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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 702

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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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 400

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)

25 pages, 389 KB

Open AccessArticle

A General Framework for the Multiplicity of Positive Solutions to Higher-Order Caputo and Hadamard Fractional Functional Differential Coupled Laplacian Systems

by Kaihong Zhao, Xiaoxia Zhao and Xiaojun Lv

Fractal Fract. 2025, 9(11), 701; https://doi.org/10.3390/fractalfract9110701 - 30 Oct 2025

Cited by 5 | Viewed by 712

Abstract

This paper applies a general framework to explore the existence of multiple positive solutions for the fractional integral boundary value problem of high-order Caputo and Hadamard fractional coupled Laplacian systems with delayed or advanced arguments. We first focus on a generalized fractional homomorphic [...] Read more.

This paper applies a general framework to explore the existence of multiple positive solutions for the fractional integral boundary value problem of high-order Caputo and Hadamard fractional coupled Laplacian systems with delayed or advanced arguments. We first focus on a generalized fractional homomorphic coupled boundary value problem with Hilfer fractional derivatives. Then we present the Green’s function corresponding to this Hilfer fractional system and its important properties. On this basis, by constructing a positive cone and applying a generalized cone fixed point theorem, we have established some novel criteria to ensure that the generalized fractional system has at least three positive solutions. As applications, we also obtain the multiplicity of the positive solutions of the Caputo and Hadamard fractional-order coupled Laplacian systems under two special Hilfer derivatives, respectively. Finally, we provide several examples to inspect the applicability of the main results. Full article

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

28 pages, 531 KB

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 833

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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25 pages, 325 KB

Open AccessReview

Advances in Fractional Lyapunov-Type Inequalities: A Comprehensive Review

by Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon

Foundations 2025, 5(2), 18; https://doi.org/10.3390/foundations5020018 - 27 May 2025

Cited by 3 | Viewed by 1143

Abstract

In this survey, we have included the recent results on Lyapunov-type inequalities for differential equations of fractional order associated with Dirichlet, nonlocal, multi-point, anti-periodic, and discrete boundary conditions. Our results involve a variety of fractional derivatives such as Riemann–Liouville, Caputo, Hilfer–Hadamard,

ψ

-Riemann–Liouville, [...] Read more.

In this survey, we have included the recent results on Lyapunov-type inequalities for differential equations of fractional order associated with Dirichlet, nonlocal, multi-point, anti-periodic, and discrete boundary conditions. Our results involve a variety of fractional derivatives such as Riemann–Liouville, Caputo, Hilfer–Hadamard,

ψ

-Riemann–Liouville, Atangana–Baleanu, tempered, half-linear, and discrete fractional derivatives. Full article

(This article belongs to the Section Mathematical Sciences)

21 pages, 370 KB

Open AccessArticle

A Study of a Nonlocal Coupled Integral Boundary Value Problem for Nonlinear Hilfer–Hadamard-Type Fractional Langevin Equations

by Bashir Ahmad, Hafed A. Saeed and Sotiris K. Ntouyas

Fractal Fract. 2025, 9(4), 229; https://doi.org/10.3390/fractalfract9040229 - 4 Apr 2025

Cited by 2 | Viewed by 950

Abstract

We discuss the existence criteria and Ulam–Hyers stability for solutions to a nonlocal integral boundary value problem of nonlinear coupled Hilfer–Hadamard-type fractional Langevin equations. Our results rely on the Leray–Schauder alternative and Banach’s fixed point theorem. Examples are included to illustrate the results [...] Read more.

We discuss the existence criteria and Ulam–Hyers stability for solutions to a nonlocal integral boundary value problem of nonlinear coupled Hilfer–Hadamard-type fractional Langevin equations. Our results rely on the Leray–Schauder alternative and Banach’s fixed point theorem. Examples are included to illustrate the results obtained. Full article

(This article belongs to the Special Issue Advances in Fractional Initial and Boundary Value Problems)

29 pages, 975 KB

Open AccessArticle

Theoretical Results on the pth Moment of ϕ-Hilfer Stochastic Fractional Differential Equations with a Pantograph Term

by Abdelhamid Mohammed Djaouti and Muhammad Imran Liaqat

Fractal Fract. 2025, 9(3), 134; https://doi.org/10.3390/fractalfract9030134 - 20 Feb 2025

Cited by 4 | Viewed by 1081

Abstract

Here, we establish significant results on the well-posedness of solutions to stochastic pantograph fractional differential equations (SPFrDEs) with the

ϕ

-Hilfer fractional derivative. Additionally, we prove the smoothness theorem for the solution and present the averaging principle result. Firstly, the contraction mapping principle [...] Read more.

Here, we establish significant results on the well-posedness of solutions to stochastic pantograph fractional differential equations (SPFrDEs) with the

ϕ

-Hilfer fractional derivative. Additionally, we prove the smoothness theorem for the solution and present the averaging principle result. Firstly, the contraction mapping principle is applied to determine the existence and uniqueness of the solution. Secondly, continuous dependence findings are presented under the condition that the coefficients satisfy the global Lipschitz criteria, along with regularity results. Thirdly, we establish results for the averaging principle by applying inequalities and interval translation techniques. Finally, we provide numerical examples and graphical results to support our findings. We make two generalizations of these findings. First, in terms of the fractional derivative, our established theorems and lemmas are consistent with the Caputo operator for

ϕ (t)

=

t

,

a = 1

. Our findings match the Riemann–Liouville fractional operator for

ϕ (t) = t

,

a = 0

. They agree with the Hadamard and Caputo–Hadamard fractional operators when

ϕ (t) = ln (t)

,

a = 0

and

ϕ (t) = ln (t)

,

a = 1

, respectively. Second, regarding the space, we are make generalizations for the case

p = 2

. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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23 pages, 393 KB

Open AccessArticle

Systems of Hilfer–Hadamard Fractional Differential Equations with Nonlocal Coupled Boundary Conditions

by Alexandru Tudorache and Rodica Luca

Fractal Fract. 2023, 7(11), 816; https://doi.org/10.3390/fractalfract7110816 - 11 Nov 2023

Cited by 6 | Viewed by 1909

Abstract

We study the existence and uniqueness of solutions for a system of Hilfer–Hadamard fractional differential equations. These equations are subject to coupled nonlocal boundary conditions that incorporate Riemann–Stieltjes integrals and a range of Hadamard fractional derivatives. To establish our key findings, we apply [...] Read more.

We study the existence and uniqueness of solutions for a system of Hilfer–Hadamard fractional differential equations. These equations are subject to coupled nonlocal boundary conditions that incorporate Riemann–Stieltjes integrals and a range of Hadamard fractional derivatives. To establish our key findings, we apply various fixed point theorems, notably including the Banach contraction mapping principle, the Krasnosel’skii fixed point theorem applied to the sum of two operators, the Schaefer fixed point theorem, and the Leray–Schauder nonlinear alternative. Full article

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

14 pages, 331 KB

Open AccessArticle

Investigation of a Coupled System of Hilfer–Hadamard Fractional Differential Equations with Nonlocal Coupled Hadamard Fractional Integral Boundary Conditions

by Bashir Ahmad and Shorog Aljoudi

Fractal Fract. 2023, 7(2), 178; https://doi.org/10.3390/fractalfract7020178 - 10 Feb 2023

Cited by 12 | Viewed by 2173

Abstract

We investigate the existence criteria for solutions of a nonlinear coupled system of Hilfer–Hadamard fractional differential equations of different orders complemented with nonlocal coupled Hadamard fractional integral boundary conditions. The desired results are accomplished with the aid of standard fixed-point theorems. We emphasize [...] Read more.

We investigate the existence criteria for solutions of a nonlinear coupled system of Hilfer–Hadamard fractional differential equations of different orders complemented with nonlocal coupled Hadamard fractional integral boundary conditions. The desired results are accomplished with the aid of standard fixed-point theorems. We emphasize that the fixed point approach is one of the effective methods to establish the existence results for boundary value problems. Examples illustrating the obtained results are constructed. Full article

(This article belongs to the Special Issue New Insights into Nonlinear Coupled Differential Equations with Its Applications)

22 pages, 433 KB

Open AccessArticle

Existence Results for Nonlinear Coupled Hilfer Fractional Differential Equations with Nonlocal Riemann–Liouville and Hadamard-Type Iterated Integral Boundary Conditions

by Sunisa Theswan, Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon

Symmetry 2022, 14(9), 1948; https://doi.org/10.3390/sym14091948 - 19 Sep 2022

Cited by 10 | Viewed by 2371

Abstract

We introduce and study a new class of nonlinear coupled Hilfer differential equations with nonlocal boundary conditions involving Riemann–Liouville and Hadamard-type iterated fractional integral operators. By applying the Leray–Schauder alternative and Krasnosel’skiĭ’s fixed point theorem, two results presenting different criteria for the existence [...] Read more.

We introduce and study a new class of nonlinear coupled Hilfer differential equations with nonlocal boundary conditions involving Riemann–Liouville and Hadamard-type iterated fractional integral operators. By applying the Leray–Schauder alternative and Krasnosel’skiĭ’s fixed point theorem, two results presenting different criteria for the existence of solutions to the given problem are proven. The third result provides a sufficient criterion for the existence of a unique solution to the problem at hand. Numerical examples are constructed to demonstrate the application of the results obtained. Two graphs show asymmetric solutions when a Hilfer parameter is varied. The work presented in this paper is novel and significantly enriches the literature on the topic. Full article

(This article belongs to the Special Issue Symmetry in Computational and Mathematical Methods of Fractional Calculus)

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35 pages, 424 KB

Open AccessReview

A Survey on Recent Results on Lyapunov-Type Inequalities for Fractional Differential Equations

by Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon

Fractal Fract. 2022, 6(5), 273; https://doi.org/10.3390/fractalfract6050273 - 18 May 2022

Cited by 13 | Viewed by 2400

Abstract

This survey paper is concerned with some of the most recent results on Lyapunov-type inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. Our work deals with Caputo, Riemann-Liouville,

ψ

-Caputo,

ψ

-Hilfer, hybrid, Caputo-Fabrizio, Hadamard, [...] Read more.

This survey paper is concerned with some of the most recent results on Lyapunov-type inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. Our work deals with Caputo, Riemann-Liouville,

ψ

-Caputo,

ψ

-Hilfer, hybrid, Caputo-Fabrizio, Hadamard, Katugampola, Hilfer-Katugampola, p-Laplacian, and proportional fractional derivative operators. Full article

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

17 pages, 314 KB

Open AccessArticle

Hermite-Hadamard Fractional Inequalities for Differentiable Functions

by Muhammad Samraiz, Zahida Perveen, Gauhar Rahman, Muhammad Adil Khan and Kottakkaran Sooppy Nisar

Fractal Fract. 2022, 6(2), 60; https://doi.org/10.3390/fractalfract6020060 - 25 Jan 2022

Cited by 14 | Viewed by 2530

Abstract

In this article, we look at a variety of mean-type integral inequalities for a well-known Hilfer fractional derivative. We consider twice differentiable convex and s-convex functions for

s \in (0, 1]

that have applications in optimization theory. In order [...] Read more.

In this article, we look at a variety of mean-type integral inequalities for a well-known Hilfer fractional derivative. We consider twice differentiable convex and s-convex functions for

s \in (0, 1]

that have applications in optimization theory. In order to infer more interesting mean inequalities, some identities are also established. The consequences for Caputo fractional derivative are presented as special cases to our general conclusions. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

25 pages, 393 KB

Open AccessArticle

Hilfer–Hadamard Fractional Boundary Value Problems with Nonlocal Mixed Boundary Conditions

by Bashir Ahmad and Sotiris K. Ntouyas

Fractal Fract. 2021, 5(4), 195; https://doi.org/10.3390/fractalfract5040195 - 3 Nov 2021

Cited by 23 | Viewed by 2992

Abstract

This paper is concerned with the existence and uniqueness of solutions for a Hilfer–Hadamard fractional differential equation, supplemented with mixed nonlocal (multi-point, fractional integral multi-order and fractional derivative multi-order) boundary conditions. The existence of a unique solution is obtained via Banach contraction mapping [...] Read more.

This paper is concerned with the existence and uniqueness of solutions for a Hilfer–Hadamard fractional differential equation, supplemented with mixed nonlocal (multi-point, fractional integral multi-order and fractional derivative multi-order) boundary conditions. The existence of a unique solution is obtained via Banach contraction mapping principle, while the existence results are established by applying the fixed point theorems due to Krasnoselskiĭ and Schaefer and Leray–Schauder nonlinear alternatives. We demonstrate the application of the main results by presenting numerical examples. We also derive the existence results for the cases of convex and non-convex multifunctions involved in the multi-valued analogue of the problem at hand. Full article

(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)

15 pages, 267 KB

Open AccessArticle

Random Coupled Hilfer and Hadamard Fractional Differential Systems in Generalized Banach Spaces

by Saïd Abbas, Nassir Al Arifi, Mouffak Benchohra and Yong Zhou

Mathematics 2019, 7(3), 285; https://doi.org/10.3390/math7030285 - 20 Mar 2019

Cited by 16 | Viewed by 3349

Abstract

This article deals with some existence and uniqueness result of random solutions for some coupled systems of Hilfer and Hilfer–Hadamard fractional differential equations with random effects. Some applications are made of generalizations of classical random fixed point theorems on generalized Banach spaces. Full article

(This article belongs to the Special Issue Fractional-Order Integral and Derivative Operators and Their Applications)

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