>*(*t*) is a twice continuously differentiable function on [−*w*, 0] and *g* (*k*) (*t*) ∈ *D*(*A* (3−*k*)/2) for *k* = 0, 1, 2.

Let us give the main theorem of paper [35].

**Theorem 1.** *The solution of Problem (1) satisfies the stability estimates:*

$$a\_1 \le da\_0 + \int\_0^w \left\| f(s) \right\|\_{D(A^{\frac{1}{2}})} ds,\\ d = 2 + |b| \, w, \tag{2}$$

$$a\_{n+1} \le da\_n + \sum\_{j=1}^{n+1} \int\_{(j-1)\omega}^{jw} \left\Vert f(s) \right\Vert\_{D(A^{\frac{1}{2}})} ds, n = 1, 2, \cdots,\tag{3}$$

$$a\_0 = \max\left\{ \max\_{-w \le t \le 0} \left\| g\_{tt}(t) \right\|\_{D(A^{\frac{1}{2}})'} \max\_{-w \le t \le 0} \left\| g\_t(t) \right\|\_{D(A)'} \max\_{-w \le t \le 0} \left\| g(t) \right\|\_{D(A^{\frac{2}{2}})} \right\},$$

$$a\_{\mathcal{U}} = \max\left\{ \max\_{(n-1)w \le t \le nw} \left\| u\_{\mathcal{U}}(t) \right\|\_{D(A^{\frac{1}{2}})'} \max\_{(n-1)w \le t \le nw} \left\| u\_t(t) \right\|\_{D(A)'} \max\_{(n-1)w \le t \le nw} \left\| u(t) \right\|\_{D(A^{\frac{2}{2}})} \right\}.$$

In practice, stability estimates for the solution of several problems for third order DPDEs were obtained.

Moreover, publications on the theory and applications of difference schemes (DSs) for third order DPDEs are not available. Thus, the construction and investigation of stable DSs for the approximate solutions of third order DPDEs is of great importance. Our aim in this paper is to construct the absolute stable three-step DS of the first order of accuracy of the third order DPDE for the approximate solution of the problem (1). We consider the uniform set of grid points:

$$(-w, \infty)\_{\tau} = \{ t\_k : t\_k = k\tau, -N \le k < \infty, N\tau = \omega \}$$

with step *τ* > 0. Applying Taylor's decomposition method on three and two points (see [36,37]), we present the DS of the first order of accuracy:

$$\begin{cases} \frac{u\_{k+2} - 3u\_{k+1} + 3u\_{k} - u\_{k-1}}{\tau^3} + A \frac{u\_{k+2} - u\_{k+1}}{\tau} = bAu\_{k-N} + f(t\_k), \; k \ge 1, \\\ u\_{k} = g(t\_k), \; -N + 1 \le k \le 0, \\\ (I + \tau^2 A) \frac{u\_{1} - u\_{0}}{\tau} = g'(0), (I + \tau^2 A) \frac{u\_{2} - 2u\_{1} + u\_{0}}{\tau^2} = g''(0), \\\ (I + \tau^2 A) \frac{u\_{mN+1} - u\_{mN}}{\tau} = \frac{u\_{mN} - u\_{mN-1}}{\tau}, \\\ (I + \tau^2 A) \frac{u\_{mN+2} - 2u\_{mN+1} + u\_{mN}}{\tau^2} = \frac{u\_{mN} - 2u\_{mN-1} + u\_{mN-2}}{\tau^2}, m = 1, 2, \dots \end{cases} (4)$$

for the approximate solution of Problem (1).

The organization of this paper is as follows. In Section 2, the main theorem on the stability of DS (4) is established. In Section 3, stability estimates of DSs for the approximate solution of three problems for third order DPDEs are obtained. Numerical results are provided for one= and two=dimensional third order DPDEs in Section 4. Finally, Section 5 gives the conclusion and our future plans.

#### **2. Stability of DS**

All over the present paper, assume that *H* is a Hilbert space and *A* is a self-adjoint positive definite operator *A* ≥ *δI* in *H* and *R* = (*I* − *iτA* 1 2 ) −1 , *R*e = (*I* + *iτA* 1 2 ) −1 .

Note that three-step DS (4) can obviously be rewritten as the system of single-step and two-step delay DS:

$$\begin{cases} \frac{\underline{u}\_{k+1} - \underline{u}\_{k}}{\tau} = v\_{k}, k \ge 0, \\\\ \frac{v\_{k+1} - 2\underline{v}\_{k} + v\_{k-1}}{\tau^{2}} + Av\_{k+1} = p\_{k}, p\_{k} = bu\_{k-N} + f(t\_{k}), \; k \ge 1, \\\\ u\_{k} = g(t\_{k}), \; -N \le k \le 0, \\\\ (I + \tau^{2}A)v\_{0} = g'(0), (I + \tau^{2}A)\frac{v\_{1} - v\_{0}}{\tau} = g''(0), \\\\ (I + \tau^{2}A)v\_{mN} = \frac{u\_{mN} - u\_{mN-1}}{\tau}, \\\\ (I + \tau^{2}A)\frac{v\_{mN+1} - v\_{mN}}{\tau} = \frac{u\_{mN} - 2u\_{mN-1} + u\_{mN-2}}{\tau^{2}}, m = 1, 2, \dots \end{cases} (5)$$

for the solution of DS (4). Applying DC(5), we can obtain the formula for the solution of DS (4). For this, we will consider two cases 1 ≤ *k* ≤ *N* and *mN* + 1 ≤ *k* ≤ (*m* + 1)*N*, *m* = 1, 2, · · · , separately.

Let 1 ≤ *k* ≤ *N*. Applying (5), we get the following DS:

$$\begin{cases} \frac{\underline{u\_{k}} - \underline{u\_{k-1}}}{\tau} = v\_{k-1}, 1 \le k \le N, \\\frac{\underline{v\_{k+1}} - 2\underline{v\_{k}} + \underline{v\_{k-1}}}{\tau^2} + Av\_{k+1} = p\_k, p\_k = bu\_{k-N} + f(t\_k), 1 \le k \le N - 1, \\\ u\_k = g(t\_k), \ -N \le k \le 0, \\\ (I + \tau^2 A)v\_0 = g'(0), (I + \tau^2 A)\frac{v\_1 - v\_0}{\tau} = g''(0). \end{cases} \tag{6}$$

Therefore, we have that (see [38]):

$$
\mu\_0 = \mathcal{g}(0), \mu\_k = \mathcal{g}(0) + \sum\_{j=0}^{k-1} \tau v\_{j\prime} \ 1 \le k \le N\prime \tag{7}
$$

$$\begin{split} v\_{0} &= R\widetilde{R}g'(0), \ v\_{1} = R\widetilde{R}g'(0) + \tau R\widetilde{R}g''(0), \\ v\_{k} &= \frac{1}{2}[R^{k-1} + \widetilde{R}^{k-1}]R\widetilde{R}g'(0) + \frac{1}{2!}A^{-\frac{1}{2}}R(R^{k} - \widetilde{R}^{k})g''(0) \\ &\quad - \sum\_{s=1}^{k-1} \frac{\tau}{2}A^{-\frac{1}{2}}[R^{k-s} - \widetilde{R}^{k-s}]p\_{s} \\ &= \frac{1}{2}[R^{k-1} + \widetilde{R}^{k-1}]R\widetilde{R}g'(0) + \frac{1}{2!}A^{-\frac{1}{2}}R(R^{k} - \widetilde{R}^{k})g''(0) \\ &+ A^{-1}\left\{\frac{1}{2}\sum\_{s=2}^{k-1}[R^{k-s} + \widetilde{R}^{k-s}](p\_{s-1} - p\_{s}) + 2p\_{k-1} - [R^{k-1} + \widetilde{R}^{k-1}]p\_{1}\right\}, 2 \le k \le N, \\ p\_{k} &= bA g(t\_{k-N}) + f(t\_{k}), \ 1 \le k \le N - 1. \end{split} \tag{8}$$

Applying Formulas (6)–(8), we obtain:

$$u\_{k} = \begin{cases} g(0) + \tau R \tilde{R} g'(0), k = 0, \\ g(0) + \tau R \tilde{R} g'(0), k = 1, \\ g(0) + 2\tau R \tilde{R} g'(0) + \tau^{2} R \tilde{R} g''(0), k = 2, \\ g(0) + \frac{1}{2i} A^{-\frac{1}{2}} \left( R^{k-2} - \tilde{R}^{k-2} \right) R \tilde{R} g'(0) - \frac{1}{2} A^{-1} R \left( 2R \tilde{R} - \left( R^{k-1} + \tilde{R}^{k-1} \right) \right) g''(0), \\ \quad + \sum\_{j=2}^{k-1} \tau \sum\_{s=1}^{j} \frac{1}{2i} A^{-\frac{1}{2}} [R^{j-s} - \tilde{R}^{j-s}] p\_{s\prime}, 3 \le k \le N. \end{cases}$$

By an interchange of the order of summation, we get:

$$u\_{k} = \begin{cases} g(0) + \tau R \tilde{\mathcal{R}} g'(0), k = 0, \\ g(0) + \tau R \tilde{\mathcal{R}} g'(0), k = 1, \\ g(0) + 2\tau R \tilde{\mathcal{R}} g'(0) + \tau^2 R \tilde{\mathcal{R}} g''(0), k = 2, \\ g(0) + \frac{1}{2!} A^{-\frac{1}{2}} \left( R^{k-2} - \tilde{R}^{k-2} \right) R \tilde{\mathcal{R}} g'(0) - \frac{1}{2} A^{-1} R \left( 2R \tilde{\mathcal{R}} - \left( R^{k-1} + \tilde{R}^{k-1} \right) \right) g''(0) \\ + A^{-1} \sum\_{s=1}^{k-2} \frac{\tau}{2} [2I - \left( R^{k-1-s} + \tilde{R}^{k-1-s} \right)] p\_{s\prime}, 3 \le k \le N \end{cases} \tag{9}$$

for the solution of DS (4).

Let 1 + *mN* ≤ *k* ≤ (*m* + 1)*N*, *m* = 1, 2, · · · . Applying (5), we can get the DS:

$$\begin{cases} \frac{u\_k - u\_{k-1}}{\tau} = v\_{k-1} mN + 1 \le k \le (m+1)N, \\\frac{v\_{k+1} - 2v\_k + v\_{k-1}}{\tau^2} + Av\_{k+1} = p\_k, p\_k = bu\_{k-N} + f(t\_k), \\\ mN + 1 \le k \le (m+1)N - 1, \\\ u\_{mN} \text{ is given}, (I + \tau^2 A)v\_{mN} = \frac{u\_{mN} - u\_{mN-1}}{\tau}, \\\ (I + \tau^2 A)\frac{v\_{mN+1} - v\_{mN}}{\tau} = \frac{u\_{mN} - 2u\_{mN-1} + u\_{mN-2}}{\tau^2}. \end{cases} \tag{10}$$

Therefore, we have that (see [38]):

$$u\_k = u\_{mN} + \sum\_{j=mN+2}^{k-1} \tau v\_{j\prime} mN + 1 \le k \le (m+1)N,\tag{11}$$

$$\begin{split} v\_{mN} &= R\widetilde{\mathcal{R}}\frac{u\_{mN} - u\_{mN-1}}{\tau}, v\_{mN+1} = R\widetilde{\mathcal{R}}\frac{u\_{mN} - u\_{mN-1}}{\tau} + \tau R\widetilde{\mathcal{R}}\frac{u\_{mN} - 2u\_{mN-1} + u\_{mN-2}}{\tau^2} \\ &\quad v\_{k} = \frac{1}{2}[R^{k-mN-1} + \widetilde{\mathcal{R}}^{k-mN-1}]R\widetilde{\mathcal{R}}\frac{u\_{mN} - u\_{mN-1}}{\tau} \\ &\quad + \frac{1}{2}A^{-\frac{1}{2}}R(R^{k-mN} - \widetilde{\mathcal{R}}^{k-mN})\frac{u\_{mN} - 2u\_{mN-1} + u\_{mN-2}}{\tau^2} \\ &\quad - \sum\_{s=mN+1}^{k-1} \frac{\tau}{2}A^{-\frac{1}{2}}[R^{k-s} - \widetilde{\mathcal{R}}^{k-s}]p\_{s} \\ &= \frac{1}{2}[R^{k-1} + \widetilde{R}^{k-1}]R\widetilde{\mathcal{R}}\frac{u\_{mN} - u\_{mN-1}}{\tau} + \frac{1}{2\tau}A^{-\frac{1}{2}}R(R^{k} - \widetilde{R}^{k})g^{\prime\prime}(0) \\ &\quad + A^{-1}\left\{\frac{1}{2}\sum\_{s=mN+2}^{k-1}[R^{k-s} + \widetilde{R}^{k-s}](p\_{s-1} - p\_{s}) \\ &\quad + 2p\_{k-mN-1} - [R^{k-mN-1} + \widetilde{R}^{k-mN-1}]p\_{mN+1}\right\}, mN+2 \le k \le (m+1)N. \end{split} \tag{12}$$

Applying Formulas (10)–(12), we can obtain:

$$u\_{k} = \begin{cases} u\_{mN} + \tau R \tilde{\mathcal{R}} \frac{u\_{mN} - u\_{mN-1}}{\tau}, k = mN + 1, \\ u\_{mN} + 2\tau R \tilde{\mathcal{R}} \frac{u\_{mN} - u\_{mN-1}}{\tau} + \tau^{2} R \tilde{\mathcal{R}} \frac{u\_{mN} - 2u\_{mN-1} + u\_{mN-2}}{\tau^{2}}, k = mN + 2, \\ u\_{mN} + \frac{1}{2i} A^{-\frac{1}{2}} \left( R^{k-mN-2} - \widetilde{\mathcal{R}}^{k-mN-2} \right) R \tilde{\mathcal{R}} \frac{u\_{mN} - u\_{mN-1}}{\tau} \\ \quad - \frac{1}{2} A^{-1} R \left( 2R \widetilde{\mathcal{R}} - \left( R^{k-mN-1} + \widetilde{\mathcal{R}}^{k-mN-1} \right) \right) \frac{u\_{mN} - 2u\_{mN-1} + u\_{mN-2}}{\tau^{2}} \\ \quad + \sum\_{j=mN+2}^{k-1} \tau \sum\_{s=mN+1}^{\top} \frac{\tau}{2i} A^{-\frac{1}{2}} [R^{j-s} - \widetilde{\mathcal{R}}^{j-s}] p\_{s\prime}, mN + 3 \le k \le (m+1)N. \end{cases}$$

By an interchange of the order of summation, we get the solution of DS (4):

$$u\_{k} = \begin{cases} u\_{mN} + \tau R \tilde{\mathcal{R}} \frac{u\_{mN} - u\_{mN-1}}{\tau}, k = mN + 1, \\ u\_{mN} + 2\tau R \tilde{\mathcal{R}} \frac{u\_{mN} - u\_{mN-1}}{\tau} + \tau^{2} R \tilde{\mathcal{R}} \frac{u\_{mN} - 2u\_{mN-1} + u\_{mN-2}}{\tau^{2}}, k = mN + 2, \\ u\_{mN} + \frac{1}{2i} A^{-\frac{1}{2}} \left( R^{k-mN-2} - \tilde{\mathcal{R}}^{k-mN-2} \right) R \tilde{\mathcal{R}} \frac{u\_{mN} - u\_{mN-1}}{\tau} \\ \quad - \frac{1}{2} A^{-1} R \left( 2R \tilde{\mathcal{R}} - \left( R^{k-mN-1} + \tilde{\mathcal{R}}^{k-mN-1} \right) \right) \frac{u\_{mN} - 2u\_{mN-1} + u\_{mN-2}}{\tau^{2}} \\ \quad + A^{-1} \sum\_{s=mN+1}^{k-2} \frac{\tau}{2} [2I - \left( R^{k-1-s} + \tilde{R}^{k-1-s} \right)] p\_{s\prime}, mN + 3 \le k \le (m+1)N. \end{cases} \tag{13}$$

The following lemma will be needed in the sequel.

**Lemma 1.** *The following estimates are fulfilled:*

$$\|\|R\|\|\_{H\to H\prime} \parallel \widetilde{R} \|\|\_{H\to H} \leq 1\,\tag{14}$$

$$\|\|\boldsymbol{R}\widetilde{\boldsymbol{R}}^{-1}\|\|\_{\boldsymbol{H}\to\boldsymbol{H}\prime} \; \|\|\widetilde{\boldsymbol{R}}\boldsymbol{R}^{-1}\|\|\_{\boldsymbol{H}\to\boldsymbol{H}} \leq 1,\tag{15}$$

$$\|\operatorname{\tau}A^{\frac{1}{2}}\mathcal{R}\|\_{H\to H'}\|\operatorname{\tau}A^{\frac{1}{2}}\tilde{\mathcal{R}}\|\_{H\to H}\le 1.\tag{16}$$

The proof of the estimates (14)–(16) is based on the spectral theory of a self-adjoint operator in a Hilbert space [39].

Now, let us study the stability of DS (4).

**Theorem 2.** *The solution of DS (4) satisfies the following stability estimates:*

$$b\_1 \le (2 + \tau |b|(N-2))b\_0 + \tau \sum\_{s=1}^{N-2} \|A^{\frac{1}{2}} f(t\_s)\|\_{H\nu} \tag{17}$$

$$b\_{m+1} \le (2+\tau|b|(N-2))b\_m + \tau \sum\_{s=mN+1}^{(m+1)N} \|A^{\frac{1}{2}}f(t\_s)\|\_{H^s} \tag{18}$$

$$b\_0 = \max\left\{ \max\_{-N \le k \le 0} \|A^{\frac{1}{2}} g\_{lt}(t\_k)\|\_{H^\prime} \max\_{-N \le k \le 0} \|Ag\_{lt}(t\_k)\|\_{H^\prime} \max\_{-N \le k \le 0} \|A^{\frac{2}{2}} g(t\_k)\|\_{H} \right\},$$

$$b\_m = \max\left\{ \max\_{(m-1)N \le k \le mN-2} \|A^{\frac{1}{2}} \frac{u\_{k+2} - 2u\_{k+1} + u\_k}{\tau^2} \|\_{H^\prime} \right\}$$

$$\max\_{(m-1)N+1 \le k \le mN} \|A \frac{u\_k - u\_{k-1}}{\tau} \|\_{H^\prime} \frac{1}{2} \max\_{(m-1)N \le k \le mN} \|A^{\frac{2}{2}} u\_k \|\_{H} \right\}, m = 1, 2, \dots, \dots$$

**Proof.** Let us estimate *b*1. Using Formula (9) and the estimates (14)–(16), we get that:

$$\|\|A^{\frac{3}{2}}u\_1\|\|\_{H} \le \|\|A^{\frac{3}{2}}\mathcal{g}(\mathbf{0})\|\|\_{H} + \|\|\tau A^{\frac{1}{2}}R\|\|\_{H \to H} \|\|\tilde{R}\|\|\_{H \to H} \|\|A\mathbf{g}\_t(\mathbf{0})\|\|\_{H} \le 2b\_{0\lambda}$$

k*A* 3 <sup>2</sup> *u*2k*<sup>H</sup>* ≤ k*A* 3 <sup>2</sup> *g*(0)k*<sup>H</sup>* + 2k*τA* 1 <sup>2</sup> *R*k*H*→*H*k*R*ek*H*→*H*k*Agt*(0)k*<sup>H</sup>* +k*τA* 1 <sup>2</sup> *R*k*H*→*H*k*τA* 1 <sup>2</sup> *R*ek*H*→*H*k*A* 1 <sup>2</sup> *gtt*(0)k*<sup>H</sup>* ≤ 4*b*0, k*A* 3 <sup>2</sup> *uk*k*<sup>H</sup>* ≤ k*A* 3 <sup>2</sup> *g*(0)k*<sup>H</sup>* + 1 2 [k*R*k *k*−2 *<sup>H</sup>*→*<sup>H</sup>* <sup>+</sup> <sup>k</sup>*R*e<sup>k</sup> *k*−2 *H*→*H* ]k*RR*ek*H*→*H*k*Agt*(0)k*<sup>H</sup>* + 1 2 k*R*k*H*→*H*[2k*RR*ek*H*→*<sup>H</sup>* + k*R*k *k*−1 *<sup>H</sup>*→*<sup>H</sup>* <sup>+</sup> <sup>k</sup>*R*e<sup>k</sup> *k*−1 *H*→*H* ]k*A* 1 <sup>2</sup> *gtt*(0)k*<sup>H</sup>* + *τ* 2 |*b*| *k*−2 ∑ *s*=1 [2 + k*R*k *k*−1−*s <sup>H</sup>*→*<sup>H</sup>* <sup>+</sup> <sup>k</sup>*R*e<sup>k</sup> *k*−1−*s H*→*H* ]k*A* 3 <sup>2</sup> *g*(*ts*−*N*)k*<sup>H</sup>* + *τ* 2 *k*−2 ∑ *s*=1 [2 + k*R*k *k*−1−*s <sup>H</sup>*→*<sup>H</sup>* <sup>+</sup> <sup>k</sup>*R*e<sup>k</sup> *k*−1−*s H*→*H* ]k*A* 1 <sup>2</sup> *f*(*ts*)k*<sup>H</sup>* ≤ k*A* 3 <sup>2</sup> *g*(0)k*<sup>H</sup>* + k*Agt*(0)k*<sup>H</sup>* +2k*A* 1 <sup>2</sup> *gtt*(0)k*<sup>H</sup>* + 2*τ*|*b*|(*N* − 2) max −*N*≤*k*≤0 k*A* 3 <sup>2</sup> *g*(*t<sup>k</sup>* )k*<sup>H</sup>* + 2*τ k*−2 ∑ *s*=1 k*A* 1 <sup>2</sup> *f*(*ts*)k*<sup>H</sup>* ≤ 2 2 + *τ*|*b*|(*N* − 2) *b*<sup>0</sup> + 2*τ N*−2 ∑ *s*=1 k*A* 1 <sup>2</sup> *f*(*ts*)k*H*, 3 ≤ *k* ≤ *N*.

From that and *u*<sup>0</sup> = *g*(0), it follows that:

$$\begin{aligned} \frac{1}{2} \max\_{0 \le k \le N} \|A^{\frac{3}{2}} u\_k\|\_H \\\\ \le (2 + \tau |b|(N-2)) b\_0 + \tau \sum\_{s=1}^{N-2} \|A^{\frac{1}{2}} f(t\_s)\|\_H \end{aligned} \tag{19}$$

for the solution of DS (4). Applying Formulas (6)–(8), we can write:

$$\frac{u\_1 - u\_0}{\tau} = v\_0 = R\widetilde{R}g\_t(0),$$

$$\frac{u\_2 - u\_1}{\tau} = v\_1 = R\widetilde{R}g\_t(0) + \tau R\widetilde{R}g\_{tt}(0),$$

$$\frac{u\_k - u\_{k-1}}{\tau} = v\_{k-1} = \frac{1}{2}[R^{k-2} + \widetilde{R}^{k-2}]R\widetilde{R}g\_t(0) + \frac{1}{2i}R(R^{k-1} - \widetilde{R}^{k-1})A^{-\frac{1}{2}}g\_{tt}(0)$$

$$-\frac{\tau}{2i}\sum\_{s=1}^{k-2}[R^{k-1-s} - \widetilde{R}^{k-1-s}]bA^{\frac{1}{2}}g(t\_{s-N}) - \frac{\tau}{2i}\sum\_{s=1}^{k-2}[R^{k-1-s} - \widetilde{R}^{k-1-s}]A^{-\frac{1}{2}}f(t\_s), \, 3 \le k \le N.$$

Using this formula and the estimates (14)–(16), we obtain:

$$\|A\frac{u\_1 - u\_0}{\tau}\|\_{H} \le \|R\tilde{R}\|\|\_{H \to H} \|g\_t(0)\|\_{H} \le b\_0.$$

$$\|A\frac{u\_2 - u\_1}{\tau}\|\_{H} \le \|R\tilde{R}\|\_{H \to H} \|Ag\_t(0)\|\_{H} + \|\tau A^{\frac{1}{2}}\mathbb{R}\|\_{H \to H} \|\tilde{R}\|\_{H \to H} \|A^{\frac{1}{2}}g\_t(0)\|\_{H} \le 2b\_0.$$

$$\|A\frac{u\_k - u\_{k-1}}{\tau}\|\_{H} \le \frac{1}{2} \|\|R\|\_{H \to H}^{k-2} + \|\tilde{R}\|\_{H \to H}^{k-2} \|\|R\tilde{R}\|\_{H \to H}\|\|g\_t(0)\|\_{H}$$

$$+ \frac{1}{2} [\|R\|\_{H \to H}^{k-1} + \|\tilde{R}\|\_{H \to H}^{k-1}] \|R\|\_{H \to H} \|A^{\frac{1}{2}}g\_t(0)\|\_{H} + \frac{\tau}{2} \|b\|\sum\_{s=1}^{N-2} (\|R\|\_{H \to H}^{k-1-s} + \|\tilde{R}\|\_{H \to H}^{k-1-s}) \|A^{\frac{3}{2}}g(s\_{s-N})\|\_{H}$$

$$+ \frac{\tau}{2} \sum\_{s=1}^{N-2} [\|R\|\_{H \to \tilde{H}}^{k-1-s} + \|\tilde{R}\|\_{H \to H}^{k-1-s}] \|A^{\frac{1}{2}}f(s\_s)\|\_{H}$$

*Symmetry* **2020**, *12*, 1033

$$\leq (2+\tau|b|(N-2))b\_0 + \tau \sum\_{s=1}^{N-2} ||A^{\frac{1}{2}}f(t\_s)||\_{H^s} \\ \mathcal{S} \leq k \leq N.$$

Combining these estimates, we obtain:

$$\max\_{1 \le k \le N} \left\| A \frac{u\_k - u\_{k-1}}{\tau} \right\|\_H \le (2 + \tau |b|(N - 2))b\_0 + \tau \sum\_{s=1}^{N-2} \|A^{\frac{1}{2}} f(t\_s)\|\_H \tag{20}$$

for the solution of DS (4). Applying Formulas (6)–(8), we can write:

$$\frac{\mu\_2 - 2\mu\_1 + \mu\_0}{\tau^2} = \tau^{-1} \left(v\_1 - v\_0 \right) = \tilde{R}R\underline{g}\_1(0),$$

$$\frac{\mu\_3 - 2\mu\_2 + \mu\_1}{\tau^2} = \tau^{-1} \left(v\_2 - v\_1\right)$$

$$= -\tau A(R\bar{R})^2 g\_{tt}(0) + \left(I - \tau^2 A\right) \left(\tilde{R}\right)^2 g\_{tt}(0) + \tau R\tilde{R}p\_t$$

$$= -\tau A(R\bar{R})^2 g\_{tt}(0) + \left(I - \tau^2 A\right) \left(\tilde{R}R\right)^2 g\_{tt}(0) + \tau R\tilde{R}[bA g(t\_{k-N}) + f(t\_1)],$$

$$\frac{u\_{k+2} - 2u\_{k+1} + u\_k}{\tau^2} = \tau^{-1} \left(v\_{k+1} - v\_k\right)$$

$$= \tau^{-1} \left\{\frac{1}{2} [R^k + \tilde{R}^k] R\tilde{R}g\_t(0) + \frac{1}{2} A^{-\frac{1}{2}} [R^{k+1} - \tilde{R}^{k+1}] Rg\_{tt}(0)\right\}$$

$$- \sum\_{k=1}^k \frac{\tau}{2} A^{-\frac{1}{2}} [R^{k+1-s} - \tilde{R}^{k+1-s}] p\_s = \frac{1}{2} [R^{k+1} + \tilde{R}^{k+1}] R\tilde{R}g\_t(0)$$

$$- \frac{1}{2i} A^{-\frac{1}{2}} [R^k - \tilde{R}^k] Rg\_{tt}(0) + \sum\_{s=1}^k \frac{\tau}{2} A^{-\frac{1}{2}} [R^{k-s} - \tilde{R}^{k-s}] p\_s\right\}$$

$$= \tau^{-1} \left\{\frac{1}{2} [R^k + \tilde{R}^k - R$$

Using this formula and the estimates (14)–(16), we obtain:

 *A* 1 2 *u*<sup>2</sup> − 2*u*<sup>1</sup> + *u*<sup>0</sup> *τ* 2 *H* ≤ k*RR*ek*H*→*H*k*A* 1 <sup>2</sup> *gtt*(0)k*<sup>H</sup>* ≤ *b*0, *A* 1 2 *u*<sup>3</sup> − 2*u*<sup>2</sup> + *u*<sup>1</sup> *τ* 2 *H* ≤ k*τA* 1 <sup>2</sup> *R*k*H*→*H*k*R*k*H*→*H*k*R*ek 2 *<sup>H</sup>*→*H*k*Agt*(0)k*<sup>H</sup>* <sup>+</sup> <sup>k</sup>*RR*ek*H*→*H*k*<sup>A</sup>* 1 <sup>2</sup> *gtt*(0)k*<sup>H</sup>* +*τ*|*b*|k*RR*ek*H*→*H*k*A* 3 <sup>2</sup> *g*(*t*1−*N*)k*<sup>H</sup>* + *τ*k*RR*ek*H*→*H*k*A* 3 <sup>2</sup> *f*(*t*1)k*<sup>H</sup>* ≤ (2 + *τ*|*b*|)*b*<sup>0</sup> + *τ*k*A* 1 <sup>2</sup> *f*(*t*1)k*H*, *A* 1 2 *uk*+<sup>2</sup> − 2*uk*+<sup>1</sup> + *u<sup>k</sup> τ* 2 *H* ≤ 1 2 [k*R*k *k <sup>H</sup>*→*<sup>H</sup>* <sup>+</sup> <sup>k</sup>*R*e<sup>k</sup> *k <sup>H</sup>*→*H*]k*RR*ek*H*→*H*k*Agt*(0)k*<sup>H</sup>*

$$\begin{split} \frac{1}{2} + \frac{1}{2} \| \| R \| \_{H \to H} \| \| R \| \_{H \to H}^{k+1} + \| \tilde{R} \| \_{H \to H}^{k+1} \| \| A^{\frac{1}{2}} g\_H(0) \| \_{H} + \frac{\tau}{2} \| b \| \sum\_{s=1}^{k-1} \| \| R \| \_{H \to H}^{k-s+1} + \| \tilde{R} \| \_{H \to H}^{k-s+1} \| \| \| A^{\frac{2}{2}} g(t\_{s-N}) \| \_{H} \\ + \frac{\tau}{2} \sum\_{s=1}^{k-1} \| \| R \| \_{H \to H}^{k-s+1} + \| \tilde{R} \| \_{H \to H}^{k-s+1} \| \| A^{\frac{1}{2}} f(t\_{s}) \| \_{H} + \tau \| b \| \| R \tilde{R} \| \_{H \to H} \| \| A^{\frac{2}{2}} g(t\_{k-N}) \| \_{H} \\ + \tau \| R \tilde{R} \| \_{H \to H} \| \| A^{\frac{1}{2}} f(t\_{k}) \| \_{H} \\ \leq (2 + \tau \| b \| (N - 2) ) b\_{0} + \tau \sum\_{s=1}^{N-2} \| \| A^{\frac{1}{2}} f(t\_{s}) \| \_{H}, 2 \leq k \leq N - 2. \end{split}$$

Combining these estimates, we can get:

$$\max\_{0 \le k \le N-2} \left\| A^{\frac{1}{2}} \frac{u\_{k+2} - 2u\_{k+1} + u\_k}{\tau^2} \right\|\_H$$

$$\le (2 + \tau |b|(N-2))b\_0 + \tau \sum\_{s=1}^{N-2} \|A^{\frac{1}{2}} f(t\_s)\|\_H \tag{21}$$

for the solution of DS (4). Estimate (17) follows from (19)–(21).

Now, let us estimate *bm*+1. Using Formula (13) and the estimates (14)–(16), we can obtain:

k*A* 3 <sup>2</sup> *umN*+1k*<sup>H</sup>* ≤ k*A* 3 <sup>2</sup> *umN*k*<sup>H</sup>* + k*τA* 1 <sup>2</sup> *R*k*H*→*H*k*R*ek*H*→*H*k*A umN* − *umN*−<sup>1</sup> *τ* k*<sup>H</sup>* ≤ 2*bm*, k*A* 3 <sup>2</sup> *umN*+2k*<sup>H</sup>* ≤ k*A* 3 <sup>2</sup> *umN*k*<sup>H</sup>* + 2k*τA* 1 <sup>2</sup> *R*k*H*→*H*k*R*ek*H*→*H*k*A umN* − *umN*−<sup>1</sup> *τ* k*H* +k*τA* 1 <sup>2</sup> *R*k*H*→*H*k*τA* 1 <sup>2</sup> *R*ek*H*→*H*k*A* 1 2 *umN* − 2*umN*−<sup>1</sup> + *umN*−<sup>2</sup> *τ* 2 k*<sup>H</sup>* ≤ 4*bm*, k*A* 3 <sup>2</sup> *uk*k*<sup>H</sup>* ≤ k*A* 3 <sup>2</sup> *umN*k*<sup>H</sup>* + 1 2 [k*R*k *k*−*mN*−2 *<sup>H</sup>*→*<sup>H</sup>* <sup>+</sup> <sup>k</sup>*R*e<sup>k</sup> *k*−*mN*−2 *H*→*H* ]k*RR*ek*H*→*H*k*A umN* − *umN*−<sup>1</sup> *τ* k*H* + 1 2 k*R*k*H*→*H*[2k*RR*ek*H*→*<sup>H</sup>* + k*R*k *k*−*mN*−1 *<sup>H</sup>*→*<sup>H</sup>* <sup>+</sup> <sup>k</sup>*R*e<sup>k</sup> *k*−*mN*−1 *H*→*H* ]k*A* 1 2 *umN* − 2*umN*−<sup>1</sup> + *umN*−<sup>2</sup> *τ* 2 k*H* + *τ* 2 |*b*| *k*−2 ∑ *s*=*mN*+1 [2 + k*R*k *k*−1−*s <sup>H</sup>*→*<sup>H</sup>* <sup>+</sup> <sup>k</sup>*R*e<sup>k</sup> *k*−1−*s H*→*H* ]k*A* 3 <sup>2</sup> *u*(*ts*−*N*)k*<sup>H</sup>* + *τ* 2 *k*−2 ∑ *s*=*mN*+1 [2 + k*R*k *k*−1−*s <sup>H</sup>*→*<sup>H</sup>* <sup>+</sup> <sup>k</sup>*R*e<sup>k</sup> *k*−1−*s H*→*H* ]k*A* 1 <sup>2</sup> *f*(*ts*)k*<sup>H</sup>* ≤ 2 2 + *τ*|*b*|(*N* − 2) *b<sup>m</sup>* + 2*τ* (*m*+1)*N*−2 ∑ *s*=*mN*+1 k*A* 1 <sup>2</sup> *f*(*ts*)k*H*, *mN* + 3 ≤ *k* ≤ (*m* + 1)*N*.

Combining these estimates, we obtain:

$$\frac{1}{2} \max\_{mN+1 \le k \le (m+1)N} ||A^{\frac{3}{2}} u\_k||\_H$$

$$\le (2 + \tau |b|(N-2))b\_m + \tau \sum\_{s=mN+1}^{(m+1)N-2} ||A^{\frac{1}{2}} f(t\_s)||\_H \tag{22}$$

for the solution of DS (4). Applying Formulas (10)–(12), we can write:

$$\frac{\mathfrak{u}\_{mN+1} - \mathfrak{u}\_{mN}}{\tau} = v\_{mN} = R\widetilde{R}\frac{\mathfrak{u}\_{mN} - \mathfrak{u}\_{mN-1}}{\tau}.$$

$$\begin{split} \frac{u\_{mN+2} - u\_{mN+1}}{\tau} &= v\_{mN+1} = R\widetilde{\bar{R}} \frac{u\_{mN} - u\_{mN-1}}{\tau} + \tau R\widetilde{\bar{R}} \frac{u\_{mN} - 2u\_{mN-1} + u\_{mN-2}}{\tau^2} \\ &\frac{u\_k - u\_{k-1}}{\tau} = v\_{k-1} = \frac{1}{2} [R^{k-mN-2} + \widetilde{\bar{R}}^{k-mN-2}] R\widetilde{\bar{R}} \frac{u\_{mN} - u\_{mN-1}}{\tau} \\ &+ \frac{1}{2i} R (R^{k-mN-1} - \widetilde{R}^{k-mN-1}) A^{-\frac{1}{2}} \frac{u\_{mN} - 2u\_{mN-1} + u\_{mN-2}}{\tau^2} \\ &- \frac{\tau}{2i} \sum\_{s=mN+1}^{k-2} [R^{k-1-s} - \widetilde{R}^{k-1-s}] b A^{\frac{1}{2}} u (t\_{s-N}) \\ &- \frac{\tau}{2i} \sum\_{s=mN+1}^{k-2} [R^{k-1-s} - \widetilde{R}^{k-1-s}] A^{-\frac{1}{2}} f(t\_{\bar{s}}), \; mN+3 \le k \le (m+1)N. \end{split}$$

Using this formula and the estimates (14)–(16), we can obtain:

k*A umN*+<sup>1</sup> − *umN τ* k*<sup>H</sup>* ≤ k*RR*ek*H*→*H*k*A umN* − *umN*−<sup>1</sup> *τ* k*<sup>H</sup>* ≤ *bm*, k*A umN*+<sup>2</sup> − *umN*+<sup>1</sup> *τ* k*<sup>H</sup>* ≤ k*RR*ek*H*→*H*k*A umN* − *umN*−<sup>1</sup> *τ* k*H* +k*τA* 1 <sup>2</sup> *R*k*H*→*H*k*R*ek*H*→*H*k*A* 1 2 *umN* − 2*umN*−<sup>1</sup> + *umN*−<sup>2</sup> *τ* 2 k*<sup>H</sup>* ≤ 2*bm*, k*A u<sup>k</sup>* − *uk*−<sup>1</sup> *τ* k*H* ≤ 1 2 [k*R*k *k*−*mN*−2 *<sup>H</sup>*→*<sup>H</sup>* <sup>+</sup> <sup>k</sup>*R*e<sup>k</sup> *k*−*mN*−2 *H*→*H* ]k*RR*ek*H*→*H*k*A umN* − *umN*−<sup>1</sup> *τ* k*H* + 1 2 [k*R*k *k*−1 *<sup>H</sup>*→*<sup>H</sup>* <sup>+</sup> <sup>k</sup>*R*e<sup>k</sup> *k*−1 *H*→*H* ]k*R*k*H*→*H*k*A* 1 2 *umN* − 2*umN*−<sup>1</sup> + *umN*−<sup>2</sup> *τ* 2 k*H* + *τ* 2 |*b*| (*m*+1)*N*−2 ∑ *s*=*mN*+1 [k*R*k *k*−1−*s <sup>H</sup>*→*<sup>H</sup>* <sup>+</sup> <sup>k</sup>*R*e<sup>k</sup> *k*−1−*s H*→*H* ]k*A* 3 <sup>2</sup> *u*(*ts*−*N*)k*<sup>H</sup>* + *τ* 2 (*m*+1)*N*−2 ∑ *s*=*mN*+1 [k*R*k *k*−1−*s <sup>H</sup>*→*<sup>H</sup>* <sup>+</sup> <sup>k</sup>*R*e<sup>k</sup> *k*−1−*s H*→*H* ]k*A* 1 <sup>2</sup> *f*(*ts*)k*<sup>H</sup>* ≤ (2 + *τ*|*b*|(*N* − 2))*b<sup>m</sup>* + *τ* (*m*+1)*N*−2 ∑ *s*=*mN*+1 k*A* 1 <sup>2</sup> *f*(*ts*)k*H*, *mN* + 3 ≤ *k* ≤ (*m* + 1)*N*.

Combining these estimates, we obtain:

$$\max\_{m:N+1 \le k \le (m+1)N} \left\| A \frac{u\_k - u\_{k-1}}{\tau} \right\|\_H \le (2 + \tau |b|(N-2))b\_m + \tau \sum\_{s=mN+1}^{(m+1)N-2} \|A^{\frac{1}{2}} f(t\_s)\|\_H \tag{23}$$

for the solution of DS (4). Applying Formulas (10)–(12), we can write:

$$\frac{u\_{mN+2} - 2u\_{mN+1} + u\_{mN}}{\tau^2} = \tau^{-1} \left(v\_{mN+1} - v\_{mN}\right) = \tilde{R}R\frac{u\_{mN} - 2u\_{mN-1} + u\_{mN-2}}{\tau^2}$$

$$\frac{u\_{mN+3} - 2u\_{mN+2} + u\_{mN+1}}{\tau^2} = \tau^{-1} \left(v\_{mN+2} - v\_{mN+1}\right)$$

$$= -\tau A \left(R\tilde{R}\right)^2 \frac{u\_{mN} - u\_{mN-1}}{\tau} + \left(I - \tau^2 A\right) \left(\tilde{R}R\right)^2 \frac{u\_{mN} - 2u\_{mN-1} + u\_{mN-2}}{\tau^2} + \tau R \tilde{R} p\_{mN+1}$$

$$= -\tau A (R\tilde{R})^2 \frac{u\_{mN} - u\_{mN-1}}{\tau} + \left(I - \tau^2 A\right) \left(\tilde{R}R\right)^2 \frac{u\_{mN} - 2u\_{mN-1} + u\_{mN-2}}{\tau^2}$$

$$\frac{u\_{k+2} - 2u\_{k+1} + u\_k}{\tau^2} = \tau^{-1} \left(v\_{k+1} - v\_k\right)$$

+*τRR*e[*bAu*(*t*1−*N*) + *f*(*tmN*+1)], = *τ* −1 1 2 [*R <sup>k</sup>*−*mN* <sup>+</sup> *<sup>R</sup>*e*k*−*mN*]*RR*<sup>e</sup> *umN* − *umN*−<sup>1</sup> *τ* + 1 2*i A* − 1 <sup>2</sup> [*R <sup>k</sup>*−*Mn*+<sup>1</sup> <sup>−</sup> *<sup>R</sup>*e*k*−*Mn*+<sup>1</sup> ]*R umN* − 2*umN*−<sup>1</sup> + *umN*−<sup>2</sup> *τ* 2 − *k* ∑ *s*=*Mn*+1 *τ* 2*i A* − 1 <sup>2</sup> [*R <sup>k</sup>*+1−*<sup>s</sup>* <sup>−</sup> *<sup>R</sup>*e*k*+1−*<sup>s</sup>* ]*p<sup>s</sup>* − 1 2 [*R <sup>k</sup>*−*Mn*−<sup>1</sup> <sup>+</sup> *<sup>R</sup>*e*k*−*Mn*−<sup>1</sup> ]*RR*e *umN* − *umN*−<sup>1</sup> *τ* − 1 2*i A* − 1 <sup>2</sup> [*R <sup>k</sup>*−*mN* <sup>−</sup> *<sup>R</sup>*e*k*−*mN*]*<sup>R</sup> umN* − 2*umN*−<sup>1</sup> + *umN*−<sup>2</sup> *τ* 2 + *k* ∑ *s*=*mN*+1 *τ* 2*i A* − 1 <sup>2</sup> [*R <sup>k</sup>*−*<sup>s</sup>* <sup>−</sup> *<sup>R</sup>*e*k*−*<sup>s</sup>* ]*p<sup>s</sup>* = *τ* −1 1 2 [*R <sup>k</sup>*−*mN* <sup>+</sup> *<sup>R</sup>*e*k*−*mN* <sup>−</sup> *<sup>R</sup> <sup>k</sup>*−*mN*−<sup>1</sup> <sup>−</sup> *<sup>R</sup>*e*k*−*mN*−<sup>1</sup> ]*RR*e *umN* − *umN*−<sup>1</sup> *τ* + 1 2*i A* −1 [*R <sup>k</sup>*+<sup>1</sup> <sup>−</sup> *<sup>R</sup>*e*k*+<sup>1</sup> <sup>−</sup> *<sup>R</sup> <sup>k</sup>* <sup>+</sup> *<sup>R</sup>*e*<sup>k</sup>* ]*R umN* − 2*umN*−<sup>1</sup> + *umN*−<sup>2</sup> *τ* 2 − *τ* 2*i A* − 1 <sup>2</sup> [*<sup>R</sup>* − *<sup>R</sup>*e]*pk*−*mN* + *k*−1 ∑ *s*=*mN*+1 *τ* 2*i A* − 1 <sup>2</sup> [*R <sup>k</sup>*−*<sup>s</sup>* <sup>−</sup> *<sup>R</sup>*e*k*−*<sup>s</sup>* <sup>−</sup> *<sup>R</sup> <sup>k</sup>*+1−*<sup>s</sup>* <sup>+</sup> *<sup>R</sup>*e*k*+1−*<sup>s</sup>* ]*p<sup>s</sup>* = − *i* 2 [*R <sup>k</sup>*−*mN* <sup>−</sup> *<sup>R</sup>*e*k*−*mN*]*RRA*<sup>e</sup> 1 2 *umN* − *umN*−<sup>1</sup> *τ* + 1 2 *R*[*R <sup>k</sup>*+<sup>1</sup> <sup>+</sup> *<sup>R</sup>*e*k*+<sup>1</sup> ] *umN* − 2*umN*−<sup>1</sup> + *umN*−<sup>2</sup> *τ* 2 −*τRRAbu* <sup>e</sup> (*tk*−*N*) − *<sup>τ</sup>RR f* <sup>e</sup> (*tk*−*mN*) − *τ* 2 *k*−1 ∑ *s*=*mN*+1 [*R <sup>k</sup>*−*s*+<sup>1</sup> <sup>+</sup> *<sup>R</sup>*e*k*−*s*+<sup>1</sup> ]*Abu*(*ts*−*N*) − *τ* 2 *k*−1 ∑ *s*=*mN*+1 [*R <sup>k</sup>*−*s*+<sup>1</sup> <sup>+</sup> *<sup>R</sup>*e*k*−*s*+<sup>1</sup> ] *f*(*ts*), *mN* + 2 ≤ *k* ≤ (*m* + 1)*N* − 2.

Using this formula and the estimates (14)–(16), we obtain that:

 *A* 1 2 *umN*+<sup>2</sup> − 2*umN*+<sup>1</sup> + *umN τ* 2 *H* ≤ k*RR*ek*H*→*H*k*A* 1 2 *umN* − 2*umN*−<sup>1</sup> + *umN*−<sup>2</sup> *τ* 2 k*<sup>H</sup>* ≤ *bm*, *A* 1 2 *umN*+<sup>3</sup> − 2*umN*+<sup>2</sup> + *umN*+<sup>1</sup> *τ* 2 *H* ≤ k*τA* 1 <sup>2</sup> *R*k*H*→*H*k*R*k*H*→*H*k*R*ek 2 *<sup>H</sup>*→*H*k*<sup>A</sup> umN* − *umN*−<sup>1</sup> *τ* k*H* +k*RR*ek*H*→*H*k*A* 1 2 *umN* − 2*umN*−<sup>1</sup> + *umN*−<sup>2</sup> *τ* 2 k*H* +*τ*|*b*|k*RR*ek*H*→*H*k*A* 3 <sup>2</sup> *u*(*t*1−*N*)k*<sup>H</sup>* + *τ*k*RR*ek*H*→*H*k*A* 3 <sup>2</sup> *f*(*tmN*+1)k*<sup>H</sup>* ≤ (2 + *τ*|*b*|)*b<sup>m</sup>* + *τ*k*A* 1 <sup>2</sup> *f*(*tmN*+1)k*H*, *A* 1 2 *uk*+<sup>2</sup> − 2*uk*+<sup>1</sup> + *u<sup>k</sup> τ* 2 *H* ≤ 1 2 [k*R*k *k*−*mN <sup>H</sup>*→*<sup>H</sup>* <sup>+</sup> <sup>k</sup>*R*e<sup>k</sup> *k*−*mN H*→*H* ]k*RR*ek*H*→*H*k*A umN* − *umN*−<sup>1</sup> *τ* k*H*

$$\begin{split} &+\frac{1}{2}||\boldsymbol{R}||\_{H\to H} \| \| \boldsymbol{R} \|\_{H\to H}^{k+1} + \| \tilde{\boldsymbol{R}} \|\_{H\to H}^{k+1} \| \| \boldsymbol{A}^{\frac{1}{2}} \frac{u\_{mN} - 2u\_{mN-1} + u\_{mN-2}}{\tau^2} \| \boldsymbol{h} \\ &+\frac{\tau}{2} |\boldsymbol{b}| \sum\_{s=mN+1}^{k-1} \left( \| \boldsymbol{R} \|\_{H\to H}^{k-s+1} + \| \tilde{\boldsymbol{R}} \|\_{H\to H}^{k-s+1} \| \| \boldsymbol{A}^{\frac{2}{2}} \boldsymbol{u}(t\_{s-N}) \| \right) \|\_{H} \\ &+\frac{\tau}{2} \sum\_{s=mN+1}^{k-1} \left( \| \boldsymbol{R} \|\_{H\to H}^{k-s+1} + \| \tilde{\boldsymbol{R}} \|\_{H\to H}^{k-s+1} \| \| \boldsymbol{A}^{\frac{1}{2}} f(t\_{s}) \| \|\_{H} \\ &+\tau |\boldsymbol{b}| \| \boldsymbol{R} \tilde{\boldsymbol{R}} \|\_{H\to H} \| \boldsymbol{A}^{\frac{1}{2}} \boldsymbol{u}(t\_{k-(m+1)N}) \| \|\_{H} + \tau \| \boldsymbol{R} \tilde{\boldsymbol{R}} \|\_{H\to H} \| \boldsymbol{A}^{\frac{1}{2}} f(t\_{k-mN}) \| \|\_{H} \\ &\leq (2+\tau|\boldsymbol{b}|(N-2)) b\_{m} + \tau \sum\_{s=mN+1}^{(m+1)N-2} \| \boldsymbol{A}^{\frac{1}{2}} f(t\_{s}) \|\_{H'} \, mN + 2 \leq k \leq (m+1)N-2. \end{split}$$

Combining these estimates, we obtain:

$$\max\_{mN \le k \le (m+1)N-2} \left\| A^{\frac{1}{2}} \frac{u\_{k+2} - 2u\_{k+1} + u\_k}{\tau^2} \right\|\_{H}$$

$$\le (2 + \tau |b|(N-2))b\_m + \tau \sum\_{s=mN+1}^{(m+1)N-2} ||A^{\frac{1}{2}} f(t\_s)||\_H \tag{24}$$

for the solution of DS (4). Estimate (18) follows from (22)–(24). Theorem 2 is proven.

Note that applying Theorem 2, we can obtain the stability estimate:

$$\max\_{mN \le k \le (m+1)N - 2} \left\| A^{\frac{1}{2}} \frac{u\_{k+2} - 2u\_{k+1} + u\_k}{\tau^2} \right\|\_{H} + \max\_{mN + 1 \le k \le (m+1)N} \left\| A \frac{u\_k - u\_{k-1}}{\tau} \right\|\_{H} \tag{25}$$

$$+ \frac{1}{2} \max\_{mN + 1 \le k \le (m+1)N} \| A^{\frac{3}{2}} u\_k \|\_{H} \le (2 + \tau |b|(N-2))^m b\_0$$

$$+ \sum\_{j=1}^m (2 + \tau |b|(N-2))^{m-j} \tau \sum\_{s=(j-1)N+1}^{jN} \| A^{\frac{1}{2}} f(t\_s) \|\_{H} m = 0, 1, \dots$$

for the solution of DS (4).

#### **3. Applications**

Note that the generality of this approach permits studying of a general class of DPDEs. We consider the applications of Theorem 2 for three types of problems. First, the mixed problem for the one-dimensional DPDE with nonlocal conditions:

$$\begin{cases} \frac{\partial^3 u(t, \mathbf{x})}{\partial t^3} - (a(\mathbf{x}) u\_{\mathbf{x}\mathbf{x}}(t, \mathbf{x}))\_{\mathbf{x}} + \delta u\_{\mathbf{l}}(t, \mathbf{x}) \\ = b \left( - (a(\mathbf{x}) u\_{\mathbf{x}}(t - w, \mathbf{x}))\_{\mathbf{x}} + \delta u(t - w, \mathbf{x}) \right) + f(t, \mathbf{x})\_{\mathbf{x}} \\ 0 < t < \infty, \ 0 < \mathbf{x} < l, \\ u(t, 0) = u(t, l), \ u\_{\mathbf{x}}(t, 0) = u\_{\mathbf{x}}(t, l), \ 0 \le t < \infty, \\ u(t, \mathbf{x}) = g(t, \mathbf{x}), -w \le t \le 0, 0 \le \mathbf{x} \le l \end{cases} \tag{26}$$

is studied. Under compatibility conditions, Problem (26) has a unique solution *u*(*t*, *x*) for the given smooth functions *a*(*x*) ≥ *a* > 0, *x* ∈ (0, *l*), *δ* > 0, *a*(*l*) = *a*(0), *g*(*t*, *x*), −*w* ≤ *t* ≤ 0, 0 ≤ *x* ≤ *l*, *f*(*t*, *x*), 0 < *t* < ∞, 0 < *x* < *l*, and *b* ∈ *R* 1 .

The construction of full discretization to Problem (26) is completed in two stages. In the first stage, we consider the uniform grid space:

$$[0,l]\_{\mathfrak{h}} = \{ \mathfrak{x} = \mathfrak{x}\_{\mathfrak{n}} : \mathfrak{x}\_{\mathfrak{n}} = \mathfrak{n}h, \ 0 \le \mathfrak{n} \le M, \ \mathrm{Mh} = l \}$$

with step *h* > 0. Let *L*2*<sup>h</sup>* = *L*2([0, *l*]*<sup>h</sup>* ) be a Hilbert space of the grid functions *ϕ h* (*x*) = {*ϕn*} *M* 0 defined on [0, *l*]*<sup>h</sup>* , equipped with the norm:

$$||\varrho^h||\_{L\_{2h}} = \left(\sum\_{\mathbf{x}\in[0,l]\_h} |\varrho(\mathbf{x})|^2 h\right)^{1/2}.$$

Let *A x h* be the second order difference operator defined by:

$$A\_h^x \boldsymbol{\varrho}^h(\mathbf{x}) = \left\{-(a(\mathbf{x})\boldsymbol{\varrho}\_{\overline{\mathbf{x}}})\_{\mathbf{x},\mathcal{U}} + \delta \boldsymbol{\varrho}\_n\right\}\_1^{M-1} \tag{27}$$

acting in the space of grid functions *ϕ h* (*x*) = {*ϕn*} *M* 0 satisfying the conditions *ϕ*<sup>0</sup> = *ϕM*, *ϕ*<sup>1</sup> − *ϕ*<sup>0</sup> = *ϕ<sup>M</sup>* − *ϕM*−1. It is well known that *A x h* is a self-adjoint positive definite operator in *L*2*<sup>h</sup>* . Applying *A x h* in (26), we can obtain the initial value problem for an infinite system of third order differential equations:

$$\begin{cases} \begin{aligned} u\_{ttt}^h(t, \mathbf{x}) + A\_h^\mathbf{x} u\_t^h(t, \mathbf{x}) &= b A\_h^\mathbf{x} u(t - w, \mathbf{x}) + f^h(t, \mathbf{x}), \\ 0 < t < \infty, &\mathbf{x} \in [0, l]\_h \\ u^h(t, \mathbf{x}) = g^h(t, \mathbf{x}), -w \le t \le 0, \mathbf{x} \in [0, l]\_h. \end{aligned} \end{cases} \tag{28}$$

In the second stage, we use DS (4) for (28):

 *u h k*+2 (*x*)−3*u h k*+1 (*x*)+3*u h k* (*x*)−*u h k*−1 (*x*) *τ* <sup>3</sup> + *A x h u h k*+2 (*x*)−*u h k*+1 (*x*) *τ* = *bA<sup>x</sup> h u h k*−*N* (*x*) + *f h k* (*x*), *f h k* (*x*) = *f h* (*tk* , *x*), *k* ≥ 1, *x* ∈ [0, *l*]*<sup>h</sup>* , *u h k* (*x*) = *g h* (*tk* , *x*), −*N* ≤ *k* ≤ 0, (*I<sup>h</sup>* + *τ* <sup>2</sup>*A x h* ) *u h* 1 (*x*)−*u h* 0 (*x*) *<sup>τ</sup>* = *g h t* (0, *x*), (*I<sup>h</sup>* + *τ* <sup>2</sup>*A x h* ) *u h* 2 (*x*)−2*u h* 1 (*x*)+*u h* 0 (*x*) *τ* <sup>2</sup> = *g h tt*(0, *x*), *x* ∈ [0, *l*]*<sup>h</sup>* , (*I<sup>h</sup>* + *τ* <sup>2</sup>*A x h* ) *u h mN*+1 (*x*)−*u h mN*(*x*) *<sup>τ</sup>* = *u h mN*(*x*)−*u h mN*−1 (*x*) *τ* , (*I<sup>h</sup>* + *τ* <sup>2</sup>*A x h* ) *u h mN*+2 (*x*)−2*u h mN*+1 (*x*)+*u h mN*(*x*) *τ* 2 = *u h mN*(*x*)−2*u h mN*−1 (*x*)+*u h mN*−2 (*x*) *τ* 2 , *m* = 1, 2, ... (29)

**Theorem 3.** *The solutions of DS (29) obey the stability estimates:*

$$\begin{split} \max\_{mN \le k \le (m+1)N-2} \left\| \frac{\boldsymbol{u}\_{k+2}^{h} - 2\boldsymbol{u}\_{k+1}^{h} + \boldsymbol{u}\_{k}^{h}}{\tau^{2}} \right\|\_{W\_{2h}^{2}} + \max\_{mN+1 \le k \le (m+1)N} \left\| \frac{\boldsymbol{u}\_{k}^{h} - \boldsymbol{u}\_{k-1}^{h}}{\tau} \right\|\_{W\_{2h}^{2}} \\ + \frac{1}{2} \max\_{mN+1 \le k \le (m+1)N} \|\boldsymbol{u}\_{k}^{h}\|\_{W\_{2h}^{3}} \le C\_{1} \left[ (2 + \tau |b|(N-2))^{m} \boldsymbol{b}\_{0}^{h} \right] \\ + \sum\_{j=1}^{m} (2 + \tau |b|(N-2))^{m-j} \tau \sum\_{s=(j-1)N+1}^{jN} \|\boldsymbol{A}^{\frac{1}{2}} f(t\_{s})\|\_{W\_{2h}^{1}} \right], m = 0, 1, \dots \\ \boldsymbol{b}\_{0}^{h} = \max \left\{ \max\_{-N \le k \le 0} \|\boldsymbol{A}^{\frac{1}{2}} \boldsymbol{g}\_{th}^{h}(t\_{k})\|\_{W\_{2h}^{1}} \max\_{-N \le k \le 0} \|\boldsymbol{g}\_{t}^{h}(t\_{k})\|\_{W\_{2h}^{2}} \cdot \max\_{-N \le k \le 0} \|\boldsymbol{g}^{h}(t\_{k})\|\_{W\_{2h}^{3}} \right\}. \\ C\_{1} \text{ does not depend on  $\tau$  h  $ph$  and  $f$ } (k) \end{split}$$

*hold, where C*<sup>1</sup> *does not depend on τ*, *h*, *g* (*tk* ), *and f <sup>h</sup> k* (*x*). **Proof.** DS (29) can be written in abstract form:

$$\begin{cases} \frac{u\_{k+2}^h - 3u\_{k+1}^h + 3u\_k^h - u\_k^h}{\tau^3} + A\_h \frac{u\_{k+2}^h - u\_{k+1}^h}{\tau} = bA\_h u\_{k-N}^h + f\_k^h, k \ge 1, \\\\ u\_k^h = g\_{k'}^h - N \le k \le 0, \\\\ (I\_h + \tau^2 A\_h) \frac{u\_1^h - u\_0^h}{\tau} = g\_1^h(0), (I\_h + \tau^2 A\_h) \frac{u\_2^h - 2u\_1^h + u\_0^h}{\tau^2} = g\_{tt}^h(0), \\\\ (I\_h + \tau^2 A\_h) \frac{u\_{mN+1}^h - u\_{mN}^h}{\tau} = \frac{u\_{mN}^h - u\_{mN-1}^h}{\tau}, \\\\ (I\_h + \tau^2 A\_h) \frac{u\_{mN+2}^h - 2u\_{mN+1}^h + u\_{mN}^h}{\tau^2} = \frac{u\_{mN}^h - 2u\_{mN-1}^h + u\_{mN-2}^h}{\tau^2}, m = 1, 2, \dots \end{cases} (30)$$

in a Hilbert space *L*2*<sup>h</sup>* with a self-adjoint positive definite operator *A<sup>h</sup>* = *A x h* . Here, *g h <sup>k</sup>* = *g h k* (*x*), *f h <sup>k</sup>* = *f h k* (*x*), and *u h <sup>k</sup>* = *u h k* (*x*) are known and unknown abstract mesh functions defined on [0, *l*]*<sup>h</sup>* . Therefore, the estimate of Theorem 3 follows from the estimate (25). Theorem 3 is proven.

Second, let Ω be the unit open cube in the *m*-dimensional Euclidean space.

R*n* (*x* = (*x*1, · · · , *xm*) : 0 < *x<sup>k</sup>* < 1, *k* = 1, · · · , *n*) with boundary *S*, Ω = Ω ∪ *S*. In [0, ∞) × Ω, the mixed problem for the DPDE with the Dirichlet condition:

$$\begin{cases} u\_{\text{lift}}(t, \mathbf{x}) - \sum\_{r=1}^{n} \left( a\_{r}(\mathbf{x}) u\_{\text{ltx}}(t, \mathbf{x}) \right)\_{\mathbf{x}\_{r}} = -b \sum\_{r=1}^{n} \left( a\_{r}(\mathbf{x}) u\_{\text{ltx}}(t - w, \mathbf{x}) \right)\_{\mathbf{x}\_{r}}, \\ 0 < t < \infty, \mathbf{x} \in \Omega, \\ u(t, \mathbf{x}) = \mathbf{0}, \ \mathbf{x} \in \mathcal{S}, \ \mathbf{0} \le t < \infty, \\ u(t, \mathbf{x}) = \mathbf{g}(t, \mathbf{x}), -w \le t \le \mathbf{0}, \mathbf{x} \in \overline{\Omega} \end{cases} \tag{31}$$

is investigated. Under compatibility conditions, Problem (31) has a unique solution *u*(*t*, *x*) for the given smooth functions *ar*(*x*) ≥ *a* > 0, (*x* ∈ Ω), *g*(*t*, *x*), −*w* ≤ *t* ≤ 0, *x* ∈ Ω, *f*(*t*, *x*), 0 < *t* < ∞, *x* ∈ Ω, and *b* ∈ *R* 1 .

The construction of full discretization to Problem (31) is completed in two stages. In the first stage, we consider the uniform grid space:

$$\begin{aligned} \Omega\_{\mathbb{H}} &= \{ \mathbf{x} = \mathbf{x}\_{r} = (h\_{1}j\_{1}, \dots, h\_{n}j\_{n}), j = (j\_{1}, \dots, j\_{n}), \ 0 \le j\_{r} \le \mathbf{N}\_{r} \\ \mathbf{N}\_{\mathbf{r}}h\_{\mathbf{r}} &= 1, \ \mathbf{r} = 1, \dots, n \}, \ \Omega\_{\mathbb{H}} = \overline{\Omega}\_{\mathbb{H}} \cap \Omega, \mathcal{S}\_{\mathbb{H}} = \overline{\Omega}\_{\mathbb{H}} \cap \mathcal{S} \end{aligned}$$

and introduce the Hilbert space *L*2*<sup>h</sup>* = *L*2(Ω*<sup>h</sup>* ) of the grid functions *ϕ h* (*x*) = {*ϕ* (*h*<sup>1</sup> *j*1, · · · , *h<sup>n</sup> jn*)} defined on Ω*<sup>h</sup>* equipped with the norm:

$$\left\| \left| \boldsymbol{\varrho}^{h} \right| \right\|\_{L\_{2h}} = \left( \sum\_{\boldsymbol{\chi} \in \Omega\_{h}} \left| \boldsymbol{\varrho}^{h} \left( \boldsymbol{\chi} \right) \right|^{2} h\_{1} \cdot \cdots \cdot h\_{n} \right)^{\frac{1}{2}}.$$

We consider the difference operator *A x h* defined by the formula:

$$A\_{\boldsymbol{h}}^{\boldsymbol{x}}\boldsymbol{u}^{\boldsymbol{h}} = -\sum\_{\boldsymbol{r}=1}^{n} \left(a\_{\boldsymbol{r}}(\boldsymbol{x})\boldsymbol{u}\_{\boldsymbol{x}\_{\boldsymbol{r}}}^{\boldsymbol{h}}\right)\_{\boldsymbol{x}\_{\boldsymbol{r}},\boldsymbol{j}\_{\boldsymbol{r}}} \tag{32}$$

acting in the space of grid functions *u h* (*x*), which satisfy the conditions *u h* (*x*) = 0 for all *x* ∈ *S<sup>h</sup>* . It is well known that *A x h* is a self-adjoint positive definite operator in *L*2*<sup>h</sup>* . Applying *A x h* in (31), we can obtain that:

$$\begin{cases} \boldsymbol{u}\_{ttt}^{h}(t, \mathbf{x}) + \boldsymbol{A}\_{\boldsymbol{h}}^{x} \boldsymbol{u}\_{t}^{h}(t, \mathbf{x}) = \boldsymbol{b} \boldsymbol{A}\_{\boldsymbol{h}}^{x} \boldsymbol{u}(t - \boldsymbol{w}, \mathbf{x}) + \boldsymbol{f}^{h}(t, \mathbf{x}),\\ \boldsymbol{0} < t < \infty, \boldsymbol{x} \in \Omega\_{\mathbb{H}} \\\ \boldsymbol{u}^{h}(t, \mathbf{x}) = \boldsymbol{g}^{h}(t, \mathbf{x}), -\boldsymbol{w} \le t \le \mathbf{0}, \mathbf{x} \in \overline{\Omega}\_{\mathbb{H}}. \end{cases} \tag{33}$$

In the second stage, we also get the difference scheme as the one-dimensional problem case:

 *u h k*+2 (*x*)−3*u h k*+1 (*x*)+3*u h k* (*x*)−*u h k*−1 (*x*) *τ* <sup>3</sup> + *A x h u h k*+2 (*x*)−*u h k*+1 (*x*) *τ* = *bA<sup>x</sup> h u h k*−*N* (*x*) + *f h k* (*x*), *f h k* (*x*) = *f h* (*tk* , *x*), *k* ≥ 1, *x* ∈ Ω*<sup>h</sup>* , *u h k* (*x*) = *g h* (*tk* , *x*), −*N* ≤ *k* ≤ 0, (*I<sup>h</sup>* + *τ* <sup>2</sup>*A x h* ) *u h* 1 (*x*)−*u h* 0 (*x*) *<sup>τ</sup>* = *g h t* (0, *x*), (*I<sup>h</sup>* + *τ* <sup>2</sup>*A x h* ) *u h* 2 (*x*)−2*u h* 1 (*x*)+*u h* 0 (*x*) *τ* <sup>2</sup> = *g h tt*(0, *x*), *x* ∈ Ω*<sup>h</sup>* , (*I<sup>h</sup>* + *τ* <sup>2</sup>*A x h* ) *u h mN*+1 (*x*)−*u h mN*(*x*) *<sup>τ</sup>* = *u h mN*(*x*)−*u h mN*−1 (*x*) *τ* , *x* ∈ Ω*<sup>h</sup>* (*I<sup>h</sup>* + *τ* <sup>2</sup>*A x h* ) *u h mN*+2 (*x*)−2*u h mN*+1 (*x*)+*u h mN*(*x*) *τ* 2 = *u h mN*(*x*)−2*u h mN*−1 (*x*)+*u h mN*−2 (*x*) *τ* 2 , *x* ∈ Ω*<sup>h</sup>* , *m* = 1, 2, ... (34)

**Theorem 4.** *The solution of DS (34) obeys the following stability estimates:*

$$\begin{split} \max\_{mN \le k \le (m+1)N-2} \left\| \frac{u\_{k+2}^{h} - 2u\_{k+1}^{h} + u\_{k}^{h}}{\tau^{2}} \right\|\_{W\_{2h}^{1}} + \max\_{mN+1 \le k \le (m+1)N} \left\| \frac{u\_{k}^{h} - u\_{k-1}^{h}}{\tau} \right\|\_{W\_{2h}^{2}} \\ + \frac{1}{2} \max\_{mN+1 \le k \le (m+1)N} \|u\_{k}^{h}\|\_{W\_{2h}^{3}} \le C\_{2} \left[ (2 + \tau |b|(N-2))^{m} b\_{0}^{h} \right. \\ \left. \quad + \sum\_{j=1}^{m} (2 + \tau |b|(N-2))^{m-j} \tau \sum\_{s=(j-1)N+1}^{jN} \|A^{\frac{1}{2}} f(t\_{k})\|\_{W\_{2h}^{1}} \right], m = 0, 1, \dots, \\ b\_{0}^{h} = \max \left\{ \max\_{-N \le k \le 0} \|A^{\frac{1}{2}} g\_{t}^{h}(t\_{k})\|\_{W\_{2h'}^{1}} \max\_{-N \le k \le 0} \|g\_{t}^{h}(t\_{k})\|\_{W\_{2h'}^{2}} \max\_{-N \le k \le 0} \|g^{h}(t\_{k})\|\_{W\_{2h}^{3}} \right\}. \end{split}$$

*k*

**Proof.** DS (34) can be written in abstract form:

$$\begin{cases} \frac{u\_{k+2}^h - 3u\_{k+1}^h + 3u\_k^h - u\_{k-1}^h}{\tau^3} + A\_h \frac{u\_{k+2}^h - u\_{k+1}^h}{\tau} = bA\_h u\_{k-N}^h + f\_k^h, k \ge 1, \\\ u\_k^h = g\_{k'}^h - N \le k \le 0, \\\ (I\_h + \tau^2 A\_h) \frac{u\_1^h - u\_0^h}{\tau} = g\_t^h(0), (I\_h + \tau^2 A\_h) \frac{u\_2^h - 2u\_1^h + u\_0^h}{\tau^2} = g\_{tt}^h(0), \\\ (I\_h + \tau^2 A\_h) \frac{u\_{mN+1}^h - u\_{mN}^h}{\tau} = \frac{u\_{mN}^h - u\_{mN-1}^h}{\tau}, \\\ (I\_h + \tau^2 A\_h) \frac{u\_{mN+2}^h - 2u\_{mN+1}^h + u\_{mN}^h}{\tau^2} = \frac{u\_{mN}^h - 2u\_{mN-1}^h + u\_{mN-2}^h}{\tau^2}, m = 1, 2, \dots \end{cases}$$

in a Hilbert space *L*2*<sup>h</sup>* = *L*2(Ω*<sup>h</sup>* ) with self-adjoint positive definite operator *A<sup>h</sup>* = *A x h* by Formula (32). Here, *g h <sup>k</sup>* = *g h k* (*x*), *f h <sup>k</sup>* = *f h k* (*x*), and *u h <sup>k</sup>* = *u h k* (*x*) are known and unknown abstract mesh functions defined on Ω*<sup>h</sup>* with the values in *L*2*<sup>h</sup>* . Then, the estimate of Theorem 4 follows from Estimate (25) and the following theorem.

**Theorem 5.** *The solution of the difference elliptic problem: [40]*

$$A\_h^\mathbf{x} u^h(\mathbf{x}) = \omega^h(\mathbf{x}), \; \mathbf{x} \in \Omega\_h; u^h(\mathbf{x}) = 0, \; \mathbf{x} \in \mathcal{S}\_h$$

*Symmetry* **2020**, *12*, 1033

*obeys the estimate:*

$$\sum\_{r=1}^n \left\| \left. u^h \right|\_{\mathcal{X}\_r \mathcal{X}\_7^r} \right\|\_{L\_{2h}} \le \mathsf{C}\_3 ||\omega^h||\_{L\_{2h}}.$$

*where C*<sup>3</sup> *does not depend on h and ω<sup>h</sup>* .

Third, in [0, ∞) × Ω, the mixed problem for DPDE with the Neumann boundary condition:

$$\begin{cases} u\_{\rm{ff}}(t, \mathbf{x}) - \sum\_{r=1}^{n} \left( a\_{r}(\mathbf{x}) u\_{\rm{tr}\_{r}}(t, \mathbf{x}) \right)\_{\mathbf{x}\_{r}} + \delta u\_{\rm{f}}(t, \mathbf{x}) \\ = b \left( - \sum\_{r=1}^{n} \left( a\_{r}(\mathbf{x}) u\_{\rm{tr}\_{r}}(t - w, \mathbf{x}) \right)\_{\mathbf{x}\_{r}} + \delta u(t - w, \mathbf{x}) \right), \\ 0 < t < \infty, \mathbf{x} \in \Omega, \\ \frac{\partial u(t, \mathbf{x})}{\partial p} = \mathbf{0}, \ \mathbf{x} \in \mathcal{S}, \ 0 \le t < \infty \\ u(t, \mathbf{x}) = g(t, \mathbf{x}), -w \le t \le 0, \mathbf{x} \in \overline{\Omega} \end{cases} \tag{35}$$

is investigated. Here, −→*p* is the normal vector to *S*. Under compatibility conditions, Problem (31) has a unique solution *u*(*t*, *x*) for the given smooth functions *ar*(*x*) ≥ *a* > 0, (*x* ∈ Ω), *g*(*t*, *x*), −*w* ≤ *t* ≤ 0, *x* ∈ Ω, *f*(*t*, *x*), 0 < *t* < ∞, *x* ∈ Ω and *b* ∈ *R* 1 .

The construction of full discretization to Problem (35) is completed in two stages. In the first stage, we introduce the second order difference operator *A x h* defined by:

$$A\_h^\mathbf{x} u^h = -\sum\_{r=1}^n \left(\alpha\_r(\mathbf{x}) u\_{\mathbf{x}\_r}^h\right)\_{\mathbf{x}\_r, j\_r} + \delta u^h \,\tag{36}$$

acting in the space of grid functions *u h* (*x*) that satisfy the conditions *Dhu h* (*x*) = 0 for all *x* ∈ *S<sup>h</sup>* . Here, *D<sup>h</sup>* is the approximation of operator *<sup>∂</sup> ∂* −→*p* . It is known that *A x h* is the self-adjoint positive definite operator in *L*2*<sup>h</sup>* . Using the difference operator *A x h* , we get the initial value problem (33). Therefore, in the second stage, we use DS (4) for Problem (33):

 *u h k*+2 (*x*)−3*u h k*+1 (*x*)+3*u h k* (*x*)−*u h k*−1 (*x*) *τ* <sup>3</sup> + *A x h u h k*+2 (*x*)−*u h k*+1 (*x*) *τ* = *bA<sup>x</sup> h u h k*−*N* (*x*) + *f h k* (*x*), *f h k* (*x*) = *f h* (*tk* , *x*), *k* ≥ 1, *x* ∈ Ω*<sup>h</sup>* , *u h k* (*x*) = *g h* (*tk* , *x*), −*N* ≤ *k* ≤ 0, (*I<sup>h</sup>* + *τ* <sup>2</sup>*A x h* ) *u h* 1 (*x*)−*u h* 0 (*x*) *<sup>τ</sup>* = *g h t* (0, *x*), (*I<sup>h</sup>* + *τ* <sup>2</sup>*A x h* ) *u h* 2 (*x*)−2*u h* 1 (*x*)+*u h* 0 (*x*) *τ* <sup>2</sup> = *g h tt*(0, *x*), *x* ∈ Ω*<sup>h</sup>* , (*I<sup>h</sup>* + *τ* <sup>2</sup>*A x h* ) *u h mN*+1 (*x*)−*u h mN*(*x*) *<sup>τ</sup>* = *u h mN*(*x*)−*u h mN*−1 (*x*) *τ* , *x* ∈ Ω*<sup>h</sup>* (*I<sup>h</sup>* + *τ* <sup>2</sup>*A x h* ) *u h mN*+2 (*x*)−2*u h mN*+1 (*x*)+*u h mN*(*x*) *τ* 2 = *u h mN*(*x*)−2*u h mN*−1 (*x*)+*u h mN*−2 (*x*) *τ* 2 , *x* ∈ Ω*<sup>h</sup>* , *m* = 1, 2, ... (37)

**Theorem 6.** *The solution of the difference scheme (37) obeys the stability estimates in Theorem 4.*

**Proof.** DS (37) can be written in abstract form:

$$\begin{cases} \frac{u\_{k+2}^h - 3u\_{k+1}^h + 3u\_k^h - u\_{k-1}^h}{\tau^3} + A\_h \frac{u\_{k+2}^h - u\_{k+1}^h}{\tau} = bA\_h u\_{k-N}^h + f\_k^h, k \ge 1, \\\ u\_k^h = \mathcal{g}\_{k'}^h - \mathcal{N} \le k \le 0, \\\ (I\_h + \tau^2 A\_h) \frac{u\_1^h - u\_0^h}{\tau} = \mathcal{g}\_t^h(0), (I\_h + \tau^2 A\_h) \frac{u\_2^h - 2u\_1^h + u\_0^h}{\tau^2} = \mathcal{g}\_{tt}^h(0), \\\ (I\_h + \tau^2 A\_h) \frac{u\_{mN+1}^h - u\_{mN}^h}{\tau} = \frac{u\_{mN}^h - u\_{mN-1}^h}{\tau}, \\\ (I\_h + \tau^2 A\_h) \frac{u\_{mN+2}^h - 2u\_{mN+1}^h + u\_{mN}^h}{\tau^2} = \frac{u\_{mN}^h - 2u\_{mN-1}^h + u\_{mN-2}^h}{\tau^2}, m = 1, 2, \dots \end{cases}$$

in a Hilbert space *L*2*<sup>h</sup>* = *L*2(Ω*<sup>h</sup>* ) with self-adjoint positive definite operator *A<sup>h</sup>* = *A x h* by Formula (36). Here, *g h <sup>k</sup>* = *g h k* (*x*), *f h <sup>k</sup>* = *f h k* (*x*), and *u h <sup>k</sup>* = *u h k* (*x*) are known and unknown abstract mesh functions defined on Ω*<sup>h</sup>* with the values in *L*2*<sup>h</sup>* . Therefore, the estimate of Theorem 6 follows from the estimate (25) and the following theorem.

**Theorem 7.** *The solution of the elliptic difference problem: [40]*

$$A\_h^\mathbf{x} u^h(\mathfrak{x}) = \omega^h(\mathfrak{x}), \ \mathfrak{x} \in \Omega\_h; D^h u^h(\mathfrak{x}) = 0, \ \mathfrak{x} \in \mathcal{S}\_h$$

*satisfies the estimate:*

$$\sum\_{r=1}^n \left\| \left. \boldsymbol{\mu}^h \right|\_{\boldsymbol{\mathcal{X}}\_r \boldsymbol{\mathcal{X}}\_r^\*} \right\|\_{L\_{2h}} \le \mathsf{C}\_4 ||\boldsymbol{\omega}^h||\_{L\_{2h'}} $$

*where C*<sup>4</sup> *is independent of h and ω<sup>h</sup>* .

#### **4. Numerical Results**

It is well known that when the analytical methods fail to work properly, the numerical methods for getting partial differential equations' approximate solutions play a vital role in applied mathematics. In the operator approach, constants in theorems can be large; therefore, in this case a nice stability result must be supported numerically. For this reason, it is important to see that for such a type of theoretical result, we need numerical applications when one cannot know concrete values of constants in stability estimates. Therefore, the first order of accuracy DSs for the solution of one- and two-dimensional DPDEs are presented. To solve this problem, a procedure of modified Gauss elimination is applied. The result of the numerical experiment supports the theoretical statements for the solution of these DSs.

#### *4.1. One-Dimensional Problem*

First, we consider the mixed problem, with the exact solution *u* (*t*, *x*) = *e* −*t* cos *x*,

$$\begin{cases} u\_{ttt}(t, \mathbf{x}) - u\_{t\mathbf{x}\mathbf{x}}(t, \mathbf{x}) = -0.1u\_{\mathbf{x}\mathbf{x}}(t - 1, \mathbf{x}) \\ -2e^{-t}\cos\mathbf{x} - 0.1e^{-(t - 1)}\cos\mathbf{x}, t > 0, \; 0 < \mathbf{x} < \pi, \\\ u\_{\mathbf{x}}(t, 0) = u\_{\mathbf{x}}(t, \pi) = 0, \; 0 \le t < \infty, \\\ u(t, \mathbf{x}) = e^{-t}\cos\mathbf{x}, \; -1 \le t \le 0, \; 0 \le \mathbf{x} \le \pi \end{cases} \tag{38}$$

for the one-dimensional DPDE.

Applying DS (4), we get the following DS:

$$\begin{cases} \begin{aligned} &u\_{n}^{k+2} - 3u\_{n}^{k+1} + 3u\_{n}^{k} - u\_{n}^{k-1} \\ &- \sum\_{n\_{n}+1}^{\tau 3}^{\tau 3} - u\_{n+1}^{k+1} - 2\left(u\_{n}^{k+2} - u\_{n}^{k+1}\right) + u\_{n-1}^{k+2} - u\_{n-1}^{k+1} \\ &- 0.1e\_{n+1} - 2u\_{n}^{k-N} - u\_{n-1}^{k} - 2e^{-t\_{k}}\cos x\_{n} - 0.1e^{-(t\_{k-N})}\cos x\_{n}, \\ &t\_{k} = k\tau, \; IN + 1 \le k \le (l+1)N - 2, \; l = 0, 1, \dots \; 1 \le k \le N-1, \\ &N\tau = 1, \; x\_{n} = n\Lambda, \; 1 \le n \le M-1, \; Mk = \tau, \\ &u\_{n}^{k} = e^{-t\_{k}}\cos x\_{n}, \; -N \le k \le 0, \; 0 \le n \le M, \\ &u\_{n}^{k+1} - u\_{n}^{k} = -e^{-t\_{k}}\cos x\_{n}, \; -N \le k \le 0, \; 0 \le n \le M, \\ &u\_{n}^{k+2} - 2u\_{n}^{k+1} + u\_{n}^{k} = e^{-t\_{k}}\cos x\_{n}, \; -N \le k \le 0, \; 0 \le n \le M, \\ &u\_{1}^{k} - u\_{0}^{k} = u\_{M}^{k} - u\_{M-1}^{k} = 0, \; 0 \le k \le N. \end{aligned} \tag{39}$$

It can be written as the second order difference problem with matrix coefficients:

$$A\ u\_{n+1} + Bu\_n + Cu\_{n-1} = D\varphi\_{\nu\_\nu} \ 1 \le n \le M-1; \newline u\_0 = u\_1, \ \newline u\_M = u\_{M-1}.\tag{40}$$

Here and in the future, we put:

$$u\_s = \begin{bmatrix} u\_s^{IN} \\ \vdots \\ u\_s^{(l+1)N} \end{bmatrix}\_{(N+1)\times 1}, \quad s = n, n \pm 1.$$

Here,

$$A = \mathbb{C} = \begin{bmatrix} 0 & 0 & 0 & 0 & \cdot & 0 & 0 & 0 \\ 0 & 0 & a & -a & \cdot & 0 & 0 & 0 \\ 0 & 0 & 0 & a & \cdot & 0 & 0 & 0 \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ 0 & 0 & 0 & 0 & \cdot & 0 & a & -a \\ 0 & 0 & 0 & 0 & \cdot & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \cdot & 0 & 0 & 0 \end{bmatrix}\_{(N+1)\times(N+1)}$$

$$B = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & \cdot & 0 & 0 & 0 & 0 \\ b & -3b & 3b - c & b - c & 0 & \cdot & 0 & 0 & 0 & 0 \\ 0 & b & -3b & 3b - c & c - b & \cdot & 0 & 0 & 0 & 0 \\ 0 & 0 & b & -3b & 3b - c & \cdot & 0 & 0 & 0 & 0 \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ 0 & 0 & 0 & 0 & 0 & \cdot & 3b - c & c - b & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \cdot & -3b & 3b - c & c - b & 0 \\ 0 & 0 & 0 & 0 & 0 & \cdot & b & -3b & 3b - c & c - b \\ -\frac{1}{\tau} & \frac{1}{\tau} & 0 & 0 & 0 & \cdot & 0 & 0 & 0 & 0 \\ \frac{1}{\tau^2} & -\frac{2}{\tau^2} & \frac{1}{\tau^2} & 0 & 0 & \cdot & 0 & 0 & 0 & 0 \\ \end{bmatrix}\_{(N+1)\times(N+1)}$$

where:

$$a = \frac{1}{\pi h^2}, \; b = -\frac{1}{\pi^3}, \; c = \frac{2}{\pi h^2}.$$

$$\boldsymbol{\varrho}\_{n} = \left[ \begin{array}{c} \boldsymbol{\varrho}\_{n}^{IN} \\ \boldsymbol{\varrho}\_{n}^{IN} \\ \vdots \\ \boldsymbol{\varrho}\_{n}^{(l+1)N} \end{array} \right]\_{(N+1)\times 1}, \quad \left\{ \begin{array}{c} \boldsymbol{\varrho}\_{n}^{IN} = \cos \mathbf{x}\_{n} \ -M \le k \le 0, \ 0 \le n \le M, \\\\ \boldsymbol{\varrho}\_{n}^{k} = \boldsymbol{f}(t\_{k}, \mathbf{x}\_{n}) = -0.1 \frac{\boldsymbol{u}\_{n+1}^{k-N} - 2\boldsymbol{u}\_{n}^{k-N} - \boldsymbol{u}\_{n-1}^{k-N}}{h^{2}} \\ -2e^{-t\_{k}} \cos \mathbf{x}\_{n} - 0.1e^{-(t\_{k-N})} \cos \mathbf{x}\_{n}, \\\\ \boldsymbol{t}\_{k} = k\pi, lN+1 \le k \le (l+1)N-2, \\\\ \boldsymbol{l} = 0, 1, \cdots, \; 1 \le n \le M-1, \\\\ \boldsymbol{\varrho}\_{n}^{(l+1)N-1} = -\cos \mathbf{x}\_{n} - M \le k \le 0, \ 0 \le n \le M, \\\\ \boldsymbol{\varrho}\_{n}^{(l+1)N} = \cos \mathbf{x}\_{n}, -M \le k \le 0, \ 0 \le n \le M, \end{array} \right\}$$

and *D* = *IN*+<sup>1</sup> is the identity matrix.

To solve this second order difference problem, we use the following formula:

$$
\mu\_n = \mu\_{n+1}\mu\_{n+1} + \beta\_{n+1} \quad n = M-1, \dots, 1, 0,\tag{41}
$$

where *u<sup>M</sup>* = (*I* − *αM*) −1 *βM*, *α<sup>j</sup>* (*j* = 1, ..., *M* − 1) are (*N* + 1) × (*N* + 1) square matrices, *β<sup>j</sup>* (*j* = 1, ..., *M* − 1) are (*N* + 1) × 1 column matrices, *α*<sup>1</sup> is the identity, and *β*<sup>1</sup> is zero matrices, and:

$$\mathfrak{a}\_{n+1} = -\left(\mathcal{B} + \mathbb{C}\mathfrak{a}\_{\mathfrak{n}}\right)^{-1} A\_{\prime} \tag{42}$$

$$\mathfrak{f}\_{n+1} = \left(\mathcal{B} + \mathbb{C}\mathfrak{a}\_{\mathfrak{n}}\right)^{-1} \left(D\mathfrak{o}\_{\mathfrak{n}} - \mathbb{C}\mathfrak{f}\_{\mathfrak{n}}\right), n = 1, \dots, M-1.$$

The errors are computed by:

$$E\_M^N = \max\_{lN+1 \le k \le (l+1)N-1, 1 \le n \le M-1} \left| u(t\_k, \mathbf{x}\_n) - u\_n^k \right| \tag{43}$$

of the numerical solutions, where *u*(*t<sup>k</sup>* , *xn*) represents the exact solution and *u k n* represents the numerical solution at (*t<sup>k</sup>* , *xn*), and the results are given in Table 1.

**Table 1.** Errors of difference scheme (DS) (39).


#### *4.2. Two-Dimensional Problem*

Second, the mixed problem with the Dirichlet condition:

$$\begin{cases} \begin{aligned} &u\_{\rm{III}}(t,\mathbf{x},\mathbf{y}) - u\_{\rm{Ixx}}(t,\mathbf{x},\mathbf{y}) - u\_{\rm{Iyy}}(t,\mathbf{x},\mathbf{y}) = -0.1u\_{\rm{Xx}}(t-1,\mathbf{x}) - 0.1u\_{\rm{Jy}}(t-1,\mathbf{x},\mathbf{y}) \\ &-3e^{-t}\sin x \sin y - 0.2e^{-(t-1)}\sin x \sin y, \; t > 0, \; 0 < \mathbf{x}, \mathbf{y} < \pi, \\ &u(t,0,\mathbf{y}) = u(t,\pi,\mathbf{y}) = 0, \; 0 \le t < \infty, \; 0 \le \mathbf{y} \le \pi, \\ &u(t,\mathbf{x},0) = u(t,\mathbf{x},\pi) = 0, \; 0 \le t < \infty, \; 0 \le \mathbf{x} \le \pi \\ &u(t,\mathbf{x},\mathbf{y}) = e^{-t}\sin x \sin y, \; -1 \le t \le 0, \; 0 \le \mathbf{x}, \mathbf{y} \le \pi \end{aligned} \end{cases} \tag{44}$$

for the two-dimensional DPDE is considered. The exact solution of Problem (44) is *u* (*t*, *x*, *y*) = *e* −*t* sin *x* sin *y*.

Applying DS (4) to the problem (44), we get the following DS of the first order of accuracy in *t*:

 *u k*+2 *<sup>n</sup>*,*<sup>m</sup>* − 3*u k*+1 *<sup>n</sup>*,*<sup>m</sup>* + 3*u k <sup>n</sup>*,*<sup>m</sup>* − *u k*−1 *n*,*m τ* 3 − *u k*+2 *<sup>n</sup>*+1,*<sup>m</sup>* − *u k*+1 *<sup>n</sup>*+1,*<sup>m</sup>* − 2 *u k*+2 *<sup>n</sup>*,*<sup>m</sup>* − *u k*+1 *n*,*m* + *u k*+2 *<sup>n</sup>*−1,*<sup>m</sup>* <sup>−</sup> *<sup>u</sup> k*+1 *n*−1,*m τh* 2 − *u k*+2 *<sup>n</sup>*,*m*+<sup>1</sup> − *u k*+1 *<sup>n</sup>*,*m*+<sup>1</sup> − 2 *u k*+2 *<sup>n</sup>*,*<sup>m</sup>* − *u k*+1 *n*,*m* + *u k*+2 *<sup>n</sup>*,*m*−<sup>1</sup> <sup>−</sup> *<sup>u</sup> k*+1 *n*,*m*−1 *τh* 2 = −0.1 *u k*−*N <sup>n</sup>*+1,*<sup>m</sup>* − 2*u k*−*N <sup>n</sup>*,*<sup>m</sup>* − *u k*−*N n*−1,*m h* 2 − *u k*−*N <sup>n</sup>*,*m*+<sup>1</sup> − 2*u k*−*N <sup>n</sup>*,*<sup>m</sup>* − *u k*−*N n*,*m*−1 *h* 2 −3*e* <sup>−</sup>*t<sup>k</sup>* sin *<sup>x</sup><sup>n</sup>* sin *<sup>y</sup><sup>m</sup>* <sup>−</sup> 0.2*<sup>e</sup>* <sup>−</sup>*tk*−*<sup>N</sup>* sin *x<sup>n</sup>* sin *ym*, *t<sup>k</sup>* = *kτ*, *lN* + 1 ≤ *k* ≤ (*l* + 1)*N* − 2, *l* = 0, 1, ..., 1 ≤ *k* ≤ *N* − 1 , *Nτ* = 1, *x<sup>n</sup>* = *nh*, *y<sup>m</sup>* = *mh* 1 ≤ *n*, *m* ≤ *M* − 1, *Mh* = *π*, *u k <sup>n</sup>*,*<sup>m</sup>* = sin *x<sup>n</sup>* sin *ym*, −*N* ≤ *k* ≤ 0, 0 ≤ *n*, *m* ≤ *M*, *u k*+1 *<sup>n</sup>*,*<sup>m</sup>* − *u k n*,*m τ* = sin *x<sup>n</sup>* sin *ym*, −*N* ≤ *k* ≤ 0, 0 ≤ *n*, *m* ≤ *M*, *u k*+2 *<sup>n</sup>*,*<sup>m</sup>* − 2*u k*+1 *<sup>n</sup>*,*<sup>m</sup>* + *u k n*,*m τ* 2 = sin *x<sup>n</sup>* sin *ym*, −*N* ≤ *k* ≤ 0, 0 ≤ *n*, *m* ≤ *M*, *u k* 0,*<sup>m</sup>* = *u k <sup>M</sup>*,*<sup>m</sup>* = 0, 0 ≤ *k* ≤ *N*, 0 ≤ *m* ≤ *M*, *u k <sup>n</sup>*,0 = *u k <sup>n</sup>*,*<sup>M</sup>* = 0, 0 ≤ *k* ≤ *N*, 0 ≤ *n* ≤ *M*. (45)

It can be written as the second order difference problem with the matrix coefficients' form:

$$A\,u\_{n+1} + Bu\_n + Cu\_{n-1,\*} = D\varphi\_n, \quad 1 \le n \le M-1; \\ u\_0 = 0, \; u\_M = 0. \tag{46}$$

Here and in the future, we put:

$$\mathbf{u}\_{\mathbf{s}} = \left[ \mathbf{u}\_{0,\mathbf{s'}}^{lN} \cdots , \mathbf{u}\_{0,\mathbf{s}}^{(l+1)N}, \mathbf{u}\_{1,\mathbf{s'}}^{lN} \cdots , \mathbf{u}\_{1,\mathbf{s}}^{(l+1)N} \right.\\ \left. \text{ } \cdots \text{ } \right., \left. \mathbf{u}\_{M,\mathbf{s'}}^{lN} \cdots , \mathbf{u}\_{M,\mathbf{s}}^{(l+1)N} \right]\_{(N+1)(M+1)\times 1}^{T} \text{ s} = \mathbf{n}\_{\mathbf{s}} n \pm 1 \ldots n \pm 1.$$

*A*, *B*, *C*, *I* are (*N* + 1)(*M* + 1) × (*N* + 1)(*M* + 1) square matrices, and *I*, *R* are identity matrices.

Here,

$$a = \frac{1}{\tau h^2}, \ b = -\frac{1}{\tau^3}, \ c = \frac{2}{\tau h^2},$$

$$A = \mathbb{C} = \begin{bmatrix} 0 & 0 & \cdot & 0 & 0 \\ 0 & E & \cdot & 0 & 0 \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ 0 & 0 & \cdot & E & 0 \\ 0 & 0 & \cdot & 0 & 0 \end{bmatrix},$$

$$B = \begin{bmatrix} Q & O & O & O & \cdot & O & O & O & O \\ E & D & E & O & \cdot & O & O & O & O \\ O & E & D & E & \cdot & O & O & O & O \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ O & O & O & O & \cdot & 0 & E & D & E \\ O & O & O & O & O & O & O & O & O \end{bmatrix},$$

where:

$$E = \begin{bmatrix} 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 0 & a & -a & \cdots & 0 & 0 & 0 \\ 0 & 0 & a & -a & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & 0 & a & -a \\ 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \end{bmatrix},$$

$$D = \begin{bmatrix} 1 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\ b & -3b & 3b - c & b - c & \cdots & 0 & 0 & 0 \\ 0 & b & -3b & 3b - c & \cdots & 0 & 0 & 0 \\ 0 & 0 & b & -3b & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & c - b & 0 & 0 \\ 0 & 0 & 0 & 0 & \cdots & -3b & 3b - c & c - b & 0 \\ 0 & 0 & 0 & 0 & \cdots & -3b & 3b - c & c - b \\ -\frac{1}{\tau} & \frac{1}{\tau} & 0 & 0 & \cdots & 0 & 0 & 0 \\ \frac{1}{\tau^2} & -\frac{2}{\tau^2} & \frac{1}{\tau^2} & 0 & \cdots & 0 & 0 & 0 \end{bmatrix}.$$

*Q* = *I* (*N*+1)×(*N*+1) , *<sup>O</sup>* = *<sup>O</sup>*(*N*+1)×(*N*+1) ,

*ϕ<sup>n</sup>* = *ϕ lN* 0,*n* . . . *ϕ* (*l*+1)*N* 0,*n ϕ lN* 1,*n* . . . *ϕ* (*l*+1)*N* 1,*n* . . . *ϕ lN M*,*n* . . . *ϕ* (*l*+1)*N M*,*n* (*M*+1)(*N*+1)×1 , *ϕ lN <sup>m</sup>*,*<sup>n</sup>* = sin *x<sup>n</sup>* sin *ym*, −*M* ≤ *k* ≤ 0, 0 ≤ *n*, *m* ≤ *M*, *ϕ k <sup>m</sup>*,*<sup>n</sup>* = −0.1 *u k*−*N <sup>n</sup>*+1,*<sup>m</sup>* − 2*u k*−*N <sup>n</sup>*,*<sup>m</sup>* − *u k*−*N n*−1,*m h* 2 − *u k*−*N <sup>n</sup>*,*m*+<sup>1</sup> − 2*u k*−*N <sup>n</sup>*,*<sup>m</sup>* − *u k*−*N n*,*m*−1 *h* 2 −3*e* <sup>−</sup>*t<sup>k</sup>* sin *x<sup>n</sup>* sin *y<sup>m</sup>* −0.2*e* <sup>−</sup>*tk*−*<sup>N</sup>* sin *x<sup>n</sup>* sin *ym*, *t<sup>k</sup>* = *kτ*, *lN* + 1 ≤ *k* ≤ (*l* + 1)*N* − 2, *l* = 0, 1, · · · , 1 ≤ *n*, *m* ≤ *M* − 1, *ϕ* (*l*+1)*N*−1 *<sup>m</sup>*,*<sup>n</sup>* = − sin *x<sup>n</sup>* sin *ym*, −*M* ≤ *k* ≤ 0, 0 ≤ *n*, *m* ≤ *M*, *ϕ* (*l*+1)*N <sup>m</sup>*,*<sup>n</sup>* = sin *x<sup>n</sup>* sin *ym*, −*M* ≤ *k* ≤ 0, 0 ≤ *n*, *m* ≤ *M*, 

To solve this second order difference problem, we use the following formula:

$$
\mu\_n = \alpha\_{n+1}\mu\_{n+1} + \beta\_{n+1}, \quad n = M-1, \dots, 1,\\
\mu\_M = 0,\tag{47}
$$

where *α<sup>j</sup>* (*j* = 1, ..., *M* − 1) are (*N* + 1)(*M* + 1) × (*N* + 1)(*M* + 1) square matrices, *β<sup>j</sup>* (*j* = 1, ..., *M* − 1) are (*N* + 1)(*M* + 1) × 1 column matrices, and *α*<sup>1</sup> and *β*<sup>1</sup> are zero matrices and:

$$\mathfrak{a}\_{n+1} = -\left(\mathcal{B} + \mathbb{C}\mathfrak{a}\_{\mathfrak{n}}\right)^{-1} A\_{\mathfrak{n}\prime} \tag{48}$$

$$\mathfrak{F}\_{n+1} = \left(\mathcal{B} + \mathbb{C}\mathfrak{a}\_{\mathfrak{n}}\right)^{-1} \left(\mathcal{D}\mathfrak{q}\_{\mathfrak{n}} - \mathbb{C}\mathfrak{f}\_{\mathfrak{n}}\right), n = 1, \dots, \mathcal{M} - 1.$$

The errors are computed by:

$$lE\_M^N = \max\_{lN+1 \le k \le (l+1)N-1, 1 \le n, m \le M-1} \left| u(t\_k, \mathbf{x}\_{n\nu} y\_m) - u\_{n,m}^k \right| \tag{49}$$

of the numerical solutions, where *u*(*t<sup>k</sup>* , *xn*, *ym*) represents the exact solution and *u k <sup>n</sup>*,*<sup>m</sup>* represents the numerical solution at (*t<sup>k</sup>* , *xn*, *ym*), and the results are given in Table 2.

**Table 2.** Errors of the difference scheme (45).


As seen in Tables 1 and 2, we obtained some numerical results. If *M* and *N* are doubled, the values of the errors decrease by a factor of approximately 1/2 for DS (39) and (45), respectively.

#### **5. Conclusions**


$$\begin{cases} \ u\_{ttt}(t) + Au\_t(t) = f\left(t, u(t - w)\right), \ 0 < t < \infty, \\\ u(t) = g(t), -w \le t \le 0 \end{cases} \tag{50}$$

in a Hilbert space *H* with an unbounded operator *A*. Moreover, applying the method of [28], we can investigate the convergence of DSs for the numerical solution of Problem (50).

**Author Contributions:** Investigation, A.A., E.H. and S.I.. All authors read and approved the final version of this manuscript.

**Acknowledgments:** The authors thank the reviewers and A. L.Skubachevskii (RUDN) and Ch. Ashyralyyev (CU) for their helpful suggestions.

**Funding:** This research was funded by "Russian Foundation for Basic Research (RFBR) grant number 16–01–00450.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Sample Availability:** Samples of the compounds are available from the authors.

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