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22 pages, 753 KB

Open AccessArticle

Existence and Global Exponential Stability of Equilibrium for an Epidemic Model with Piecewise Constant Argument of Generalized Type

by Kuo-Shou Chiu and Fernando Córdova-Lepe

Axioms 2025, 14(7), 514; https://doi.org/10.3390/axioms14070514 - 3 Jul 2025

Viewed by 464

Abstract

The authors investigate an epidemic model described by a differential equation, which includes a piecewise constant argument of the generalized type (DEPCAG). In this work, the main goal is to find an invariant region for the system and prove the existence and uniqueness [...] Read more.

The authors investigate an epidemic model described by a differential equation, which includes a piecewise constant argument of the generalized type (DEPCAG). In this work, the main goal is to find an invariant region for the system and prove the existence and uniqueness of solutions with the defined conditions using integral equations. On top of that, an auxiliary result is established, outlining the relationship between the unknown function values in the deviation argument and the time parameter. The stability analysis is conducted using the Lyapunov–Razumikhin method, adapted for differential equations with a piecewise constant argument of the generalized type. The trivial equilibrium’s stability is examined, and the stability of the positive equilibrium is assessed by transforming it into a trivial form. Finally, sufficient conditions for the uniform asymptotic stability of both the trivial and positive equilibria are established. Full article

(This article belongs to the Section Mathematical Analysis)

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25 pages, 1117 KB

Open AccessArticle

Instantaneously Impulsive Stabilization of Mittag–Leffler Numerical Chua’s Oscillator

by Huizhen Qu, Tianwei Zhang and Jianwen Zhou

Fractal Fract. 2025, 9(6), 332; https://doi.org/10.3390/fractalfract9060332 - 23 May 2025

Viewed by 447

Abstract

The Euler difference approach has become a prevalent tool in the research of integral order differential equations. Nevertheless, a review of the literature reveals a dearth of studies examining fractional order models using the exponential Euler difference approach. The present study employs an [...] Read more.

The Euler difference approach has become a prevalent tool in the research of integral order differential equations. Nevertheless, a review of the literature reveals a dearth of studies examining fractional order models using the exponential Euler difference approach. The present study employs an exponential Euler difference approach to examine the properties of nonlocal discrete-time oscillators with Mittag–Leffler kernels and piecewise features, with the aim of providing insights into a continuous-time nonlocal nonlinear system. By employing impulsive equations of variations in constants with different forms in conjunction with the Gronwall inequality, a controller that is capable of instantaneously responding and stabilizing the nonlocal discrete-time oscillator is devised. This controller is realized through an associated algorithm. As a case study, the primary outcome is applied to a problem of impulsive stabilization in nonlocal discrete-time Chua’s oscillator. This article presents a stabilizing algorithm for piecewise nonlocal discrete-time oscillators developed using a novel impulsive approach. Full article

(This article belongs to the Special Issue Fractional-Order Systems: Modeling and Control Applications in Engineering)

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17 pages, 751 KB

Open AccessArticle

Finite-Time Stability of a Class of Nonstationary Nonlinear Fractional Order Time Delay Systems: New Gronwall–Bellman Inequality Approach

by Mihailo P. Lazarević, Stjepko Pišl and Darko Radojević

Mathematics 2025, 13(9), 1490; https://doi.org/10.3390/math13091490 - 30 Apr 2025

Viewed by 350

Abstract

This paper aims to analyze finite-time stability (FTS) for a class of nonstationary nonlinear two-term fractional-order time-delay systems with

α, β \in (0, 2)

. Using a new type of generalized Gronwall–Bellman inequality, we derive new FTS stability criteria for these [...] Read more.

This paper aims to analyze finite-time stability (FTS) for a class of nonstationary nonlinear two-term fractional-order time-delay systems with

α, β \in (0, 2)

. Using a new type of generalized Gronwall–Bellman inequality, we derive new FTS stability criteria for these systems in terms of the Mittag–Leffler function. We demonstrate that our theoretical results are less conservative than those presented in the existing literature. Finally, we provide three numerical examples using a modified Adams–Bashforth–Moulton algorithm to illustrate the applicability of the proposed stability conditions. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing for Applied Mathematics)

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12 pages, 222 KB

Open AccessArticle

An Asymptotic Behavior Property of High-Order Nonlinear Dynamic Equations on Time Scales

by Yuan Yuan and Qinghua Ma

Axioms 2025, 14(4), 270; https://doi.org/10.3390/axioms14040270 - 2 Apr 2025

Viewed by 297

Abstract

In this work, by using one dynamic Gronwall–Bihari-type integral inequality on time scales, an interesting asymptotic behavior property of high-order nonlinear dynamic equations on time scales was obtained, which also generalized two classical results belong to Máté and Nevai’s and Agarwal and Bohner’s, [...] Read more.

In this work, by using one dynamic Gronwall–Bihari-type integral inequality on time scales, an interesting asymptotic behavior property of high-order nonlinear dynamic equations on time scales was obtained, which also generalized two classical results belong to Máté and Nevai’s and Agarwal and Bohner’s, respectively. Full article

(This article belongs to the Special Issue Infinite Dynamical System and Differential Equations)

16 pages, 376 KB

Open AccessArticle

Linear Sixth-Order Conservation Difference Scheme for KdV Equation

by Jie He, Jinsong Hu and Zhong Chen

Mathematics 2025, 13(7), 1132; https://doi.org/10.3390/math13071132 - 30 Mar 2025

Viewed by 365

Abstract

A numerical investigation is conducted for the initial boundary value problem of the Korteweg–de Vries (KdV) equation with homogeneous boundary conditions. Using the average implicit difference discretization, a second-order theoretical accuracy in time is achieved. For the spatial direction, a center-symmetric discretization coupled [...] Read more.

A numerical investigation is conducted for the initial boundary value problem of the Korteweg–de Vries (KdV) equation with homogeneous boundary conditions. Using the average implicit difference discretization, a second-order theoretical accuracy in time is achieved. For the spatial direction, a center-symmetric discretization coupled with the extrapolation technique is employed, yielding a three-level linear difference method with sixth-order accuracy. Consequently, the integration of these methods results in a linear finite difference scheme that accurately simulates the two conserved quantities of the original problem. Furthermore, theoretical results, including the convergence and stability of the proposed scheme, are proved using the discrete Sobolev inequality and the discrete Gronwall inequality. Numerical experiments validate the reliability of the scheme. Full article

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24 pages, 362 KB

Open AccessArticle

Stability and Controllability Analysis of Stochastic Fractional Differential Equations Under Integral Boundary Conditions Driven by Rosenblatt Process with Impulses

by Mohamed S. Algolam, Sadam Hussain, Bakri A. I. Younis, Osman Osman, Blgys Muflh, Khaled Aldwoah and Nidal Eljaneid

Fractal Fract. 2025, 9(3), 146; https://doi.org/10.3390/fractalfract9030146 - 26 Feb 2025

Cited by 1 | Viewed by 1079

Abstract

Differential equations are frequently used to mathematically describe many problems in real life, but they are always subject to intrinsic phenomena that are neglected and could influence how the model behaves. In some cases like ecosystems, electrical circuits, or even economic models, the [...] Read more.

Differential equations are frequently used to mathematically describe many problems in real life, but they are always subject to intrinsic phenomena that are neglected and could influence how the model behaves. In some cases like ecosystems, electrical circuits, or even economic models, the model may suddenly change due to outside influences. Occasionally, such changes start off impulsively and continue to exist for specific amounts of time. Non-instantaneous impulses are used in the creation of the models for this kind of scenario. In this paper, a new class of non-instantaneous impulsive

ψ

-Caputo fractional stochastic differential equations under integral boundary conditions driven by the Rosenblatt process was examined. Semigroup theory, stochastic theory, the Banach fixed-point theorem, and fractional calculus were applied to investigating the existence of piecewise continuous mild solutions for the systems under consideration. The impulsive Gronwall’s inequality was employed to establish the unique stability conditions for the system under consideration. Furthermore, we examined the controllability results of the proposed system. Finally, some examples were provided to demonstrate the validity of the presented work. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

32 pages, 453 KB

Open AccessArticle

Almost Periodic Solutions of Differential Equations with Generalized Piecewise Constant Delay

by Kuo-Shou Chiu

Mathematics 2024, 12(22), 3528; https://doi.org/10.3390/math12223528 - 12 Nov 2024

Cited by 1 | Viewed by 1241

Abstract

In this paper, we investigate differential equations with generalized piecewise constant delay, DEGPCD in short, and establish the existence and stability of a unique almost periodic solution that is exponentially stable. Our results are derived by utilizing the properties of the [...] Read more.

In this paper, we investigate differential equations with generalized piecewise constant delay, DEGPCD in short, and establish the existence and stability of a unique almost periodic solution that is exponentially stable. Our results are derived by utilizing the properties of the

(μ_{1}, μ_{2})

-exponential dichotomy, Cauchy and Green matrices, a Gronwall-type inequality for DEGPCD, and the Banach fixed point theorem. We apply these findings to derive new criteria for the existence, uniqueness, and convergence dynamics of almost periodic solutions in both the linear inhomogeneous and quasilinear DEGPCD systems through the

(μ_{1}, μ_{2})

-exponential dichotomy for difference equations. These results are novel and serve to recover, extend, and improve upon recent research. Full article

(This article belongs to the Special Issue The Delay Differential Equations and Their Applications)

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18 pages, 309 KB

Open AccessArticle

New Nonlinear Retarded Integral Inequalities and Their Applications to Nonlinear Retarded Integro-Differential Equations

by Mahvish Samar, Xinzhong Zhu, Abdul Shakoor and Mawia Osman

Axioms 2024, 13(6), 356; https://doi.org/10.3390/axioms13060356 - 27 May 2024

Cited by 1 | Viewed by 1076

Abstract

The purpose of this article is to present some new nonlinear retarded integral inequalities which can be utilized to study the existence, stability, boundedness, uniqueness, and asymptotic behavior of solutions of nonlinear retarded integro-differential equations, and these inequalities can be used in the [...] Read more.

The purpose of this article is to present some new nonlinear retarded integral inequalities which can be utilized to study the existence, stability, boundedness, uniqueness, and asymptotic behavior of solutions of nonlinear retarded integro-differential equations, and these inequalities can be used in the symmetrical properties of functions. These inequalities also generalize some former famous inequalities in the literature. Two examples as applications will be provided to demonstrate the strength of our inequalities in estimating the boundedness and global existence of the solution to initial value problems for nonlinear integro-differential equations and differential equations which can be seen in graphs. This research work will ensure opening new opportunities for studying nonlinear dynamic inequalities on a time-scale structure of a varying nature. Full article

(This article belongs to the Special Issue Advances in Difference Equations)

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16 pages, 323 KB

Open AccessArticle

A New Nonlinear Integral Inequality with a Tempered Ψ–Hilfer Fractional Integral and Its Application to a Class of Tempered Ψ–Caputo Fractional Differential Equations

by Milan Medved’, Michal Pospíšil and Eva Brestovanská

Axioms 2024, 13(5), 301; https://doi.org/10.3390/axioms13050301 - 1 May 2024

Cited by 3 | Viewed by 1660

Abstract

In this paper, the tempered

Ψ

–Riemann–Liouville fractional derivative and the tempered

Ψ

–Caputo fractional derivative of order

n - 1 < α < n \in N

are introduced for

C^{n - 1}

–functions. A nonlinear version of the second Henry–Gronwall inequality [...] Read more.

In this paper, the tempered

Ψ

–Riemann–Liouville fractional derivative and the tempered

Ψ

–Caputo fractional derivative of order

n - 1 < α < n \in N

are introduced for

C^{n - 1}

–functions. A nonlinear version of the second Henry–Gronwall inequality for integral inequalities with the tempered

Ψ

–Hilfer fractional integral is derived. By using this inequality, an existence and uniqueness result and a sufficient condition for the non-existence of blow-up solutions of nonlinear tempered

Ψ

–Caputo fractional differential equations are proved. Illustrative examples are given. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inequalities)

14 pages, 509 KB

Open AccessArticle

The Existence and Averaging Principle for a Class of Fractional Hadamard Itô–Doob Stochastic Integral Equations

by Mohamed Rhaima, Lassaad Mchiri and Abdellatif Ben Makhlouf

Symmetry 2023, 15(10), 1910; https://doi.org/10.3390/sym15101910 - 12 Oct 2023

Cited by 2 | Viewed by 1318

Abstract

In this paper, we investigate the existence and uniqueness properties pertaining to a class of fractional Hadamard Itô–Doob stochastic integral equations (FHIDSIE). Our study centers around the utilization of the Picard iteration technique (PIT), which not only establishes these fundamental properties but also [...] Read more.

In this paper, we investigate the existence and uniqueness properties pertaining to a class of fractional Hadamard Itô–Doob stochastic integral equations (FHIDSIE). Our study centers around the utilization of the Picard iteration technique (PIT), which not only establishes these fundamental properties but also unveils the remarkable averaging principle within FHIDSIE. To accomplish this, we harness powerful mathematical tools, including the Hölder and Gronwall inequalities. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus III)

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17 pages, 356 KB

Open AccessArticle

Nonlinear Integral Inequalities Involving Tempered Ψ-Hilfer Fractional Integral and Fractional Equations with Tempered Ψ-Caputo Fractional Derivative

by Milan Medveď, Michal Pospíšil and Eva Brestovanská

Fractal Fract. 2023, 7(8), 611; https://doi.org/10.3390/fractalfract7080611 - 8 Aug 2023

Cited by 5 | Viewed by 1408

Abstract

In this paper, the nonlinear version of the Henry–Gronwall integral inequality with the tempered

Ψ

-Hilfer fractional integral is proved. The particular cases, including the linear one and the nonlinear integral inequality of this type with multiple tempered

Ψ

-Hilfer fractional integrals, are [...] Read more.

In this paper, the nonlinear version of the Henry–Gronwall integral inequality with the tempered

Ψ

-Hilfer fractional integral is proved. The particular cases, including the linear one and the nonlinear integral inequality of this type with multiple tempered

Ψ

-Hilfer fractional integrals, are presented as corollaries. To illustrate the results, the problem of the nonexistence of blowing-up solutions of initial value problems for fractional differential equations with tempered

Ψ

-Caputo fractional derivative of order

0 < α < 1

, where the right side may depend on time, the solution, or its tempered

Ψ

-Caputo fractional derivative of lower order, is investigated. As another application of the integral inequalities, sufficient conditions for the

Ψ

-exponential stability of trivial solutions are proven for these kinds of differential equations. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

22 pages, 355 KB

Open AccessArticle

Uncertain Stochastic Hybrid Age-Dependent Population Equation Based on Subadditive Measure: Existence, Uniqueness and Exponential Stability

by Zhifu Jia and Xinsheng Liu

Symmetry 2023, 15(8), 1512; https://doi.org/10.3390/sym15081512 - 31 Jul 2023

Cited by 2 | Viewed by 1640

Abstract

The existing literature lacks a study on age-dependent population equations based on subadditive measures. In this paper, we propose a hybrid age-dependent population dynamic system (referred to as APDS) that incorporates uncertain random perturbations driven by both the well-known Wiener process and the [...] Read more.

The existing literature lacks a study on age-dependent population equations based on subadditive measures. In this paper, we propose a hybrid age-dependent population dynamic system (referred to as APDS) that incorporates uncertain random perturbations driven by both the well-known Wiener process and the Liu process associated with belief degree, which have similar symmetry in terms of form. Firstly, we redefine the Liu integral in a mean square sense and then extend Liu’s lemma and the Itô-Liu formula. We then utilize the extensions of the Itô-Liu formula, Barkholder-Davis-Gundy (BDG) inequality, the Liu’s lemma, the Gronwall’s lemma and the symmetric nature of calculus itself to establish the uniqueness of a strong solution for the hybrid APDS. Additionally, we prove the existence of the hybrid APDS by combining the proof of uniqueness with some important lemmas. Finally, under appropriate assumptions, we demonstrate the exponential stability of the hybrid system. Full article

(This article belongs to the Section Mathematics)

10 pages, 274 KB

Open AccessArticle

A Second-Order Time Discretization for Second Kind Volterra Integral Equations with Non-Smooth Solutions

by Boya Zhou and Xiujun Cheng

Mathematics 2023, 11(12), 2594; https://doi.org/10.3390/math11122594 - 6 Jun 2023

Viewed by 1286

Abstract

In this paper, a novel second-order method based on a change of variable and the symmetrical and repeated quadrature formula is presented for numerical solving second kind Volterra integral equations with non-smooth solutions. Applying the discrete Grönwall inequality with weak singularity, the convergence [...] Read more.

In this paper, a novel second-order method based on a change of variable and the symmetrical and repeated quadrature formula is presented for numerical solving second kind Volterra integral equations with non-smooth solutions. Applying the discrete Grönwall inequality with weak singularity, the convergence order

O (N^{- 2})

in

L^{\infty}

norm is proved, where N refers to the number of time steps. Numerical results are conducted to verify the efficiency and accuracy of the method. Full article

(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)

17 pages, 520 KB

Open AccessArticle

Finite-Interval Stability Analysis of Impulsive Fractional-Delay Dynamical System

by K. Kaliraj, P. K. Lakshmi Priya and Juan J. Nieto

Fractal Fract. 2023, 7(6), 447; https://doi.org/10.3390/fractalfract7060447 - 31 May 2023

Cited by 3 | Viewed by 1343

Abstract

Stability analysis over a finite time interval is a well-formulated technique to study the dynamical behaviour of a system. This article provides a novel analysis on the finite-time stability of a fractional-order system using the approach of the delayed-type matrix Mittag-Leffler function. At [...] Read more.

Stability analysis over a finite time interval is a well-formulated technique to study the dynamical behaviour of a system. This article provides a novel analysis on the finite-time stability of a fractional-order system using the approach of the delayed-type matrix Mittag-Leffler function. At first, we discuss the solution’s existence and uniqueness for our considered fractional model. Then standard form of integral inequality of Gronwall’s type is used along with the application of the delayed Mittag-Leffler argument to derive the sufficient bounds for the stability of the dynamical system. The analysis of the system is extended and studied with impulsive perturbations. Further, we illustrate the numerical simulations of our analytical study using relevant examples. Full article

(This article belongs to the Section General Mathematics, Analysis)

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16 pages, 312 KB

Open AccessArticle

On Some Generalizations of Integral Inequalities in n Independent Variables and Their Applications

by Waleed Abuelela, Ahmed A. El-Deeb and Dumitru Baleanu

Symmetry 2022, 14(11), 2257; https://doi.org/10.3390/sym14112257 - 27 Oct 2022

Cited by 1 | Viewed by 1328

Abstract

Throughout this article, generalizations of some Grónwall–Bellman integral inequalities for two real-valued unknown functions in n independent variables are introduced. We are looking at some novel explicit bounds of a particular class of Young and Pachpatte integral inequalities. The results in this paper [...] Read more.

Throughout this article, generalizations of some Grónwall–Bellman integral inequalities for two real-valued unknown functions in n independent variables are introduced. We are looking at some novel explicit bounds of a particular class of Young and Pachpatte integral inequalities. The results in this paper can be utilized as a useful way to investigate the uniqueness, boundedness, continuousness, dependence and stability of nonlinear hyperbolic partial integro-differential equations. To highlight our research advantages, several implementations of these findings will be presented. Young’s method, which depends on a Riemann method, will follow to prove the key results. Symmetry plays an essential role in determining the correct methods for solving dynamic inequalities. Full article

(This article belongs to the Section Mathematics)

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