Fractal and Fractional

Research

19 pages, 665 KiB

Open AccessArticle

Solutions to Variable-Order Fractional BVPs with Multipoint Data in W^s,p Spaces

by Zineb Bellabes, Kadda Maazouz, Naima Boussekkine and Rosana Rodríguez-López

Fractal Fract. 2025, 9(7), 461; https://doi.org/10.3390/fractalfract9070461 - 15 Jul 2025

Viewed by 54

This study explores the existence of positive solutions within a Sobolev space for a boundary value problem that involves Riemann–Liouville fractional derivatives of variable order. The analysis utilizes the method of upper and lower solutions in combination with the Schauder fixed-point theorem. To [...] Read more.

This study explores the existence of positive solutions within a Sobolev space for a boundary value problem that involves Riemann–Liouville fractional derivatives of variable order. The analysis utilizes the method of upper and lower solutions in combination with the Schauder fixed-point theorem. To illustrate the theoretical findings, a numerical example is included. Full article

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

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)

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)

34 pages, 435 KiB

Open AccessArticle

A Hadamard Fractional Boundary Value Problem on an Infinite Interval at Resonance

by Alexandru Tudorache and Rodica Luca

Fractal Fract. 2025, 9(6), 378; https://doi.org/10.3390/fractalfract9060378 - 13 Jun 2025

Viewed by 334

Abstract

This paper addresses the existence of solutions to a Hadamard fractional differential equation of arbitrary order on an infinite interval, subject to integral boundary conditions that incorporate both Riemann–Stieltjes integrals and Hadamard fractional derivatives. Due to the presence of nontrivial solutions in the [...] Read more.

This paper addresses the existence of solutions to a Hadamard fractional differential equation of arbitrary order on an infinite interval, subject to integral boundary conditions that incorporate both Riemann–Stieltjes integrals and Hadamard fractional derivatives. Due to the presence of nontrivial solutions in the associated homogeneous boundary value problem, the problem is classified as resonant. The Mawhin continuation theorem is utilized to derive the main findings. Full article

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

14 pages, 336 KiB

Open AccessArticle

The Existence and Stability of Integral Fractional Differential Equations

by Rahman Ullah Khan and Ioan-Lucian Popa

Fractal Fract. 2025, 9(5), 295; https://doi.org/10.3390/fractalfract9050295 - 1 May 2025

Viewed by 546

Abstract

The main goal of this research is to study integro-fractional differential equations and simulate their dynamic behavior using ABC-fractional derivatives. We investigate the Hyers–Ulam stability of the proposed system and further expand the prerequisites for the existence and uniqueness of the solutions. The [...] Read more.

The main goal of this research is to study integro-fractional differential equations and simulate their dynamic behavior using ABC-fractional derivatives. We investigate the Hyers–Ulam stability of the proposed system and further expand the prerequisites for the existence and uniqueness of the solutions. The Schauder fixed-point theorem and the Banach contraction principle are employed to obtain the results. Finally, we present an example to demonstrate the practical application of our theoretical conclusions. Full article

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

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17 pages, 22156 KiB

Open AccessArticle

Variable-Fractional-Order Nosé–Hoover System: Chaotic Dynamics and Numerical Simulations

by D. K. Almutairi, Dalal M. AlMutairi, Nidal E. Taha, Mohammed E. Dafaalla and Mohamed A. Abdoon

Fractal Fract. 2025, 9(5), 277; https://doi.org/10.3390/fractalfract9050277 - 25 Apr 2025

Viewed by 465

Abstract

This study explores the variable-order fractional Nosé–Hoover system, investigating the evolution of its chaotic and stable states under variable-order derivatives. Variable-order derivatives introduce greater complexity and adaptability into a system’s dynamics. The main objective is to examine these effects through numerical simulations, showcasing [...] Read more.

This study explores the variable-order fractional Nosé–Hoover system, investigating the evolution of its chaotic and stable states under variable-order derivatives. Variable-order derivatives introduce greater complexity and adaptability into a system’s dynamics. The main objective is to examine these effects through numerical simulations, showcasing how changes in the order function influence a system’s behavior. The variable-order behavior is shown by phase space orbits and time series for various variable orders

α

. We look at how the system acts by using numerical solutions and numerical simulations. The phase space orbits and time series for different

α

show variable-order effects. The findings emphasize the role of variable-order derivatives in enhancing chaotic behavior, offering novel insights into their impact on dynamical systems. Full article

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

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15 pages, 293 KiB

Open AccessArticle

Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its Applications

by Yongqing Wang

Fractal Fract. 2025, 9(4), 261; https://doi.org/10.3390/fractalfract9040261 - 19 Apr 2025

Viewed by 326

Abstract

In this article, we study a fractional lower-order differential equation, [...] Read more.

In this article, we study a fractional lower-order differential equation,

- D_{0 +}^{α} Υ (ξ) + a (ξ) Υ (ξ) = y (ξ), ξ \in (0, 1), α \in (1, 2),

with a Dirichlet-type boundary condition, where

a (ξ) \in L^{1} [0, 1]

permits singularity. When the coefficient of perturbation term

a (ξ)

is continuous on

[0, 1]

, Graef et al. derived the associated Green’s function under certain conditions on

a

, but failed to obtain its positivity. We obtain some positive properties of the Green’s function of this problem under certain conditions. The results will fill the gap in the above literature. As for application of the theoretical results, a singular fractional differential equation with a perturbation term is considered. The unique positive solution is obtained by using the fixed point theorem, and an iterative sequence is established to approximate it. In addition, a semipositone fractional differential equation,

- D_{0 +}^{α} Υ (ξ) = μ F (ξ, Υ (ξ)), ξ \in (0, 1), α \in (1, 2),

is also considered. The existence of positive solutions is determined under a more general condition,

F (ξ, x) \geq - b (ξ) x - e (ξ)

, where

b (ξ), e (ξ) \in L^{1} [0, 1]

are non-negative functions. Relevant examples are listed to manifest the theoretical results. Full article

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

14 pages, 292 KiB

Open AccessArticle

Positive Normalized Solutions to a Kind of Fractional Kirchhoff Equation with Critical Growth

by Shiyong Zhang and Qiongfen Zhang

Fractal Fract. 2025, 9(3), 193; https://doi.org/10.3390/fractalfract9030193 - 20 Mar 2025

Viewed by 272

Abstract

In this paper, we have investigated the existence of normalized solutions for a class of fractional Kirchhoff equations involving nonlinearity and critical nonlinearity. The nonlinearity satisfies

L^{2}

-supercritical conditions. We transform the problem into an extremal problem within the framework of Lagrange [...] Read more.

In this paper, we have investigated the existence of normalized solutions for a class of fractional Kirchhoff equations involving nonlinearity and critical nonlinearity. The nonlinearity satisfies

L^{2}

-supercritical conditions. We transform the problem into an extremal problem within the framework of Lagrange multipliers by utilizing the energy functional of the equation in the fractional Sobolev space and applying the mass constraint condition (i.e., for given

m > 0,

\int_{R^{N}} {| u |}^{2} d x = m^{2}

). We introduced a new set and proved that it is a natural constraint. The proof is based on a constrained minimization method and some characterizations of the mountain pass levels are given in order to prove the existence of ground state normalized solutions. Full article

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

21 pages, 374 KiB

Open AccessArticle

A Class of ψ-Hilfer Fractional Pantograph Equations with Functional Boundary Data at Resonance

by Bingzhi Sun, Shuqin Zhang, Tianhu Yu and Shanshan Li

Fractal Fract. 2025, 9(3), 186; https://doi.org/10.3390/fractalfract9030186 - 17 Mar 2025

Viewed by 334

Abstract

In this paper, we explore the outcomes related to the existence of nonlocal functional boundary value problems associated with pantograph equations utilizing

ψ

-Hilfer fractional derivatives. The nonlinear term relies on unknown functions which contain a proportional delay term and their fractional derivatives [...] Read more.

In this paper, we explore the outcomes related to the existence of nonlocal functional boundary value problems associated with pantograph equations utilizing

ψ

-Hilfer fractional derivatives. The nonlinear term relies on unknown functions which contain a proportional delay term and their fractional derivatives in a higher order. We discuss various existence results for the different “smoothness” requirements of the unknown function by means of Mawhin’s coincidence theory at resonance. We wrap up by providing a detailed explanation accompanied by an illustration of one of the outcomes. Full article

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

13 pages, 289 KiB

Open AccessArticle

Existence, Uniqueness, and Stability of Solutions for Nabla Fractional Difference Equations

by Nikolay D. Dimitrov and Jagan Mohan Jonnalagadda

Fractal Fract. 2024, 8(10), 591; https://doi.org/10.3390/fractalfract8100591 - 8 Oct 2024

Cited by 4 | Viewed by 1125

Abstract

In this paper, we study a class of nabla fractional difference equations with multipoint summation boundary conditions. We obtain the exact expression of the corresponding Green’s function and deduce some of its properties. Then, we impose some sufficient conditions in order to ensure [...] Read more.

In this paper, we study a class of nabla fractional difference equations with multipoint summation boundary conditions. We obtain the exact expression of the corresponding Green’s function and deduce some of its properties. Then, we impose some sufficient conditions in order to ensure existence and uniqueness results. Also, we establish some conditions under which the solution to the considered problem is generalized Ulam–Hyers–Rassias stable. In the end, some examples are included in order to illustrate our main results. Full article

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

27 pages, 406 KiB

Open AccessArticle

Positive Solutions to a System of Coupled Hadamard Fractional Boundary Value Problems

by Alexandru Tudorache and Rodica Luca

Fractal Fract. 2024, 8(9), 543; https://doi.org/10.3390/fractalfract8090543 - 19 Sep 2024

Cited by 3 | Viewed by 789

Abstract

We explore the existence, uniqueness, and multiplicity of positive solutions to a system of Hadamard fractional differential equations that contain fractional integral terms. Defined on a finite interval, this system is subject to general coupled nonlocal boundary conditions encompassing Riemann–Stieltjes integrals and Hadamard [...] Read more.

We explore the existence, uniqueness, and multiplicity of positive solutions to a system of Hadamard fractional differential equations that contain fractional integral terms. Defined on a finite interval, this system is subject to general coupled nonlocal boundary conditions encompassing Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main results, we employ several fixed-point theorems, namely the Banach contraction mapping principle, the Schauder fixed-point theorem, the Leggett–Williams fixed-point theorem, and the Guo–Krasnosel’skii fixed-point theorem. Full article

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

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