Mathematics

Research

8 pages, 255 KiB

Open AccessFeature PaperArticle

Radially Symmetric Positive Solutions of the Dirichlet Problem for the p-Laplace Equation

by Bo Yang

Mathematics 2024, 12(15), 2351; https://doi.org/10.3390/math12152351 - 27 Jul 2024

Viewed by 544

We consider the p-Laplace boundary value problem with the Dirichlet boundary condition. A new lower estimate for positive solutions of the problem is obtained. As an application of this new lower estimate, some sufficient conditions for the existence and nonexistence of positive [...] Read more.

We consider the p-Laplace boundary value problem with the Dirichlet boundary condition. A new lower estimate for positive solutions of the problem is obtained. As an application of this new lower estimate, some sufficient conditions for the existence and nonexistence of positive solutions for the p-Laplace problem are obtained. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

13 pages, 292 KiB

Open AccessArticle

Minimum Principles for Sturm–Liouville Inequalities and Applications

by Phuc Ngo and Kunquan Lan

Mathematics 2024, 12(13), 2088; https://doi.org/10.3390/math12132088 - 3 Jul 2024

Viewed by 618

Abstract

A minimum principle for a Sturm–Liouville (S-L) inequality is obtained, which shows that the minimum value of a nonconstant solution of a S-L inequality never occurs in the interior of the domain (a closed interval) of the solution. The minimum principle is then [...] Read more.

A minimum principle for a Sturm–Liouville (S-L) inequality is obtained, which shows that the minimum value of a nonconstant solution of a S-L inequality never occurs in the interior of the domain (a closed interval) of the solution. The minimum principle is then applied to prove that any nonconstant solutions of S-L inequalities subject to separated inequality boundary conditions (IBCs) must be strictly positive in the interiors of their domains and are increasing or decreasing for some of these IBCs. These positivity results are used to prove the uniqueness of the solutions of linear S-L equations with separated BCs. All of these results hold for the corresponding second-order differential inequalities (or equations), which are special cases of S-L inequalities (or equations). These results are applied to two models arising from the source distribution of the human head and chemical reactor theory. The first model is governed by a nonlinear S-L equation, while the second one is governed by a nonlinear second-order differential equation. For the first model, the explicit solutions are not available, and there are no results on the existence of solutions of the first model. Our results show that all the nonconstant solutions are increasing and are strictly positive solutions. For the second model, many results on the uniqueness of the solutions and the existence of multiple solutions have been obtained before. Our results are applied to prove that all the nonconstant solutions are decreasing and strictly positive. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

20 pages, 321 KiB

Open AccessArticle

Global Existence of Small Data Solutions to Weakly Coupled Systems of Semi-Linear Fractional σ–Evolution Equations with Mass and Different Nonlinear Memory terms

by Seyyid Ali Saiah, Abdelatif Kainane Mezadek, Mohamed Kainane Mezadek, Abdelhamid Mohammed Djaouti, Ashraf Al-Quran and Ali M. A. Bany Awad

Mathematics 2024, 12(13), 1942; https://doi.org/10.3390/math12131942 - 22 Jun 2024

Viewed by 566

Abstract

We study in this paper the long-term existence of solutions to the system of weakly coupled equations with fractional evolution and various nonlinearities. Our objective is to determine the connection between the regularity assumptions on the initial data, the memory terms, and the [...] Read more.

We study in this paper the long-term existence of solutions to the system of weakly coupled equations with fractional evolution and various nonlinearities. Our objective is to determine the connection between the regularity assumptions on the initial data, the memory terms, and the permissible range of exponents in a specific equation. Using

L^{p} - L^{q}

estimates for solutions to the corresponding linear fractional

σ

–evolution equations with vanishing right-hand sides, and applying a fixed-point argument, the existence of small data solutions is established for some admissible range of powers

(p_{1}, p_{2}, \dots, p_{k})

. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

16 pages, 1127 KiB

Open AccessArticle

Stability and Bifurcation Analysis in a Discrete Predator–Prey System of Leslie Type with Radio-Dependent Simplified Holling Type IV Functional Response

by Luyao Lv and Xianyi Li

Mathematics 2024, 12(12), 1803; https://doi.org/10.3390/math12121803 - 10 Jun 2024

Cited by 1 | Viewed by 597

Abstract

In this paper, we use a semi-discretization method to consider the predator–prey model of Leslie type with ratio-dependent simplified Holling type IV functional response. First, we discuss the existence and stability of the positive fixed point in total parameter space. Subsequently, through using [...] Read more.

In this paper, we use a semi-discretization method to consider the predator–prey model of Leslie type with ratio-dependent simplified Holling type IV functional response. First, we discuss the existence and stability of the positive fixed point in total parameter space. Subsequently, through using the central manifold theorem and bifurcation theory, we obtain sufficient conditions for the flip bifurcation and Neimark–Sacker bifurcation of this system to occur. Finally, the numerical simulations illustrate the existence of Neimark–Sacker bifurcation and obtain some new dynamical phenomena of the system—the existence of a limit cycle. Corresponding biological meanings are also formulated. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

► Show Figures

Figure 1

9 pages, 243 KiB

Open AccessFeature PaperArticle

Differentiation of Solutions of Caputo Boundary Value Problems with Respect to Boundary Data

by Jeffrey W. Lyons

Mathematics 2024, 12(12), 1790; https://doi.org/10.3390/math12121790 - 8 Jun 2024

Viewed by 728

Abstract

Under suitable continuity and uniqueness conditions, solutions of an

α

order Caputo fractional boundary value problem are differentiated with respect to boundary values and boundary points. This extends well-known results for nth order boundary value problems. The approach used applies a standard algorithm [...] Read more.

Under suitable continuity and uniqueness conditions, solutions of an

α

order Caputo fractional boundary value problem are differentiated with respect to boundary values and boundary points. This extends well-known results for nth order boundary value problems. The approach used applies a standard algorithm to achieve the result and makes heavy use of recent results for differentiation of solutions of Caputo fractional intial value problems with respect to initial conditions and continuous dependence for Caputo fractional boundary value problems. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

16 pages, 298 KiB

Open AccessArticle

Kamenev-Type Criteria for Testing the Asymptotic Behavior of Solutions of Third-Order Quasi-Linear Neutral Differential Equations

by Hail S. Alrashdi, Wedad Albalawi, Ali Muhib, Osama Moaaz and Elmetwally M. Elabbasy

Mathematics 2024, 12(11), 1734; https://doi.org/10.3390/math12111734 - 3 Jun 2024

Viewed by 504

Abstract

This paper aims to study the asymptotic properties of nonoscillatory solutions (eventually positive or negative) of a class of third-order canonical neutral differential equations. We use Riccati substitution to reduce the order of the considered equation, and then we use the Philos function [...] Read more.

This paper aims to study the asymptotic properties of nonoscillatory solutions (eventually positive or negative) of a class of third-order canonical neutral differential equations. We use Riccati substitution to reduce the order of the considered equation, and then we use the Philos function class to obtain new criteria of the Kamenev type, which guarantees that all nonoscillatory solutions converge to zero. This approach is characterized by the possibility of applying its conditions to a wider area of equations. This is not the only aspect that distinguishes our results; we also use improved relationships between the solution and the corresponding function, which in turn is reflected in a direct improvement of the criteria. The findings in this article extend and generalize previous findings in the literature and also improve some of these findings. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

24 pages, 365 KiB

Open AccessFeature PaperArticle

Existence of Solutions to a System of Fractional

q

-Difference Boundary Value Problems

by Alexandru Tudorache and Rodica Luca

Mathematics 2024, 12(9), 1335; https://doi.org/10.3390/math12091335 - 27 Apr 2024

Viewed by 880

Abstract

We are investigating the existence of solutions to a system of two fractional

q

-difference equations containing fractional

q

-integral terms, subject to multi-point boundary conditions that encompass

q

-derivatives and fractional

q

-derivatives of different orders. In our main results, we rely [...] Read more.

We are investigating the existence of solutions to a system of two fractional

q

-difference equations containing fractional

q

-integral terms, subject to multi-point boundary conditions that encompass

q

-derivatives and fractional

q

-derivatives of different orders. In our main results, we rely on various fixed point theorems, such as the Leray–Schauder nonlinear alternative, the Schaefer fixed point theorem, the Krasnosel’skii fixed point theorem for the sum of two operators, and the Banach contraction mapping principle. Finally, several examples are provided to illustrate our findings. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

17 pages, 946 KiB

Open AccessFeature PaperArticle

Bifurcation Analysis for an OSN Model with Two Delays

by Liancheng Wang and Min Wang

Mathematics 2024, 12(9), 1321; https://doi.org/10.3390/math12091321 - 26 Apr 2024

Viewed by 779

Abstract

In this research, we introduce and analyze a mathematical model for online social networks, incorporating two distinct delays. These delays represent the time it takes for active users within the network to begin disengaging, either with or without contacting non-users of online social [...] Read more.

In this research, we introduce and analyze a mathematical model for online social networks, incorporating two distinct delays. These delays represent the time it takes for active users within the network to begin disengaging, either with or without contacting non-users of online social platforms. We focus particularly on the user prevailing equilibrium (UPE), denoted as

P^{*}

, and explore the role of delays as parameters in triggering Hopf bifurcations. In doing so, we find the conditions under which Hopf bifurcations occur, then establish stable regions based on the two delays. Furthermore, we delineate the boundaries of stability regions wherein bifurcations transpire as the delays cross these thresholds. We present numerical simulations to illustrate and validate our theoretical findings. Through this interdisciplinary approach, we aim to deepen our understanding of the dynamics inherent in online social networks. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

► Show Figures

Figure 1

9 pages, 241 KiB

Open AccessArticle

The Blow-Up of the Local Energy Solution to the Wave Equation with a Nontrivial Boundary Condition

by Yulong Liu

Mathematics 2024, 12(9), 1317; https://doi.org/10.3390/math12091317 - 25 Apr 2024

Viewed by 827

Abstract

In this study, we examine the wave equation with a nontrivial boundary condition. The main target of this study is to prove the local-in-time existence and the blow-up in finite time of the energy solution. Through the construction of an auxiliary function and [...] Read more.

In this study, we examine the wave equation with a nontrivial boundary condition. The main target of this study is to prove the local-in-time existence and the blow-up in finite time of the energy solution. Through the construction of an auxiliary function and the imposition of appropriate conditions on the initial data, we establish the both lower and upper bounds for the blow-up time of the solution. Meanwhile, based on these estimates, we obtain the result of the local-in-time existence and the blow-up of the energy solution. This approach enhances our understanding of the dynamics leading to blow-up in the considered condition. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

20 pages, 316 KiB

Open AccessFeature PaperArticle

A Signed Maximum Principle for Boundary Value Problems for Riemann–Liouville Fractional Differential Equations with Analogues of Neumann or Periodic Boundary Conditions

by Paul W. Eloe, Yulong Li and Jeffrey T. Neugebauer

Mathematics 2024, 12(7), 1000; https://doi.org/10.3390/math12071000 - 27 Mar 2024

Viewed by 746

Abstract

Sufficient conditions are obtained for a signed maximum principle for boundary value problems for Riemann–Liouville fractional differential equations with analogues of Neumann or periodic boundary conditions in neighborhoods of simple eigenvalues. The primary objective is to exhibit four specific boundary value problems for [...] Read more.

Sufficient conditions are obtained for a signed maximum principle for boundary value problems for Riemann–Liouville fractional differential equations with analogues of Neumann or periodic boundary conditions in neighborhoods of simple eigenvalues. The primary objective is to exhibit four specific boundary value problems for which the sufficient conditions can be verified. To show an application of the signed maximum principle, a method of upper and lower solutions coupled with monotone methods is developed to obtain sufficient conditions for the existence of a maximal solution and a minimal solution of a nonlinear boundary value problem. A specific example is provided to show that sufficient conditions for the nonlinear problem can be realized. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

14 pages, 304 KiB

Open AccessFeature PaperArticle

Periodic Solutions to Nonlinear Second-Order Difference Equations with Two-Dimensional Kernel

by Daniel Maroncelli

Mathematics 2024, 12(6), 849; https://doi.org/10.3390/math12060849 - 14 Mar 2024

Viewed by 879

Abstract

In this work, we provide conditions for the existence of periodic solutions to nonlinear, second-order difference equations of the form

y (t + 2) + b y (t + 1) + c y (t) = g (y (t))

, where b and c are real parameters,

c \neq 0

, and

g : R \to R

is continuous. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

28 pages, 398 KiB

Open AccessArticle

Chains with Connections of Diffusion and Advective Types

by Sergey Kashchenko

Mathematics 2024, 12(6), 790; https://doi.org/10.3390/math12060790 - 7 Mar 2024

Viewed by 670

Abstract

The local dynamics of a system of oscillators with a large number of elements and with diffusive- and advective-type couplings containing a large delay are studied. Critical cases in the problem of the stability of the zero equilibrium state are singled out, and [...] Read more.

The local dynamics of a system of oscillators with a large number of elements and with diffusive- and advective-type couplings containing a large delay are studied. Critical cases in the problem of the stability of the zero equilibrium state are singled out, and it is shown that all of them have infinite dimensions. Applying special methods of infinite normalization, we construct quasinormal forms, namely, nonlinear boundary value problems of the parabolic type, whose nonlocal dynamics determine the behavior of the solutions of the initial system in a small neighborhood of the equilibrium state. These quasinormal forms contain either two or three spatial variables, which emphasizes the complexity of the dynamical properties of the original problem. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

14 pages, 300 KiB

Open AccessArticle

Existence and Limit Behavior of Constraint Minimizers for a Varying Non-Local Kirchhoff-Type Energy Functional

by Xincai Zhu and Hanxiao Wu

Mathematics 2024, 12(5), 661; https://doi.org/10.3390/math12050661 - 23 Feb 2024

Viewed by 691

Abstract

In this paper, we study the constrained minimization problem for an energy functional which is related to a Kirchhoff-type equation. For

s = 1

, there many articles have analyzed the limit behavior of minimizers when

η > 0

as [...] Read more.

In this paper, we study the constrained minimization problem for an energy functional which is related to a Kirchhoff-type equation. For

s = 1

, there many articles have analyzed the limit behavior of minimizers when

η > 0

as

b \to 0^{+}

or

b > 0

as

η \to 0^{+}

. When the equation involves a varying non-local term

{(\int_{R^{3}} {| \nabla u |}^{2} d x)}^{s}

, we give a detailed limit behavior analysis of constrained minimizers for any positive sequence

{η_{k}}

with

η_{k} \to 0^{+}

. The present paper obtains an interesting result on this topic and enriches the conclusions of previous works. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

11 pages, 235 KiB

Open AccessArticle

Multivalued Contraction Fixed-Point Theorem in b-Metric Spaces

by Bachir Slimani, John R. Graef and Abdelghani Ouahab

Mathematics 2024, 12(4), 567; https://doi.org/10.3390/math12040567 - 13 Feb 2024

Viewed by 985

Abstract

The authors explore fixed-point theory in b-metric spaces and strong b-metric spaces. They wish to prove some new extensions of the Covitz and Nadler fixed-point theorem in b-metric spaces. In so doing, they wish to answer a question proposed by [...] Read more.

The authors explore fixed-point theory in b-metric spaces and strong b-metric spaces. They wish to prove some new extensions of the Covitz and Nadler fixed-point theorem in b-metric spaces. In so doing, they wish to answer a question proposed by Kirk and Shahzad about Nadler’s theorem holding in strong b-metric spaces. In addition, they offer an improvement to the fixed-point theorem proven by Dontchev and Hager. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

35 pages, 444 KiB

Open AccessArticle

Multiplicity Results of Solutions to the Double Phase Problems of Schrödinger–Kirchhoff Type with Concave–Convex Nonlinearities

by Yun-Ho Kim and Taek-Jun Jeong

Mathematics 2024, 12(1), 60; https://doi.org/10.3390/math12010060 - 24 Dec 2023

Cited by 2 | Viewed by 999

Abstract

The present paper is devoted to establishing several existence results for infinitely many solutions to Schrödinger–Kirchhoff-type double phase problems with concave–convex nonlinearities. The first aim is to demonstrate the existence of a sequence of infinitely many large-energy solutions by applying the fountain theorem [...] Read more.

The present paper is devoted to establishing several existence results for infinitely many solutions to Schrödinger–Kirchhoff-type double phase problems with concave–convex nonlinearities. The first aim is to demonstrate the existence of a sequence of infinitely many large-energy solutions by applying the fountain theorem as the main tool. The second aim is to obtain that our problem admits a sequence of infinitely many small-energy solutions. To obtain these results, we utilize the dual fountain theorem. In addition, we prove the existence of a sequence of infinitely many weak solutions converging to 0 in

L^{\infty}

-space. To derive this result, we exploit the dual fountain theorem and the modified functional method. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

28 pages, 374 KiB

Open AccessArticle

Global Existence, Blowup, and Asymptotic Behavior for a Kirchhoff-Type Parabolic Problem Involving the Fractional Laplacian with Logarithmic Term

by Zihao Guan and Ning Pan

Mathematics 2024, 12(1), 5; https://doi.org/10.3390/math12010005 - 19 Dec 2023

Cited by 1 | Viewed by 872

Abstract

In this paper, we studied a class of semilinear pseudo-parabolic equations of the Kirchhoff type involving the fractional Laplacian with logarithmic nonlinearity: [...] Read more.

In this paper, we studied a class of semilinear pseudo-parabolic equations of the Kirchhoff type involving the fractional Laplacian with logarithmic nonlinearity:

\{\begin{matrix} u_{t} + M ({[u]}_{s}^{2}) {(- Δ)}^{s} u + {(- Δ)}^{s} u_{t} = {| u |}^{p - 2} u \ln | u |, & in Ω \times (0, T), \\ u (x, 0) = u_{0} (x), & in Ω, \\ u (x, t) = 0, & on \partial Ω \times (0, T), \end{matrix}

, where

{[u]}_{s}

is the Gagliardo semi-norm of u,

{(- Δ)}^{s}

is the fractional Laplacian,

s \in (0, 1)

,

2 λ < p < 2_{s}^{*} = 2 N / (N - 2 s)

,

Ω \in R^{N}

is a bounded domain with

N > 2 s

, and

u_{0}

is the initial function. To start with, we combined the potential well theory and Galerkin method to prove the existence of global solutions. Finally, we introduced the concavity method and some special inequalities to discuss the blowup and asymptotic properties of the above problem and obtained the upper and lower bounds on the blowup at the sublevel and initial level. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

Journal Menu

Journal Browser

Advances in Differential and Difference Equations and Their Applications

Share This Special Issue

Special Issue Editors

Special Issue Information

Keywords

Benefits of Publishing in a Special Issue

Published Papers (16 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI