MDPI - Publisher of Open Access Journals

22 pages, 753 KiB

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 345

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)

► Show Figures

Figure 1

16 pages, 1058 KiB

Open AccessArticle

Ulam–Hyers Stability of Fractional Difference Equations with Hilfer Derivatives

by Marko Kostić, Halis Can Koyuncuoğlu and Jagan Mohan Jonnalagadda

Fractal Fract. 2025, 9(7), 417; https://doi.org/10.3390/fractalfract9070417 - 26 Jun 2025

Viewed by 377

Abstract

This paper investigates the Ulam–Hyers stability of both linear and nonlinear delayed neutral Hilfer fractional difference equations. We utilize the nabla Laplace transform, known as the N-transform, along with a generalized discrete Gronwall inequality to derive sufficient conditions for stability. For the [...] Read more.

This paper investigates the Ulam–Hyers stability of both linear and nonlinear delayed neutral Hilfer fractional difference equations. We utilize the nabla Laplace transform, known as the N-transform, along with a generalized discrete Gronwall inequality to derive sufficient conditions for stability. For the linear case, we provide an explicit solution formula involving discrete Mittag-Leffler functions and establish its stability properties. In the nonlinear case, we concentrate on delayed neutral Hilfer fractional difference equations, a class of systems that appears to be unexplored in the existing literature with respect to Ulam–Hyers stability. In particular, for the linear case, the absolute difference between the solution of the linear Hilfer fractional difference equation and the solution of the corresponding perturbed equation is bounded by the function of

ε

when the perturbed term is bounded by

ε

. In the case of the neutral fractional delayed Hilfer difference equation, the absolute difference is bounded by a constant multiple of

ε

. Our results fill this gap by offering novel stability criteria. We support our theoretical findings with illustrative numerical examples and simulations, which visually confirm the predicted stability behavior and demonstrate the applicability of the results in discrete fractional dynamic systems. Full article

(This article belongs to the Special Issue Recent Developments in Multidimensional Fractional Differential Equations and Fractional Difference Equations)

► Show Figures

Figure 1

36 pages, 544 KiB

Open AccessArticle

Well-Posedness of Cauchy-Type Problems for Nonlinear Implicit Hilfer Fractional Differential Equations with General Order in Weighted Spaces

by Jakgrit Sompong, Samten Choden, Ekkarath Thailert and Sotiris K. Ntouyas

Symmetry 2025, 17(7), 986; https://doi.org/10.3390/sym17070986 - 22 Jun 2025

Viewed by 216

Abstract

This paper establishes the well-posedness of Cauchy-type problems with non-symmetric initial conditions for nonlinear implicit Hilfer fractional differential equations of general fractional orders in weighted function spaces. Using fixed-point techniques, we first prove the existence of solutions via Schaefer’s fixed-point theorem. The uniqueness [...] Read more.

This paper establishes the well-posedness of Cauchy-type problems with non-symmetric initial conditions for nonlinear implicit Hilfer fractional differential equations of general fractional orders in weighted function spaces. Using fixed-point techniques, we first prove the existence of solutions via Schaefer’s fixed-point theorem. The uniqueness and Ulam–Hyers stability are then derived using Banach’s contraction principle. By introducing a novel singular-kernel Gronwall inequality, we extend the analysis to Ulam–Hyers–Rassias stability and continuous dependence on initial data. The theoretical framework is unified for general fractional orders and validated through examples, demonstrating its applicability to implicit systems with memory effects. Key contributions include weighted-space analysis and stability criteria for this class of equations. Full article

(This article belongs to the Special Issue Symmetry in Fixed Point Theory and Optimization: Computations and Applications)

► Show Figures

Figure 1

17 pages, 751 KiB

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 273

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)

► Show Figures

Figure 1

23 pages, 380 KiB

Open AccessArticle

Generalized Grönwall Inequality and Ulam–Hyers Stability in ℒ^p Space for Fractional Stochastic Delay Integro-Differential Equations

by Abdelhamid Mohammed Djaouti and Muhammad Imran Liaqat

Mathematics 2025, 13(8), 1252; https://doi.org/10.3390/math13081252 - 10 Apr 2025

Viewed by 364

Abstract

In this work, we derive novel theoretical results concerning well-posedness and Ulam–Hyers stability. Specifically, we investigate the well-posedness of Caputo–Katugampola fractional stochastic delay integro-differential equations. Additionally, we develop a generalized Grönwall inequality and apply it to prove Ulam–Hyers stability in

L^{p}

space. [...] Read more.

In this work, we derive novel theoretical results concerning well-posedness and Ulam–Hyers stability. Specifically, we investigate the well-posedness of Caputo–Katugampola fractional stochastic delay integro-differential equations. Additionally, we develop a generalized Grönwall inequality and apply it to prove Ulam–Hyers stability in

L^{p}

space. Our findings generalize existing results for fractional derivatives and space, as we formulate them in the Caputo–Katugampola fractional derivative and

L^{p}

space. To support our theoretical results, we present an illustrative example. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

► Show Figures

Figure 1

12 pages, 222 KiB

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 279

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)

32 pages, 453 KiB

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 1145

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)

► Show Figures

Figure 1

23 pages, 367 KiB

Open AccessArticle

Quantum Laplace Transforms for the Ulam–Hyers Stability of Certain q-Difference Equations of the Caputo-like Type

by Sina Etemad, Ivanka Stamova, Sotiris K. Ntouyas and Jessada Tariboon

Fractal Fract. 2024, 8(8), 443; https://doi.org/10.3390/fractalfract8080443 - 28 Jul 2024

Cited by 6 | Viewed by 1418

Abstract

We aim to investigate the stability property for the certain linear and nonlinear fractional

q

-difference equations in the Ulam–Hyers and Ulam–Hyers–Rassias sense. To achieve this goal, we prove that three types of the linear

q

-difference equations of the

q

-Caputo-like type [...] Read more.

We aim to investigate the stability property for the certain linear and nonlinear fractional

q

-difference equations in the Ulam–Hyers and Ulam–Hyers–Rassias sense. To achieve this goal, we prove that three types of the linear

q

-difference equations of the

q

-Caputo-like type are Ulam–Hyers stable by using the quantum Laplace transform and quantum Mittag–Leffler function. Moreover, after proving the existence property for a nonlinear Cauchy

q

-difference initial value problem, we use the same quantum Laplace transform and the

q

-Gronwall inequality to show that it is generalized Ulam–Hyers–Rassias stable. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

21 pages, 1994 KiB

Open AccessArticle

Iterative Learning Formation Control via Input Sharing for Fractional-Order Singular Multi-Agent Systems with Local Lipschitz Nonlinearity

by Guangxu Wang, Rui Wang, Danhu Yi, Xingyu Zhou and Shuyu Zhang

Fractal Fract. 2024, 8(6), 347; https://doi.org/10.3390/fractalfract8060347 - 11 Jun 2024

Cited by 3 | Viewed by 1325

Abstract

For a class of fractional-order singular multi-agent systems (FOSMASs) with local Lipschitz nonlinearity, this paper proposes a closed-loop

D^{α}

-type iterative learning formation control law via input sharing to achieve the stable formation of FOSMASs in a finite time. Firstly, the formation [...] Read more.

For a class of fractional-order singular multi-agent systems (FOSMASs) with local Lipschitz nonlinearity, this paper proposes a closed-loop

D^{α}

-type iterative learning formation control law via input sharing to achieve the stable formation of FOSMASs in a finite time. Firstly, the formation control issue of FOSMASs with local Lipschitz nonlinearity under the fixed communication topology (FCT) is transformed into the consensus tracking control scenario. Secondly, by virtue of utilizing the characteristics of fractional calculus and the generalized Gronwall inequality, sufficient conditions for the convergence of formation error are given. Then, drawing upon the FCT, the iteration-varying switching communication topology is considered and examined. Ultimately, the validity of the

D^{α}

-type learning method is showcased through two numerical cases. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Multiagent Systems: Theory and Applications)

► Show Figures

Figure 1

21 pages, 356 KiB

Open AccessArticle

Investigation of Well-Posedness for a Direct Problem for a Nonlinear Fractional Diffusion Equation and an Inverse Problem

by Özge Arıbaş, İsmet Gölgeleyen and Mustafa Yıldız

Fractal Fract. 2024, 8(6), 315; https://doi.org/10.3390/fractalfract8060315 - 26 May 2024

Cited by 1 | Viewed by 1299

Abstract

In this paper, we consider a direct problem and an inverse problem involving a nonlinear fractional diffusion equation, which can be applied to many physical situations. The equation contains a Caputo fractional derivative, a symmetric uniformly elliptic operator and a source term consisting [...] Read more.

In this paper, we consider a direct problem and an inverse problem involving a nonlinear fractional diffusion equation, which can be applied to many physical situations. The equation contains a Caputo fractional derivative, a symmetric uniformly elliptic operator and a source term consisting of the sum of two terms, one of which is linear and the other is nonlinear. The well-posedness of the direct problem is examined and the results are used to investigate the stability of an inverse problem of determining a function in the linear part of the source. The main tools in our study are the generalized eigenfunction expansions theory for nonlinear fractional diffusion equations, contraction mapping, Young’s convolution and generalized Grönwall’s inequalities. We present a stability estimate for the solution of the inverse source problem by means of observation data at a given point in the domain. Full article

(This article belongs to the Special Issue Recent Advances in the Equation with Nonlinear Fractional Diffusion)

20 pages, 345 KiB

Open AccessArticle

A Study of Some Generalized Results of Neutral Stochastic Differential Equations in the Framework of Caputo–Katugampola Fractional Derivatives

by Abdelhamid Mohammed Djaouti, Zareen A. Khan, Muhammad Imran Liaqat and Ashraf Al-Quran

Mathematics 2024, 12(11), 1654; https://doi.org/10.3390/math12111654 - 24 May 2024

Cited by 8 | Viewed by 1471

Abstract

Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We [...] Read more.

Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We generalize results in two ways: first, by extending the existing result for

p = 2

to results in the

L^{p}

space; second, by incorporating the Caputo–Katugampola fractional derivatives, we extend the results established with Caputo fractional derivatives. Additionally, we provide examples to enhance the understanding of the theoretical results we establish. Full article

(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)

16 pages, 323 KiB

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 1619

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)

16 pages, 268 KiB

Open AccessArticle

Existence and Hyers–Ulam Stability for Random Impulsive Stochastic Pantograph Equations with the Caputo Fractional Derivative

by Dongdong Gao and Jianli Li

Mathematics 2024, 12(8), 1145; https://doi.org/10.3390/math12081145 - 10 Apr 2024

Viewed by 975

Abstract

In this paper, we study the existence, uniqueness and Hyers–Ulam stability of a class of fractional stochastic pantograph equations with random impulses. Firstly, we establish sufficient conditions to ensure the existence of solutions for the considered equations by applying Schaefer’s fixed point theorem [...] Read more.

In this paper, we study the existence, uniqueness and Hyers–Ulam stability of a class of fractional stochastic pantograph equations with random impulses. Firstly, we establish sufficient conditions to ensure the existence of solutions for the considered equations by applying Schaefer’s fixed point theorem under relaxed linear growth conditions. Secondly, we prove the solution for the considered equations is Hyers–Ulam stable via Gronwall’s inequality. Moreover, the previous literature will be significantly generalized in our paper. Finally, an example is given to explain the efficiency of the obtained results. Full article

(This article belongs to the Topic Mathematical Modeling)

21 pages, 396 KiB

Open AccessArticle

Existence, Uniqueness, and Averaging Principle of Fractional Neutral Stochastic Differential Equations in the

L^{p}

Space with the Framework of the Ψ-Caputo Derivative

by Abdelhamid Mohammed Djaouti, Zareen A. Khan, Muhammad Imran Liaqat and Ashraf Al-Quran

Mathematics 2024, 12(7), 1037; https://doi.org/10.3390/math12071037 - 30 Mar 2024

Cited by 9 | Viewed by 1249

Abstract

In this research work, we use the concepts of contraction mapping to establish the existence and uniqueness results and also study the averaging principle in

L^{p}

space by using Jensen’s, Grönwall–Bellman’s, Hölder’s, and Burkholder–Davis–Gundy’s inequalities, and the interval translation technique for a [...] Read more.

In this research work, we use the concepts of contraction mapping to establish the existence and uniqueness results and also study the averaging principle in

L^{p}

space by using Jensen’s, Grönwall–Bellman’s, Hölder’s, and Burkholder–Davis–Gundy’s inequalities, and the interval translation technique for a class of fractional neutral stochastic differential equations. We establish the results within the framework of the

Ψ

-Caputo derivative. We generalize the two situations of

p = 2

and the Caputo derivative with the findings that we obtain. To help with the understanding of the theoretical results, we provide two applied examples at the end. Full article

(This article belongs to the Special Issue Recent Research on Fractional Calculus: Theory and Applications)

► Show Figures

Figure 1

16 pages, 322 KiB

Open AccessArticle

A Result Regarding Finite-Time Stability for Hilfer Fractional Stochastic Differential Equations with Delay

by Man Li, Yujun Niu and Jing Zou

Fractal Fract. 2023, 7(8), 622; https://doi.org/10.3390/fractalfract7080622 - 15 Aug 2023

Cited by 5 | Viewed by 1509

Abstract

Hilfer fractional stochastic differential equations with delay are discussed in this paper. Firstly, the solutions to the corresponding equations are given using the Laplace transformation and its inverse. Afterwards, the Picard iteration technique and the contradiction method are brought up to demonstrate the [...] Read more.

Hilfer fractional stochastic differential equations with delay are discussed in this paper. Firstly, the solutions to the corresponding equations are given using the Laplace transformation and its inverse. Afterwards, the Picard iteration technique and the contradiction method are brought up to demonstrate the existence and uniqueness of understanding, respectively. Further, finite-time stability is obtained using the generalized Grönwall–Bellman inequality. As verification, an example is provided to support the theoretical results. Full article

(This article belongs to the Special Issue Analysis of Caputo-Type Fractional Derivatives and Differential Equations)

Search Results (37)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (37)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI