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36 pages, 488 KB

Open AccessArticle

Analysis of Implicit Neutral-Tempered Caputo Fractional Volterra–Fredholm Integro-Differential Equations Involving Retarded and Advanced Arguments

by Abdulrahman A. Sharif and Muath Awadalla

Mathematics 2026, 14(3), 470; https://doi.org/10.3390/math14030470 - 29 Jan 2026

Viewed by 83

Abstract

This paper investigates a class of implicit neutral fractional integro-differential equations of Volterra–Fredholm type. The equations incorporate a tempered fractional derivative in the Caputo sense, along with both retarded (delay) and advanced arguments. The problem is formulated on a time domain segmented into [...] Read more.

This paper investigates a class of implicit neutral fractional integro-differential equations of Volterra–Fredholm type. The equations incorporate a tempered fractional derivative in the Caputo sense, along with both retarded (delay) and advanced arguments. The problem is formulated on a time domain segmented into past, present, and future intervals and includes nonlinear mixed integral operators. Using Banach’s contraction mapping principle and Schauder’s fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions within the space of continuous functions. The study is then extended to general Banach spaces by employing Darbo’s fixed point theorem combined with the Kuratowski measure of noncompactness. Ulam–Hyers–Rassias stability is also analyzed under appropriate conditions. To demonstrate the practical applicability of the theoretical framework, explicit examples with specific nonlinear functions and integral kernels are provided. Furthermore, detailed numerical simulations are conducted using MATLAB-based specialized algorithms, illustrating solution convergence and behavior in both finite-dimensional and Banach space contexts. Full article

(This article belongs to the Special Issue Recent Developments in Theoretical and Applied Mathematics—In Memory of Prof. Dr. Ravi P. Agarwal)

19 pages, 329 KB

Open AccessArticle

Ulam-Type Stability Results for Fractional Integro-Delay Differential and Integral Equations via the ψ-Hilfer Operator

by Cemil Tunç and Osman Tunç

Fractal Fract. 2026, 10(1), 57; https://doi.org/10.3390/fractalfract10010057 - 14 Jan 2026

Viewed by 229

Abstract

In this article, we investigate a nonlinear

ψ

-Hilfer fractional order Volterra integro-delay differential equation (

ψ

-Hilfer FRVIDDE) and a nonlinear

ψ

-Hilfer fractional Volterra delay integral equation (

ψ

-Hilfer FRVDIE), both of which incorporate multiple variable time delays. We establish [...] Read more.

In this article, we investigate a nonlinear

ψ

-Hilfer fractional order Volterra integro-delay differential equation (

ψ

-Hilfer FRVIDDE) and a nonlinear

ψ

-Hilfer fractional Volterra delay integral equation (

ψ

-Hilfer FRVDIE), both of which incorporate multiple variable time delays. We establish sufficient conditions for the existence of a unique solution and the Ulam–Hyers stability (U-H stability) of both the

ψ

-Hilfer FRVIDDE and

ψ

-the Hilfer FRVDIE through two new main results. The proof technique relies on the Banach contraction mapping principle, properties of the Hilfer operator, and some additional analytical tools. The considered

ψ

-Hilfer FRVIDDE and

ψ

-Hilfer FRVDIE are new fractional mathematical models in the relevant literature. They extend and improve some available related fractional mathematical models from cases without delay to models incorporating multiple variable time delays, and they also provide new contributions to the qualitative theory of fractional delay differential and fractional delay integral equations. We also give two new examples to verify the applicability of main results of the article. Finally, the article presents substantial and novel results with new examples, contributing to the relevant literature. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

18 pages, 338 KB

Open AccessArticle

Unified Fixed-Point Theorems for Generalized p-Reich and p-Sehgal Contractions in Complete Metric Spaces with Application to Fractal and Fractional Systems

by Zouaoui Bekri, Nicola Fabiano, Amir Baklouti and Abdullah Assiry

Fractal Fract. 2026, 10(1), 27; https://doi.org/10.3390/fractalfract10010027 - 4 Jan 2026

Viewed by 283

Abstract

This paper introduces new generalized forms of contractive mappings in the framework of complete metric spaces. By extending the classical Reich and Sehgal contractions to their iterated counterparts in Singh’s sense, we establish unified fixed-point theorems that ensure both existence and uniqueness under [...] Read more.

This paper introduces new generalized forms of contractive mappings in the framework of complete metric spaces. By extending the classical Reich and Sehgal contractions to their iterated counterparts in Singh’s sense, we establish unified fixed-point theorems that ensure both existence and uniqueness under constant and variable contractive parameters. The proposed p-Reich and p-Sehgal contractions encompass several well-known results, including those of Banach, Kannan, Chatterjea, Reich, and Sehgal, as special cases. Convergence of the associated Picard iterative process is rigorously analyzed, revealing deeper insights into the iterative stability and asymptotic behavior of nonlinear mappings in metric spaces. The practical utility of our unified fixed-point theorems is illustrated through concrete applications in fractal and fractional calculus. Full article

(This article belongs to the Special Issue Mathematical and Computational Approaches to Fractal and Fractional Systems Using Fixed Point Theory)

21 pages, 328 KB

Open AccessArticle

Analytic Study on Φ-Hilfer Fractional Neutral-Type Functional Integro-Differential Equations with Terminal Conditions

by Ravichandran Vivek, Abdulah A. Alghamdi, Mohamed M. El-Dessoky, Dhandapani Maheswari and Natarajan Bharath

Mathematics 2026, 14(1), 182; https://doi.org/10.3390/math14010182 - 3 Jan 2026

Viewed by 247

Abstract

The current manuscript is concerned with the uniqueness and existence of a solution for a new class of

Φ

-Hilfer fractional neutral functional integro-differential equations (

Φ

-HFNFIDEs) with terminal conditions. Firstly, employing Babenko’s approach, we convert the aforesaid equation under consideration into [...] Read more.

The current manuscript is concerned with the uniqueness and existence of a solution for a new class of

Φ

-Hilfer fractional neutral functional integro-differential equations (

Φ

-HFNFIDEs) with terminal conditions. Firstly, employing Babenko’s approach, we convert the aforesaid equation under consideration into an analogous integral equation. More precisely, using the multivariate Mittag-Leffler function, Banach contraction principle, and Krasnoselskii’s fixed-point theorem, we derive some conditions that guarantee the uniqueness and the existence of the solutions. For an illustration of our results in this manuscript, two examples are provided as well. Full article

21 pages, 776 KB

Open AccessArticle

Solvability, Ulam–Hyers Stability, and Kernel Analysis of Multi-Order σ-Hilfer Fractional Systems: A Unified Theoretical Framework

by Yasir A. Madani, Mohammed Almalahi, Osman Osman, Ahmed M. I. Adam, Haroun D. S. Adam, Ashraf A. Qurtam and Khaled Aldwoah

Fractal Fract. 2026, 10(1), 21; https://doi.org/10.3390/fractalfract10010021 - 29 Dec 2025

Viewed by 638

Abstract

This paper establishes a rigorous analytical framework for a nonlinear multi-order fractional differential system governed by the generalized

σ

-Hilfer operator in weighted Banach spaces. In contrast to existing studies that often treat specific kernels or fixed fractional orders in isolation, our approach [...] Read more.

This paper establishes a rigorous analytical framework for a nonlinear multi-order fractional differential system governed by the generalized

σ

-Hilfer operator in weighted Banach spaces. In contrast to existing studies that often treat specific kernels or fixed fractional orders in isolation, our approach provides a unified treatment that simultaneously handles multiple fractional orders, a tunable kernel

σ (ς)

, weighted integral conditions, and a nonlinearity depending on a fractional integral of the solution. By converting the hierarchical differential structure into an equivalent Volterra integral equation, we derive sufficient conditions for the existence and uniqueness of solutions using the Banach contraction principle and Mönch’s fixed-point theorem with measures of non-compactness. The analysis is extended to Ulam–Hyers stability, ensuring robustness under modeling perturbations. A principal contribution is the systematic classification of the system’s symmetric reductions—specifically the Riemann–Liouville, Caputo, Hadamard, and Katugampola forms—all governed by a single spectral condition dependent on

σ (ς)

. The theoretical results are illustrated by numerical examples that highlight the sensitivity of solutions to the memory kernel and the fractional orders. This work provides a cohesive analytical tool for a broad class of fractional systems with memory, thereby unifying previously disparate fractional calculi under a single, consistent framework. Full article

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

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19 pages, 334 KB

Open AccessArticle

On a Nonlinear Proportional Fractional Integro-Differential Equation with Functional Boundary Conditions: Existence, Uniqueness, and Ulam–Hyers Stability

by Sahar Mohammad A. Abusalim, Raouf Fakhfakh and Abdellatif Ben Makhlouf

Fractal Fract. 2026, 10(1), 16; https://doi.org/10.3390/fractalfract10010016 - 27 Dec 2025

Viewed by 750

Abstract

This work introduces a new category of proportional fractional integro-differential equations (PFIDEs) governed by functional boundary conditions. We derive verifiable sufficient criteria that guarantee the Ulam–Hyers Stability, existence and uniqueness of solutions to this problem. Our analytical approach leverages Babenko’s method to construct [...] Read more.

This work introduces a new category of proportional fractional integro-differential equations (PFIDEs) governed by functional boundary conditions. We derive verifiable sufficient criteria that guarantee the Ulam–Hyers Stability, existence and uniqueness of solutions to this problem. Our analytical approach leverages Babenko’s method to construct an inverse operator, which allows us to reformulate the differential problem into an equivalent integral equation. The analysis is then conducted using key mathematical tools, including contraction mapping principle of Banach, the Leray–Schauder alternative, and properties of multivariate Mittag–Leffler functions. The Ulam–Hyers Stability is rigorously examined to assess the system’s resilience to small perturbations. The applicability and effectiveness of the established theoretical results are demonstrated through two illustrative examples. This research provides a unified and adaptable framework that advances the analysis of complex fractional-order dynamical systems subject to nonlocal constraints. Full article

10 pages, 244 KB

Open AccessArticle

On p-Hardy–Rogers and p-Zamfirescu Contractions in Complete Metric Spaces: Existence and Uniqueness Results

by Zouaoui Bekri, Nicola Fabiano, Mohammed Ahmed Alomair and Abdulaziz Khalid Alsharidi

Mathematics 2025, 13(24), 4011; https://doi.org/10.3390/math13244011 - 16 Dec 2025

Viewed by 259

Abstract

In this paper, we introduce and investigate two generalized forms of classical contraction mappings, namely the p-Hardy–Rogers and p-Zamfirescu contractions. By incorporating the integer parameter

p \geq 1

, these new definitions extend the traditional Hardy–Rogers and Zamfirescu conditions to iterated [...] Read more.

In this paper, we introduce and investigate two generalized forms of classical contraction mappings, namely the p-Hardy–Rogers and p-Zamfirescu contractions. By incorporating the integer parameter

p \geq 1

, these new definitions extend the traditional Hardy–Rogers and Zamfirescu conditions to iterated mappings

ħ^{p}

. We establish fixed-point theorems, ensuring both existence and uniqueness of fixed points for continuous self-maps on complete metric spaces that satisfy these p-contractive conditions. The proofs are constructed via geometric estimates on the iterates and by transferring the fixed point from the p-th iterate

ħ^{p}

to the original mapping ħ. Our results unify and broaden several well-known fixed-point theorems reported in previous studies, including those of Banach, Hardy–Rogers, and Zamfirescu as special cases. Full article

(This article belongs to the Section C: Mathematical Analysis)

38 pages, 488 KB

Open AccessArticle

Existence and Uniqueness of Solutions for Singular Fractional Integro-Differential Equations with p-Laplacian and Two Kinds of Fractional Derivatives

by Fang Wang, Lishan Liu, Haibo Gu, Lina Ma and Yonghong Wu

Axioms 2025, 14(12), 890; https://doi.org/10.3390/axioms14120890 - 30 Nov 2025

Viewed by 338

Abstract

The paper is devoted to the study of a class of singular high-order fractional integro-differential equations with p-Laplacian operator, involving both the Riemann–Liouville fractional derivative and the Caputo fractional derivative. First, we investigate the problem by the method of reducing the order [...] Read more.

The paper is devoted to the study of a class of singular high-order fractional integro-differential equations with p-Laplacian operator, involving both the Riemann–Liouville fractional derivative and the Caputo fractional derivative. First, we investigate the problem by the method of reducing the order of fractional derivative. Then, by using the Schauder fixed point theorem, the existence of solutions is proved. The upper and lower bounds for the unique solution of the problem are established under various conditions by employing the Banach contraction mapping principle. Furthermore, four numerical examples are presented to illustrate the applications of our main results. Full article

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

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23 pages, 463 KB

Open AccessArticle

The Existence and Uniqueness of Mild Solutions for Fuzzy Hilfer Fractional Evolution Equations with Non-Local Conditions

by Kholoud N. Alharbi and Sanaa Alotaibi

Axioms 2025, 14(11), 855; https://doi.org/10.3390/axioms14110855 - 20 Nov 2025

Viewed by 301

Abstract

In this paper, we investigate a fuzzy Hilfer fractional evolution equation of type

0 < β < 1

and order

1 < α < 2

subject to nonlocal conditions. Using the infinitesimal generator of a strongly continuous cosine family, we define a mild [...] Read more.

In this paper, we investigate a fuzzy Hilfer fractional evolution equation of type

0 < β < 1

and order

1 < α < 2

subject to nonlocal conditions. Using the infinitesimal generator of a strongly continuous cosine family, we define a mild solution for the proposed system. The existence and uniqueness of such mild solutions are established through Schauder’s fixed-point theorem and the Banach contraction principle. An illustrative application to a fuzzy fractional wave equation is presented to demonstrate the effectiveness of the developed approach. The main contribution of this study lies in the unified treatment of fuzzy Hilfer fractional evolution equations under nonlocal conditions, which generalizes and extends several existing results and provides a solid analytical foundation for modeling systems with memory and uncertainty. Full article

(This article belongs to the Special Issue Fractional Differentiation and Applied Mathematics: Recent Advances and Applications)

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21 pages, 334 KB

Open AccessArticle

Square-Mean S-Asymptotically (ω,c)-Periodic Solutions to Neutral Stochastic Impulsive Equations

by Belkacem Chaouchi, Wei-Shih Du, Marko Kostić and Daniel Velinov

Symmetry 2025, 17(11), 1938; https://doi.org/10.3390/sym17111938 - 12 Nov 2025

Viewed by 455

Abstract

This paper investigates the existence of square-mean S-asymptotically

(ω, c)

-periodic solutions for a class of neutral impulsive stochastic differential equations driven by fractional Brownian motion, addressing the challenge of modeling long-range dependencies, delayed feedback, and abrupt changes in [...] Read more.

This paper investigates the existence of square-mean S-asymptotically

(ω, c)

-periodic solutions for a class of neutral impulsive stochastic differential equations driven by fractional Brownian motion, addressing the challenge of modeling long-range dependencies, delayed feedback, and abrupt changes in systems like biological networks or mechanical oscillators. By employing semigroup theory to derive mild solution representations and the Banach contraction principle, we establish sufficient conditions–such as Lipschitz continuity of nonlinear terms and growth bounds on the resolvent operator—that guarantee the uniqueness and existence of such solutions in the space

{S A P}_{ω, c} ([0, \infty), L^{2} (Ω, H))

. The important results demonstrate that under these assumptions, the mild solution exhibits square-mean S-asymptotic

(ω, c)

-periodicity, enabling robust asymptotic analysis beyond classical periodicity. We illustrate these findings with examples, such as a neutral stochastic heat equation with impulses, revealing stability thresholds and decay rates and highlighting the framework’s utility in predicting long-term dynamics. These outcomes advance stochastic analysis by unifying neutral, impulsive, and fractional noise effects, with potential applications in control theory and engineering. Full article

(This article belongs to the Special Issue Advance in Functional Equations, Second Edition)

26 pages, 389 KB

Open AccessArticle

On Hilfer–Hadamard Tripled System with Symmetric Nonlocal Riemann–Liouville Integral Boundary Conditions

by Shorog Aljoudi, Hind Alamri and Alanoud Alotaibi

Symmetry 2025, 17(11), 1867; https://doi.org/10.3390/sym17111867 - 4 Nov 2025

Viewed by 393

Abstract

The objective of this manuscript is to investigate the existence, uniqueness criteria and Ulam–Hyers stability of solutions to tripled systems of the Hilfer–Hadamard type supplemented with symmetric nonlocal multi-point Riemann–Liouville integral boundary conditions. By converting the considered problem into an equivalent fixed-point problem, [...] Read more.

The objective of this manuscript is to investigate the existence, uniqueness criteria and Ulam–Hyers stability of solutions to tripled systems of the Hilfer–Hadamard type supplemented with symmetric nonlocal multi-point Riemann–Liouville integral boundary conditions. By converting the considered problem into an equivalent fixed-point problem, the existence and uniqueness are proven by application of the Leray–Schauder nonlinear alternative and Banach’s contraction principle, respectively. In addition, we discuss the Ulam–Hyers stability and generalized Ulam–Hyers stability of the results, and illustrative examples are also presented to demonstrate their correctness and effectiveness. Full article

(This article belongs to the Special Issue Symmetry in Fractional Derivatives, Fractional Equations and Fractional Order Systems)

15 pages, 264 KB

Open AccessFeature PaperArticle

Monotone Picard Maps in Relational Metric Spaces

by Mihai Turinici

Mathematics 2025, 13(21), 3518; https://doi.org/10.3390/math13213518 - 3 Nov 2025

Viewed by 359

Abstract

The 2008 fixed point result on rs-relational metric spaces due to Jachymski is equivalent with the classical 1922 Banach Contraction Principle. This is also valid for the 1961 statement in metric spaces due to Edelstein, or the 2005 fixed point result in quasi-ordered [...] Read more.

The 2008 fixed point result on rs-relational metric spaces due to Jachymski is equivalent with the classical 1922 Banach Contraction Principle. This is also valid for the 1961 statement in metric spaces due to Edelstein, or the 2005 fixed point result in quasi-ordered metric spaces obtained by Nieto and Rodriguez-Lopez. Full article

16 pages, 313 KB

Open AccessArticle

Reformulation of Fixed Point Existence: From Banach to Kannan and Chatterjea Contractions

by Zouaoui Bekri, Nicola Fabiano, Mohammed Ahmed Alomair and Abdulaziz Khalid Alsharidi

Axioms 2025, 14(10), 717; https://doi.org/10.3390/axioms14100717 - 23 Sep 2025

Cited by 2 | Viewed by 608

Abstract

This paper presents a reformulation of classical existence and uniqueness results for second-order boundary value problems (BVPs) using the Kannan fixed point theorem, extending beyond the Banach contraction principle. We shift focus from the nonlinearity j to the solution operator T defined via [...] Read more.

This paper presents a reformulation of classical existence and uniqueness results for second-order boundary value problems (BVPs) using the Kannan fixed point theorem, extending beyond the Banach contraction principle. We shift focus from the nonlinearity j to the solution operator T defined via Green’s function and establish a sufficient condition under which T satisfies the Kannan contraction criterion. Specifically, if the derivative of j is bounded by K and

K \cdot {(η - ζ)}^{2} / 8 < 1 / 3

, then T is a Kannan contraction, ensuring a unique solution. This condition applies even when the Banach contraction principle fails. We also explore the plausibility of applying the Chatterjea contraction, though rigorous verification remains open. Examples illustrate the applicability of the results. This work highlights the utility of generalized contractions in differential equations. Full article

(This article belongs to the Special Issue Research in Fixed Point Theory and Its Applications)

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22 pages, 350 KB

Open AccessArticle

Coupled System of (k, ψ)-Hilfer and (k, ψ)-Caputo Sequential Fractional Differential Equations with Non-Separated Boundary Conditions

by Furkan Erkan, Nuket Aykut Hamal, Sotiris K. Ntouyas, Jessada Tariboon and Phollakrit Wongsantisuk

Axioms 2025, 14(9), 685; https://doi.org/10.3390/axioms14090685 - 7 Sep 2025

Viewed by 656

Abstract

This paper is concerned with the existence and uniqueness of solutions for a coupled system of

(k, ψ)

-Hilfer and

(k, ψ)

-Caputo sequential fractional differential equations with non-separated boundary conditions. We make use of the Banach [...] Read more.

This paper is concerned with the existence and uniqueness of solutions for a coupled system of

(k, ψ)

-Hilfer and

(k, ψ)

-Caputo sequential fractional differential equations with non-separated boundary conditions. We make use of the Banach contraction mapping principle to obtain the uniqueness result, while two existence results are proved by using Leray–Schauder nonlinear alternative and Krasnosel’skiĭ’s fixed point theorem. The obtained results are illustrated by numerical examples. Full article

(This article belongs to the Special Issue Advances in Nonlinear Analysis and Boundary Value Problems)

25 pages, 347 KB

Open AccessArticle

Fixed-Point Theorems in Branciari Distance Spaces

by Seong-Hoon Cho

Axioms 2025, 14(8), 635; https://doi.org/10.3390/axioms14080635 - 14 Aug 2025

Viewed by 586

Abstract

In this study, the concepts of

σ

-Caristi maps and generalized

σ

-contraction maps are introduced, and fixed-point theorems for such maps are established. A generalization of Caristi’s fixed-point theorem and Banach’s contraction principle is proved. The relationships among the various contraction conditions [...] Read more.

In this study, the concepts of

σ

-Caristi maps and generalized

σ

-contraction maps are introduced, and fixed-point theorems for such maps are established. A generalization of Caristi’s fixed-point theorem and Banach’s contraction principle is proved. The relationships among the various contraction conditions introduced in this paper are examined. Examples are provided to elucidate the main theorem, and applications to integral and differential equations are also discussed. Full article

(This article belongs to the Special Issue Trends in Fixed Point Theory and Fractional Calculus)

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