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26 pages, 394 KiB

Open AccessArticle

Existence and Uniqueness Analysis for (k, ψ)-Hilfer and (k, ψ)-Caputo Sequential Fractional Differential Equations and Inclusions with Non-Separated Boundary Conditions

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

Fractal Fract. 2025, 9(7), 437; https://doi.org/10.3390/fractalfract9070437 - 2 Jul 2025

Viewed by 224

Abstract

This paper investigates the existence and uniqueness of solutions to a class of sequential fractional differential equations and inclusions involving the

(k, ψ)

-Hilfer and

(k, ψ)

-Caputo derivatives under non-separated boundary conditions. By reformulating the problems [...] Read more.

This paper investigates the existence and uniqueness of solutions to a class of sequential fractional differential equations and inclusions involving the

(k, ψ)

-Hilfer and

(k, ψ)

-Caputo derivatives under non-separated boundary conditions. By reformulating the problems into equivalent fixed-point systems, several classical fixed-point theorems, including those of Banach, Krasnosel’ski

\overset{˘}{i}

’s, Schaefer, and the Leray–Schauder alternative, are employed to derive rigorous results. The study is further extended to the multi-valued setting, where existence results are established for both convex- and nonconvex-valued multi-functions using appropriate fixed-point techniques. Numerical examples are provided to illustrate the applicability and effectiveness of the theoretical findings. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition)

13 pages, 265 KiB

Open AccessArticle

Stability and Hyperstability of Ternary Hom-Multiplier on Ternary Banach Algebra

by Vahid Keshavarz, Mohammad Taghi Heydari and Douglas R. Anderson

Axioms 2025, 14(7), 494; https://doi.org/10.3390/axioms14070494 - 25 Jun 2025

Viewed by 189

Abstract

In this article, we investigate the 3D additive-type functional equation. Next, we introduce the ternary hom-multiplier in ternary Banach algebras using the concepts of ternary homomorphisms and ternary multipliers. We first establish proof that solutions to the 3D additive-type functional equation are additive [...] Read more.

In this article, we investigate the 3D additive-type functional equation. Next, we introduce the ternary hom-multiplier in ternary Banach algebras using the concepts of ternary homomorphisms and ternary multipliers. We first establish proof that solutions to the 3D additive-type functional equation are additive mappings. We further demonstrate that these solutions are

C

-linear mappings. The final portion of our work examines both the stability and hyperstability properties of the 3D additive-type functional equation, ternary hom-multiplier, and ternary Jordan hom-multiplier on ternary Banach algebras. Our analysis employs the fixed-point theorem using control functions developed by Gǎvruta and Rassias. Full article

(This article belongs to the Section Algebra and Number Theory)

22 pages, 303 KiB

Open AccessArticle

Remarks on a New Variable-Coefficient Integro-Differential Equation via Inverse Operators

by Chenkuan Li, Nate Fingas and Ying Ying Ou

Fractal Fract. 2025, 9(7), 404; https://doi.org/10.3390/fractalfract9070404 - 23 Jun 2025

Viewed by 216

Abstract

In this paper, we investigate functional inverse operators associated with a class of fractional integro-differential equations. We further explore the existence, uniqueness, and stability of solutions to a new integro-differential equation featuring variable coefficients and a functional boundary condition. To demonstrate the applicability [...] Read more.

In this paper, we investigate functional inverse operators associated with a class of fractional integro-differential equations. We further explore the existence, uniqueness, and stability of solutions to a new integro-differential equation featuring variable coefficients and a functional boundary condition. To demonstrate the applicability of our main theorems, we provide several examples in which we compute values of the two-parameter Mittag–Leffler functions. The proposed approach is particularly effective for addressing a wide range of integral and fractional nonlinear differential equations with initial or boundary conditions—especially those involving variable coefficients, which are typically challenging to treat using classical integral transform methods. Finally, we demonstrate a significant application of the inverse operator approach by solving a Caputo fractional convection partial differential equation in

R^{n}

with an initial condition. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition)

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 164

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)

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28 pages, 13895 KiB

Open AccessArticle

Solvability of Fuzzy Partially Differentiable Models for Caputo–Hadamard-Type Goursat Problems Involving Generalized Hukuhara Difference

by Si-Yuan Lin, Heng-You Lan and Ji-Hong Li

Fractal Fract. 2025, 9(6), 395; https://doi.org/10.3390/fractalfract9060395 - 19 Jun 2025

Viewed by 249

Abstract

In this paper, we investigate a class of fuzzy partially differentiable models for Caputo–Hadamard-type Goursat problems with generalized Hukuhara difference, which have been widely recognized as having a significant role in simulating and analyzing various kinds of processes in engineering and physical sciences. [...] Read more.

In this paper, we investigate a class of fuzzy partially differentiable models for Caputo–Hadamard-type Goursat problems with generalized Hukuhara difference, which have been widely recognized as having a significant role in simulating and analyzing various kinds of processes in engineering and physical sciences. By transforming the fuzzy partially differentiable models into equivalent integral equations and employing classical Banach and Schauder fixed-point theorems, we establish the existence and uniqueness of solutions for the fuzzy partially differentiable models. Furthermore, in order to overcome the complexity of obtaining exact solutions of systems involving Caputo–Hadamard fractional derivatives, we explore numerical approximations based on trapezoidal and Simpson’s rules and propose three numerical examples to visually illustrate the main results presented in this paper. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

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19 pages, 330 KiB

Open AccessArticle

On the Existence of (p,q)-Solutions for the Post-Quantum Langevin Equation: A Fixed-Point-Based Approach

by Mohammed Jasim Mohammed, Ali Ghafarpanah, Sina Etemad, Sotiris K. Ntouyas and Jessada Tariboon

Axioms 2025, 14(6), 474; https://doi.org/10.3390/axioms14060474 - 19 Jun 2025

Viewed by 281

Abstract

The two-parameter

(p, q)

-operators are a new family of operators in calculus that have shown their capabilities in modeling various systems in recent years. Following this path, in this paper, we present a new construction of the Langevin equation [...] Read more.

The two-parameter

(p, q)

-operators are a new family of operators in calculus that have shown their capabilities in modeling various systems in recent years. Following this path, in this paper, we present a new construction of the Langevin equation using two-parameter

(p, q)

-Caputo derivatives. For this new Langevin equation, equivalently, we obtain the solution structure as a post-quantum integral equation and then conduct an existence analysis via a fixed-point-based approach. The use of theorems such as the Krasnoselskii and Leray–Schauder fixed-point theorems will guarantee the existence of solutions to this equation, whose uniqueness is later proven by Banach’s contraction principle. Finally, we provide three examples in different structures and validate the results numerically. Full article

(This article belongs to the Special Issue Fractional Order Functional Differential Equations and Fixed Point Theory)

13 pages, 330 KiB

Open AccessArticle

Existence of Solutions to Fractional Differential Equations with Mixed Caputo–Riemann Derivative

by Mahir Almatarneh, Sonuc Zorlu and Nazim I. Mahmudov

Fractal Fract. 2025, 9(6), 374; https://doi.org/10.3390/fractalfract9060374 - 12 Jun 2025

Viewed by 482

Abstract

The study of fractional differential equations is gaining increasing significance due to their wide-ranging applications across various fields. Different methods, including fixed-point theory, variational approaches, and the lower and upper solutions method, are employed to analyze the existence and uniqueness of solutions to [...] Read more.

The study of fractional differential equations is gaining increasing significance due to their wide-ranging applications across various fields. Different methods, including fixed-point theory, variational approaches, and the lower and upper solutions method, are employed to analyze the existence and uniqueness of solutions to fractional differential equations. This paper investigates the existence and uniqueness of solutions to a class of nonlinear fractional differential equations involving mixed Caputo–Riemann fractional derivatives with integral initial conditions, set within a Banach space. Sufficient conditions are provided for the existence and uniqueness of solutions based on the problem’s parameters. The results are derived by constructing the Green’s function for the initial value problem. Schauder’s fixed-point theorem is used to prove existence, while Banach’s contraction mapping principle ensures uniqueness. Finally, an example is given to demonstrate the practical application of the results. Full article

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12 pages, 274 KiB

Open AccessArticle

Existence and Stability Analysis of Nonlinear Systems with Hadamard Fractional Derivatives

by Mouataz Billah Mesmouli, Ioan-Lucian Popa and Taher S. Hassan

Mathematics 2025, 13(11), 1869; https://doi.org/10.3390/math13111869 - 3 Jun 2025

Viewed by 305

Abstract

This paper investigates the existence, uniqueness, and finite-time stability of solutions to a class of nonlinear systems governed by the Hadamard fractional derivative. The analysis is carried out using two fundamental tools from fixed point theory: the Krasnoselskii fixed point theorem and the [...] Read more.

This paper investigates the existence, uniqueness, and finite-time stability of solutions to a class of nonlinear systems governed by the Hadamard fractional derivative. The analysis is carried out using two fundamental tools from fixed point theory: the Krasnoselskii fixed point theorem and the Banach contraction principle. These methods provide rigorous conditions under which solutions exist and are unique. Furthermore, criteria ensuring the finite-time stability of the system are derived. To demonstrate the practicality of the theoretical results, a detailed example is presented. This paper also discusses certain assumptions and presents corollaries that naturally emerge from the main theorems. Full article

11 pages, 261 KiB

Open AccessArticle

A Result Regarding the Existence and Attractivity for a Class of Nonlinear Fractional Difference Equations with Time-Varying Delays

by Shihan Wang and Danfeng Luo

Fractal Fract. 2025, 9(6), 362; https://doi.org/10.3390/fractalfract9060362 - 31 May 2025

Viewed by 316

Abstract

In this paper, we are studying a class of nonlinear fractional difference equations with time-varying delays in Banach space. By means of mathematical induction and the Picard iteration method, we first obtain the existence result of this fractional difference system. Under some new [...] Read more.

In this paper, we are studying a class of nonlinear fractional difference equations with time-varying delays in Banach space. By means of mathematical induction and the Picard iteration method, we first obtain the existence result of this fractional difference system. Under some new criteria along with the Schauder’s fixed point theorem, we then derive the attractivity conclusions. Subsequently, with the aid of Grönwall’s inequality, we prove that the system is globally attractive. Finally, we give two examples to prove the validity of our theorems. Full article

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

17 pages, 285 KiB

Open AccessArticle

Fixed Point Approximation for Enriched Suzuki Nonexpansive Mappings in Banach Spaces

by Doaa Filali, Fahad Maqbul Alamrani, Esmail Alshaban, Adel Alatawi, Amid Yousef Alanazi and Faizan Ahmad Khan

Axioms 2025, 14(6), 426; https://doi.org/10.3390/axioms14060426 - 30 May 2025

Viewed by 276

Abstract

This paper investigates the approximation of fixed points for mappings that satisfy the enriched (C) condition using a modified iterative process in a Banach space framework. We first establish a weak convergence result and then derive strong convergence theorems under suitable assumptions. To [...] Read more.

This paper investigates the approximation of fixed points for mappings that satisfy the enriched (C) condition using a modified iterative process in a Banach space framework. We first establish a weak convergence result and then derive strong convergence theorems under suitable assumptions. To illustrate the applicability of our findings, we present a numerical example involving mappings that satisfy the enriched (C) condition but not the standard (C) condition. Additionally, numerical computations and graphical representations demonstrate that the proposed iterative process achieves a faster convergence rate compared to several existing methods. As a practical application, we introduce a projection based an iterative process for solving split feasibility problems (SFPs) in a Hilbert space setting. Our findings contribute to the ongoing development of iterative processes for solving optimization and feasibility problems in mathematical and applied sciences. Full article

(This article belongs to the Special Issue Fixed-Point Theory and Its Related Topics, 5th Edition)

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28 pages, 531 KiB

Open AccessArticle

Representation Formulas and Stability Analysis for Hilfer–Hadamard Proportional Fractional Differential Equations

by Safoura Rezaei Aderyani, Reza Saadati and Donal O’Regan

Fractal Fract. 2025, 9(6), 359; https://doi.org/10.3390/fractalfract9060359 - 29 May 2025

Viewed by 383

Abstract

This paper introduces a novel version of the Gronwall inequality specifically related to the Hilfer–Hadamard proportional fractional derivative. By utilizing Picard’s method of successive approximations along with the definition of Mittag–Leffler functions, we derive a representation formula for the solution of the Hilfer–Hadamard [...] Read more.

This paper introduces a novel version of the Gronwall inequality specifically related to the Hilfer–Hadamard proportional fractional derivative. By utilizing Picard’s method of successive approximations along with the definition of Mittag–Leffler functions, we derive a representation formula for the solution of the Hilfer–Hadamard proportional fractional differential equation featuring constant coefficients, expressed in the form of the Mittag–Leffler kernel. We establish the uniqueness of the solution through the application of Banach’s fixed-point theorem, leveraging several properties of the Mittag–Leffler kernel. The current study outlines optimal stability, a new Ulam-type concept based on classical special functions. It aims to improve approximation accuracy by optimizing perturbation stability, offering flexible solutions to various fractional systems. While existing Ulam stability concepts have gained interest, extending and optimizing them for control and stability analysis in science and engineering remains a new challenge. The proposed approach not only encompasses previous ideas but also emphasizes the enhancement and optimization of model stability. The numerical results, presented in tables and charts, are provided in the application section to facilitate a better understanding. Full article

(This article belongs to the Special Issue Advances in Fractional Operators: Mathematical Perspectives and Multidisciplinary Applications)

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16 pages, 1304 KiB

Open AccessArticle

Enhancing Stability in Fractional-Order Systems: Criteria and Applications

by Safoura Rezaei Aderyani, Reza Saadati, Chenkuan Li and Donal O’Regan

Fractal Fract. 2025, 9(6), 345; https://doi.org/10.3390/fractalfract9060345 - 26 May 2025

Viewed by 317

Abstract

This study investigates the stability of fractional-order systems with infinite delay, which are prevalent in many fields due to their effectiveness in modeling complex dynamic behaviors. Recent advancements concerning the existence and various categories of stability for solutions to the given problem are [...] Read more.

This study investigates the stability of fractional-order systems with infinite delay, which are prevalent in many fields due to their effectiveness in modeling complex dynamic behaviors. Recent advancements concerning the existence and various categories of stability for solutions to the given problem are also highlighted. This investigation utilizes tools such as the Picard operator approach, the Banach fixed-point theorem, an extended form of Gronwall’s inequality, and several well-known special functions. We establish key stability criteria for fractional differential equations using Hadamard fractional derivatives and illustrate these concepts using a numerical example. Specifically, graphical representations of the system’s responses demonstrate how fractional-order control enhances stability compared to traditional integer-order approaches. Our results emphasize the value of fractional systems in improving system performance and robustness. Full article

(This article belongs to the Special Issue Harmonic and Geometric Analysis for Fractional Equations)

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19 pages, 285 KiB

Open AccessArticle

Extensions of Göhde and Kannan Fixed Point Theorems in Strictly Convex Banach Spaces

by Moosa Gabeleh and Maggie Aphane

Axioms 2025, 14(6), 400; https://doi.org/10.3390/axioms14060400 - 23 May 2025

Viewed by 337

Abstract

Let nonempty subsets E and F of a Banach space X be given, along with a mapping

S : E \cup F \to E \cup F

defined as noncyclic when

S (E) \subseteq E

and

S (F) \subseteq F

[...] Read more.

Let nonempty subsets E and F of a Banach space X be given, along with a mapping

S : E \cup F \to E \cup F

defined as noncyclic when

S (E) \subseteq E

and

S (F) \subseteq F

. In this case, an optimal pair of fixed points is defined as a point

(p, q) \in E \times F

where p and q are fixed points of

S

that estimate the distance between E and F. This article explores an extended version of Göhde’s fixed point problem to identify optimal fixed point pairs for noncyclic relatively nonexpansive maps in strictly convex Banach spaces, while introducing new classes of noncyclic Kannan contractions, noncyclic relatively Kannan nonexpansive contractions using the proximal projection mapping defined on union of proximal pairs, and proving additional existence results with supporting examples. Full article

20 pages, 386 KiB

Open AccessArticle

Some Fixed Point Results for Novel Contractions with Applications in Fractional Differential Equations for Market Equilibrium and Economic Growth

by Min Wang, Muhammad Din and Mi Zhou

Fractal Fract. 2025, 9(5), 324; https://doi.org/10.3390/fractalfract9050324 - 19 May 2025

Viewed by 366

Abstract

In this study, we introduce two new classes of contractions, namely enriched

(I, ρ, χ)

-contractions and generalized enriched

(I, ρ, χ)

-contractions, within the context of normed spaces. These classes generalize several well-known contraction [...] Read more.

In this study, we introduce two new classes of contractions, namely enriched

(I, ρ, χ)

-contractions and generalized enriched

(I, ρ, χ)

-contractions, within the context of normed spaces. These classes generalize several well-known contraction types, including

χ

-contractions, Banach contractions, enriched contractions, Kannan contractions, Bianchini contractions, Zamfirescu contractions, non-expansive mappings, and

(ρ, χ)

-enriched contractions. We establish related fixed point results for the novel contractions in normed spaces endowed with the binary relations preserving key symmetric properties, ensuring consistency and applicability. The Krasnoselskij iteration method is refined to incorporate symmetric constraints, facilitating fixed point identification within these spaces. By appropriately selecting constants in the definition of enriched

(I, ρ, χ)

-contractions, employing a suitable binary relation, or control function

χ \in Θ

, our framework generalizes and extends classical fixed point theorems. Illustrative examples highlight the significance of our findings in reinforcing fixed point conditions and demonstrating their broader applicability. Additionally, this paper explores how these ideas guarantee the stability of the production–consumption markets equilibrium and the economic growth model. Full article

(This article belongs to the Special Issue Fractional Order Modelling of Dynamical Systems)

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12 pages, 236 KiB

Open AccessArticle

The Solvability of an Infinite System of Nonlinear Integral Equations Associated with the Birth-And-Death Stochastic Process

by Szymon Dudek and Leszek Olszowy

Symmetry 2025, 17(5), 757; https://doi.org/10.3390/sym17050757 - 14 May 2025

Viewed by 278

Abstract

One of the methods for studying the solvability of infinite systems of integral or differential equations is the application of various fixed-point theorems to operators acting in appropriate functional Banach spaces. This method is fairly well developed, frequently used, and effective in many [...] Read more.

One of the methods for studying the solvability of infinite systems of integral or differential equations is the application of various fixed-point theorems to operators acting in appropriate functional Banach spaces. This method is fairly well developed, frequently used, and effective in many situations. However, there are cases in which certain infinite systems of differential equations arise—linked to the modeling of significant real-world phenomena—where this method, based on situating considerations within Banach spaces, fails and cannot be applied. In this paper, we propose a slightly different approach, which involves conducting the analysis within appropriate functional Fréchet spaces. We discuss the fundamental properties of these spaces and formulate compactness criteria. The main result of this paper is a positive answer, using the proposed method, to an open problem concerning the modeling of a stochastic birth-and-death process, as formulated in one of the cited publications. The most important conclusion is that the presented computational technique, based on functional Fréchet spaces, can be regarded as a more effective alternative to methods based on Banach spaces. Full article

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

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