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10 pages, 229 KiB

Open AccessArticle

Existence of Global Mild Solutions for Nonautonomous Abstract Evolution Equations

by Mian Zhou, Yong Liang and Yong Zhou

Mathematics 2025, 13(11), 1722; https://doi.org/10.3390/math13111722 - 23 May 2025

Viewed by 266

In this paper, we investigate the Cauchy problem for nonautonomous abstract evolution equations of the form [...] Read more.

In this paper, we investigate the Cauchy problem for nonautonomous abstract evolution equations of the form

y^{'} (t) = A (t) y (t) + f (t, y (t)), t \geq 0

,

y (0) = y_{0}

. We obtain new existence theorems for global mild solutions under both compact and noncompact evolution families

U (t, s)

. Our key method relies on the generalized Ascoli–Arzela theorem we previously obtained. Finally, an example is provided to illustrate the applicability of our results. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

15 pages, 512 KiB

Open AccessFeature PaperArticle

Numerical Solution of Nonlinear Quadratic Integral Equation of Hammerstein Type Based on Fixed-Point Scheme

by Reza Mollapourasl and Joseph Siebor

Mathematics 2025, 13(9), 1413; https://doi.org/10.3390/math13091413 - 25 Apr 2025

Viewed by 290

Abstract

Existence of the solution for the nonlinear quadratic integral equation of the Hammerstein type in the Banach space BC(

R_{+}

) has been proved by using the technique of measure of noncompactness and fixed-point theorem. In this article, we obtain an approximate [...] Read more.

Existence of the solution for the nonlinear quadratic integral equation of the Hammerstein type in the Banach space BC(

R_{+}

) has been proved by using the technique of measure of noncompactness and fixed-point theorem. In this article, we obtain an approximate solution for the quadratic integral equation by using the Sinc method and the fixed-point technique. Moreover, the convergence of the numerical scheme for the solution of the integral equation is demonstrated by a theorem, and numerical experiments are presented to show the accuracy of the numerical scheme and guarantee the analytical results. Full article

(This article belongs to the Section E: Applied Mathematics)

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18 pages, 293 KiB

Open AccessArticle

Existence and Controls for Fractional Evolution Equations

by Ying Chen and Yong Zhou

Axioms 2025, 14(5), 329; https://doi.org/10.3390/axioms14050329 - 24 Apr 2025

Viewed by 334

Abstract

In this paper, we investigate the existence and uniqueness of mild solutions for non-autonomous fractional evolution equations (NFEEs) using the technique of non-compactness measure, focusing on scenarios where the semigroup is non-compact. Furthermore, the optimal control of nonlinear NFEEs with integral index functionals [...] Read more.

In this paper, we investigate the existence and uniqueness of mild solutions for non-autonomous fractional evolution equations (NFEEs) using the technique of non-compactness measure, focusing on scenarios where the semigroup is non-compact. Furthermore, the optimal control of nonlinear NFEEs with integral index functionals is studied, and the existence of optimal control pairs is proven. Finally, by constructing a corresponding Gramian controllability operator using the solution operator, a sufficient condition is provided for the existence of approximate controllability of the corresponding problem. Full article

(This article belongs to the Special Issue Nonlinear Fractional Differential Equations: Theory and Applications)

17 pages, 289 KiB

Open AccessArticle

Existence of Hilfer Fractional Evolution Inclusions with Almost Sectorial Operators

by Mian Zhou and Yong Zhou

Mathematics 2025, 13(9), 1370; https://doi.org/10.3390/math13091370 - 22 Apr 2025

Viewed by 249

Abstract

In this paper, we mainly focus on the the existence of Hilfer fractional evolution inclusions with almost sectorial operators. For two cases in which the almost sector operators are compact and noncompact, we obtain existence criteria for mild solutions, which extend and improve [...] Read more.

In this paper, we mainly focus on the the existence of Hilfer fractional evolution inclusions with almost sectorial operators. For two cases in which the almost sector operators are compact and noncompact, we obtain existence criteria for mild solutions, which extend and improve some related results in the literature. Full article

18 pages, 286 KiB

Open AccessFeature PaperArticle

Existence of Mild Solutions for Fractional Integrodifferential Equations with Hilfer Derivatives

by Mian Zhou and Yong Zhou

Mathematics 2025, 13(9), 1369; https://doi.org/10.3390/math13091369 - 22 Apr 2025

Viewed by 260

Abstract

In this paper, we study the existence of solutions for fractional integrodifferential equations with Hilfer derivatives. We establish some new existence theorems for mild solutions by using Schaefer’s fixed-point theorem, a measure of noncompactness, and the resolvent operators associated with almost sectorial operators. [...] Read more.

In this paper, we study the existence of solutions for fractional integrodifferential equations with Hilfer derivatives. We establish some new existence theorems for mild solutions by using Schaefer’s fixed-point theorem, a measure of noncompactness, and the resolvent operators associated with almost sectorial operators. Our results improve and extend many known results in the relevant references by removing some strong assumptions. Furthermore, we propose new nonlocal initial conditions for Hilfer evolution equations and study the existence of mild solutions to nonlocal problems. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

12 pages, 287 KiB

Open AccessArticle

Infinite Systems of Differential and Integral Equations: Current State and Some Open Problems

by Józef Banaś, Agnieszka Chlebowicz and Beata Rzepka

Symmetry 2025, 17(4), 575; https://doi.org/10.3390/sym17040575 - 10 Apr 2025

Viewed by 449

Abstract

This paper presents some topics of the theory of infinite systems of differential and integral equations. Our considerations focus on showing the symmetries that can be encountered in the theory of nonlinear differential and integral equations from the viewpoint of initial conditions, such [...] Read more.

This paper presents some topics of the theory of infinite systems of differential and integral equations. Our considerations focus on showing the symmetries that can be encountered in the theory of nonlinear differential and integral equations from the viewpoint of initial conditions, such as the symmetry of the behaviour of solutions of differential equations with respect to initial conditions, the symmetry of the behaviour of solutions in

+ \infty

and

- \infty

and some other essential properties of solutions of differential and integral equations. First of all, we describe the fundamental facts connected with the theory of infinite systems of both differential and integral equations. Particular attention is paid to the location of infinite systems of the mentioned equations in a suitable Banach space. Indeed, we define the spaces in question and describe the basic properties of those spaces. Next, we discuss conditions imposed on terms of equations of the considered infinite systems that guarantee the existence of solutions of those systems and allow us to obtain some essential information on those solutions. Moreover, after the description of the current state of investigations concerning the theory of infinite systems of differential and integral equations, we formulate a few open problems concerning the mentioned systems of equations. Full article

(This article belongs to the Special Issue Nonlinear Differential and Integral Equations and Their Infinite Systems)

19 pages, 314 KiB

Open AccessArticle

Nonlocal Conformable Differential Inclusions Generated by Semigroups of Linear Bounded Operators or by Sectorial Operators with Impulses in Banach Spaces

by Faryal Abdullah Al-Adsani and Ahmed Gamal Ibrahim

Axioms 2025, 14(4), 230; https://doi.org/10.3390/axioms14040230 - 21 Mar 2025

Viewed by 358

Abstract

This paper aims to explore sufficient conditions for the existence of mild solutions to two types of nonlocal, non-instantaneous, impulsive semilinear differential inclusions involving a conformable fractional derivative, where the linear part is the infinitesimal generator of a

C_{0}

-semigroup or a [...] Read more.

This paper aims to explore sufficient conditions for the existence of mild solutions to two types of nonlocal, non-instantaneous, impulsive semilinear differential inclusions involving a conformable fractional derivative, where the linear part is the infinitesimal generator of a

C_{0}

-semigroup or a sectorial operator and the nonlinear part is a multi-valued function with convex or nonconvex values. We provide a definition of the mild solutions, and then, by using appropriate fixed-point theorems for multi-valued functions and the properties of both the conformable derivative and the measure of noncompactness, we achieve our findings. We did not assume that the semigroup generated by the linear part is compact, and this makes our work novel and interesting. We give examples of the application of our theoretical results. Full article

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

19 pages, 4920 KiB

Open AccessArticle

Analytical and Computational Investigations of Stochastic Functional Integral Equations: Solution Existence and Euler–Karhunen–Loève Simulation

by Manochehr Kazemi, AliReza Yaghoobnia, Behrouz Parsa Moghaddam and Alexandra Galhano

Mathematics 2025, 13(3), 427; https://doi.org/10.3390/math13030427 - 27 Jan 2025

Viewed by 799

Abstract

This paper presents a comprehensive investigation into the solution existence of stochastic functional integral equations within real separable Banach spaces, emphasizing the establishment of sufficient conditions. Leveraging advanced mathematical tools including probability measures of noncompactness and Petryshyn’s fixed-point theorem adapted for stochastic processes, [...] Read more.

This paper presents a comprehensive investigation into the solution existence of stochastic functional integral equations within real separable Banach spaces, emphasizing the establishment of sufficient conditions. Leveraging advanced mathematical tools including probability measures of noncompactness and Petryshyn’s fixed-point theorem adapted for stochastic processes, a robust analytical framework is developed. Additionally, this paper introduces the Euler–Karhunen–Loève method, which utilizes the Karhunen–Loève expansion to represent stochastic processes, particularly suited for handling continuous-time processes with an infinite number of random variables. By conducting thorough analysis and computational simulations, which also involve implementing the Euler–Karhunen–Loève method, this paper effectively highlights the practical relevance of the proposed methodology. Two specific instances, namely, the Delay Cox–Ingersoll–Ross process and modified Black–Scholes with proportional delay model, are utilized as illustrative examples to underscore the effectiveness of this approach in tackling real-world challenges encountered in the realms of finance and stochastic dynamics. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

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16 pages, 304 KiB

Open AccessArticle

A Study of p-Laplacian Nonlocal Boundary Value Problem Involving Generalized Fractional Derivatives in Banach Spaces

by Madeaha Alghanmi

Mathematics 2025, 13(1), 138; https://doi.org/10.3390/math13010138 - 1 Jan 2025

Viewed by 723

Abstract

The aim of this article is to introduce and study a new class of fractional integro nonlocal boundary value problems involving the p-Laplacian operator and generalized fractional derivatives. The existence of solutions in Banach spaces is investigated with the aid of the [...] Read more.

The aim of this article is to introduce and study a new class of fractional integro nonlocal boundary value problems involving the p-Laplacian operator and generalized fractional derivatives. The existence of solutions in Banach spaces is investigated with the aid of the properties of Kuratowski’s noncompactness measure and Sadovskii’s fixed-point theorem. Two illustrative examples are constructed to guarantee the applicability of our results. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

13 pages, 324 KiB

Open AccessArticle

φ−Hilfer Fractional Cauchy Problems with Almost Sectorial and Lie Bracket Operators in Banach Algebras

by Faten H. Damag, Amin Saif and Adem Kiliçman

Fractal Fract. 2024, 8(12), 741; https://doi.org/10.3390/fractalfract8120741 - 16 Dec 2024

Cited by 1 | Viewed by 977

Abstract

In the theory of Banach algebras, we use the Schauder fixed-point theorem to obtain some results that concern the existence property for mild solutions of fractional Cauchy problems that involve the Lie bracket operator, the almost sectorial operator, and the

φ -

Hilfer [...] Read more.

In the theory of Banach algebras, we use the Schauder fixed-point theorem to obtain some results that concern the existence property for mild solutions of fractional Cauchy problems that involve the Lie bracket operator, the almost sectorial operator, and the

φ -

Hilfer derivative operator. For any Banach algebra and in two types of non-compact associated semigroups and compact associated semigroups, we prove some properties of the existence of these mild solutions using the Hausdorff measure of a non-compact associated semigroup in the collection of bounded sets. That is, we obtain the existence property of mild solutions when the semigroup associated with an almost sectorial operator is compact as well as non-compact. Some examples are introduced as applications for our results in commutative real Banach algebra

R

and commutative Banach algebra of the collection of continuous functions in

R

. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)

21 pages, 366 KiB

Open AccessArticle

Approximate and Exact Controllability for Hilfer Fractional Stochastic Evolution Equations

by Qien Li and Danfeng Luo

Fractal Fract. 2024, 8(12), 733; https://doi.org/10.3390/fractalfract8120733 - 13 Dec 2024

Cited by 2 | Viewed by 905

Abstract

This paper investigates the controllability of Hilfer fractional stochastic evolution equations (HFSEEs). Initially, we obtain a conclusion regarding the approximate controllability of HFSEEs by employing the Tikhonov-type regularization method and Schauder′s fixed-point theorem. Additionally, the conditions for the exact controllability of HFSEEs are [...] Read more.

This paper investigates the controllability of Hilfer fractional stochastic evolution equations (HFSEEs). Initially, we obtain a conclusion regarding the approximate controllability of HFSEEs by employing the Tikhonov-type regularization method and Schauder′s fixed-point theorem. Additionally, the conditions for the exact controllability of HFSEEs are explored, utilizing the Mönch′s fixed-point theorem and measure of noncompactness. Finally, the proposed method is validated through an example, thereby demonstrating its effectiveness. Full article

21 pages, 506 KiB

Open AccessArticle

Study on Controllability for Ψ-Hilfer Fractional Stochastic Differential Equations

by Abdur Raheem, Fahad M. Alamrani, Javed Akhtar, Adel Alatawi, Esmail Alshaban, Areefa Khatoon and Faizan Ahmad Khan

Fractal Fract. 2024, 8(12), 727; https://doi.org/10.3390/fractalfract8120727 - 11 Dec 2024

Cited by 2 | Viewed by 1118

Abstract

The goal of this paper is to study the existence of a mild solution and controllability for a class of neutral stochastic differential equations (SDEs) involving the

Ψ

-Hilfer fractional derivatives, a generalization of the well-known Riemann–Liouville fractional derivative using almost sectorial operators. [...] Read more.

The goal of this paper is to study the existence of a mild solution and controllability for a class of neutral stochastic differential equations (SDEs) involving the

Ψ

-Hilfer fractional derivatives, a generalization of the well-known Riemann–Liouville fractional derivative using almost sectorial operators. Sufficient conditions for controllability are established using the notion of measure of noncompactness (MNC) and the Mönch fixed-point theorem. An example is given to illustrate the abstract findings. Full article

(This article belongs to the Special Issue Advances in Fractional Dynamical and Control Problems with Nonlocal Operators)

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13 pages, 253 KiB

Open AccessArticle

A Study on Positive Solutions to Nonlinear Fractional Differential Equations

by Haide Gou

Mathematics 2024, 12(21), 3379; https://doi.org/10.3390/math12213379 - 29 Oct 2024

Viewed by 951

Abstract

The purpose of this article concerns a type of nonlinear boundary value problem of fractional differential equations with the Caputo derivative. Using the fixed point index theory of condensing mapping in cones, the existence results of positive solutions for such a problem are [...] Read more.

The purpose of this article concerns a type of nonlinear boundary value problem of fractional differential equations with the Caputo derivative. Using the fixed point index theory of condensing mapping in cones, the existence results of positive solutions for such a problem are obtained. Full article

12 pages, 269 KiB

Open AccessArticle

Existence and Stability for Fractional Differential Equations with a ψ–Hilfer Fractional Derivative in the Caputo Sense

by Wenchang He, Yuhang Jin, Luyao Wang, Ning Cai and Jia Mu

Mathematics 2024, 12(20), 3271; https://doi.org/10.3390/math12203271 - 18 Oct 2024

Viewed by 984

Abstract

This article aims to explore the existence and stability of solutions to differential equations involving a

ψ

-Hilfer fractional derivative in the Caputo sense, which, compared to classical

ψ

-Hilfer fractional derivatives (in the Riemann–Liouville sense), provide a clear physical interpretation when dealing [...] Read more.

This article aims to explore the existence and stability of solutions to differential equations involving a

ψ

-Hilfer fractional derivative in the Caputo sense, which, compared to classical

ψ

-Hilfer fractional derivatives (in the Riemann–Liouville sense), provide a clear physical interpretation when dealing with initial conditions. We discovered that the

ψ

-Hilfer fractional derivative in the Caputo sense can be represented as the inverse operation of the

ψ

-Riemann–Liouville fractional integral, and used this property to prove the existence of solutions for linear differential equations with a

ψ

-Hilfer fractional derivative in the Caputo sense. Additionally, we applied Mönch’s fixed-point theorem and knowledge of non-compactness measures to demonstrate the existence of solutions for nonlinear differential equations with a

ψ

-Hilfer fractional derivative in the Caputo sense, and further discussed the Ulam–Hyers–Rassias stability and semi-Ulam–Hyers–Rassias stability of these solutions. Finally, we illustrated our results through case studies. Full article

11 pages, 312 KiB

Open AccessArticle

Existence of Solutions for a Coupled Hadamard Fractional System of Integral Equations in Local Generalized Morrey Spaces

by Asra Hadadfard, Mohammad Bagher Ghaemi and António M. Lopes

Axioms 2024, 13(10), 688; https://doi.org/10.3390/axioms13100688 - 3 Oct 2024

Viewed by 1016

Abstract

This paper introduces a new measure of non-compactness within a bounded domain of

R^{N}

in the generalized Morrey space. This measure is used to establish the existence of solutions for a coupled Hadamard fractional system of integral equations in generalized Morrey spaces. [...] Read more.

This paper introduces a new measure of non-compactness within a bounded domain of

R^{N}

in the generalized Morrey space. This measure is used to establish the existence of solutions for a coupled Hadamard fractional system of integral equations in generalized Morrey spaces. To illustrate the application of the main result, an example is presented. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

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