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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

Viewed by 249

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)

15 pages, 523 KiB

Open AccessArticle

Stability Analysis Study of Time-Fractional Nonlinear Modified Kawahara Equation Based on the Homotopy Perturbation Sadik Transform

by Zhihua Chen, Saeed Kosari, Jana Shafi and Mohammad Hossein Derakhshan

Fractal Fract. 2024, 8(9), 512; https://doi.org/10.3390/fractalfract8090512 - 29 Aug 2024

Viewed by 375

Abstract

In this manuscript, we survey a numerical algorithm based on the combination of the homotopy perturbation method and the Sadik transform for solving the time-fractional nonlinear modified shallow water waves (called Kawahara equation) within the frame of the Caputo–Prabhakar (CP) operator. The nonlinear [...] Read more.

In this manuscript, we survey a numerical algorithm based on the combination of the homotopy perturbation method and the Sadik transform for solving the time-fractional nonlinear modified shallow water waves (called Kawahara equation) within the frame of the Caputo–Prabhakar (CP) operator. The nonlinear terms are handled with the assistance of the homotopy polynomials. The stability analysis of the implemented method is studied by using S-stable mapping and the Banach contraction principle. Also, we use the fixed-point method to determine the existence and uniqueness of solutions in the given suggested model. Finally, some numerical simulations are illustrated to display the accuracy and efficiency of the present numerical method. Moreover, numerical behaviors are captured to validate the reliability and efficiency of the scheme. Full article

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13 pages, 267 KiB

Open AccessArticle

New Fixed Point Theorems for Generalized Meir–Keeler Type Nonlinear Mappings with Applications to Fixed Point Theory

by Shin-Yi Huang and Wei-Shih Du

Symmetry 2024, 16(8), 1088; https://doi.org/10.3390/sym16081088 - 22 Aug 2024

Viewed by 695

Abstract

In this paper, we investigate new fixed point theorems for generalized Meir–Keeler type nonlinear mappings satisfying the condition (DH). As applications, we obtain many new fixed point theorems which generalize and improve several results available in the corresponding literature. An example is [...] Read more.

In this paper, we investigate new fixed point theorems for generalized Meir–Keeler type nonlinear mappings satisfying the condition (DH). As applications, we obtain many new fixed point theorems which generalize and improve several results available in the corresponding literature. An example is provided to illustrate and support our main results. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

20 pages, 332 KiB

Open AccessArticle

β–Ulam–Hyers Stability and Existence of Solutions for Non-Instantaneous Impulsive Fractional Integral Equations

by Wei-Shih Du, Michal Fečkan, Marko Kostić and Daniel Velinov

Fractal Fract. 2024, 8(8), 469; https://doi.org/10.3390/fractalfract8080469 - 12 Aug 2024

Viewed by 780

Abstract

In this paper, we investigate a class of non-instantaneous impulsive fractional integral equations. Utilizing the Banach contraction mapping principle, we establish the existence and uniqueness of solutions for the considered problem. Additionally, employing Schauder’s fixed-point theorem, we demonstrate the existence of solutions within [...] Read more.

In this paper, we investigate a class of non-instantaneous impulsive fractional integral equations. Utilizing the Banach contraction mapping principle, we establish the existence and uniqueness of solutions for the considered problem. Additionally, employing Schauder’s fixed-point theorem, we demonstrate the existence of solutions within the framework of

β

-Banach spaces. Moreover, we examine the

β

–Ulam–Hyers stability of the solutions, providing insights into the stability behavior under small perturbations. An illustrative example is presented to demonstrate the practical applicability and effectiveness of the theoretical results obtained. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)

19 pages, 372 KiB

Open AccessArticle

Existence, Uniqueness, and Stability of a Nonlinear Tripled Fractional Order Differential System

by Yasir A. Madani, Mohammed Nour A. Rabih, Faez A. Alqarni, Zeeshan Ali, Khaled A. Aldwoah and Manel Hleili

Fractal Fract. 2024, 8(7), 416; https://doi.org/10.3390/fractalfract8070416 - 15 Jul 2024

Viewed by 685

Abstract

This manuscript investigates the existence, uniqueness, and different forms of Ulam stability for a system of three coupled differential equations involving the Riemann–Liouville (RL) fractional operator. The Leray–Schauder alternative is employed to confirm the existence of solutions, while the Banach contraction principle is [...] Read more.

This manuscript investigates the existence, uniqueness, and different forms of Ulam stability for a system of three coupled differential equations involving the Riemann–Liouville (RL) fractional operator. The Leray–Schauder alternative is employed to confirm the existence of solutions, while the Banach contraction principle is used to establish their uniqueness. Stability conditions are derived utilizing classical nonlinear functional analysis techniques. Theoretical findings are illustrated with an example. The proposed system generalizes third-order ordinary differential equations (ODEs) with different boundary conditions (BCs). Full article

(This article belongs to the Special Issue Advances in Fractional Integral and Derivative Operators with Applications)

18 pages, 1324 KiB

Open AccessArticle

Study of Quantum Difference Coupled Impulsive System with Respect to Another Function

by Nattapong Kamsrisuk, Chaiyod Kamthorncharoen, Sotiris K. Ntouyas and Jessada Tariboon

Symmetry 2024, 16(7), 897; https://doi.org/10.3390/sym16070897 - 14 Jul 2024

Viewed by 624

Abstract

In this paper, we study a quantum difference coupled impulsive system with respect to another function. Some quantum derivative and integral asymmetric graphs with respect to another function are shown to illustrate the behavior of parameters. Existence and uniqueness results are established via [...] Read more.

In this paper, we study a quantum difference coupled impulsive system with respect to another function. Some quantum derivative and integral asymmetric graphs with respect to another function are shown to illustrate the behavior of parameters. Existence and uniqueness results are established via Banach contraction mapping principle and Leray–Schauder alternative. Examples illustrating the obtained results are also included. Our results are new and significantly contribute to the literature to this new subject on quantum calculus on finite intervals with respect to another function. Full article

(This article belongs to the Special Issue Symmetries in Differential Equations and Application - Volume II)

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17 pages, 499 KiB

Open AccessArticle

Analysis of Non-Local Integro-Differential Equations with Hadamard Fractional Derivatives: Existence, Uniqueness, and Stability in the Context of RLC Models

by Manigandan Murugesan, Saravanan Shanmugam, Mohamed Rhaima and Ragul Ravi

Fractal Fract. 2024, 8(7), 409; https://doi.org/10.3390/fractalfract8070409 - 12 Jul 2024

Cited by 1 | Viewed by 709

Abstract

In this study, we focus on the stability analysis of the RLC model by employing differential equations with Hadamard fractional derivatives. We prove the existence and uniqueness of solutions using Banach’s contraction principle and Schaefer’s fixed point theorem. To facilitate our key conclusions, [...] Read more.

In this study, we focus on the stability analysis of the RLC model by employing differential equations with Hadamard fractional derivatives. We prove the existence and uniqueness of solutions using Banach’s contraction principle and Schaefer’s fixed point theorem. To facilitate our key conclusions, we convert the problem into an equivalent integro-differential equation. Additionally, we explore several versions of Ulam’s stability findings. Two numerical examples are provided to illustrate the applications of our main results. We also observe that modifications to the Hadamard fractional derivative lead to asymmetric outcomes. The study concludes with an applied example demonstrating the existence results derived from Schaefer’s fixed point theorem. These findings represent novel contributions to the literature on this topic, significantly advancing our understanding. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

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15 pages, 319 KiB

Open AccessArticle

Recent Advances in Proximity Point Theory Applied to Fractional Differential Equations

by Nabil Mlaiki, Dur-e-Shehwar Sagheer, Sana Noreen, Samina Batul and Ahmad Aloqaily

Axioms 2024, 13(6), 395; https://doi.org/10.3390/axioms13060395 - 13 Jun 2024

Viewed by 503

Abstract

This article introduces the concept of generalized

(ﬀ F, b, \overset{˘}{ϕ})

contraction in the context of

b

-metric spaces by utilizing the idea of

F

contraction introduced by Dariusz Wardowski. The main findings of the research focus on [...] Read more.

This article introduces the concept of generalized

(ﬀ F, b, \overset{˘}{ϕ})

contraction in the context of

b

-metric spaces by utilizing the idea of

F

contraction introduced by Dariusz Wardowski. The main findings of the research focus on the existence of best proximity points for multi-valued

(ﬀ F, b, \overset{˘}{ϕ})

contractions in partially ordered

b

-metric spaces. The article provides examples to illustrate the main results and demonstrates the existence of solutions to a second-order differential equation and a fractional differential equation using the established theorems. Additionally, several corollaries are presented to show that the results generalize many existing fixed-point and best proximity point theorems. Full article

(This article belongs to the Special Issue Research on Fixed Point Theory and Application)

12 pages, 287 KiB

Open AccessArticle

Existence Results and Finite-Time Stability of a Fractional (p,q)-Integro-Difference System

by Mouataz Billah Mesmouli, Loredana Florentina Iambor, Amir Abdel Menaem and Taher S. Hassan

Mathematics 2024, 12(9), 1399; https://doi.org/10.3390/math12091399 - 3 May 2024

Cited by 1 | Viewed by 643

Abstract

In this article, we mainly generalize the results in the literature for a fractional q-difference equation. Our study considers a more comprehensive type of fractional

(p, q)

-difference system of nonlinear equations. By the Banach contraction mapping principle, we obtain a [...] Read more.

In this article, we mainly generalize the results in the literature for a fractional q-difference equation. Our study considers a more comprehensive type of fractional

(p, q)

-difference system of nonlinear equations. By the Banach contraction mapping principle, we obtain a unique solution. By Krasnoselskii’s fixed-point theorem, we prove the existence of solutions. In addition, finite stability has been established too. The main results in the literature have been proven to be a particular corollary of our work. Full article

(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions, 3rd Edition)

15 pages, 290 KiB

Open AccessArticle

New Results on the Ulam–Hyers–Mittag–Leffler Stability of Caputo Fractional-Order Delay Differential Equations

by Osman Tunç

Mathematics 2024, 12(9), 1342; https://doi.org/10.3390/math12091342 - 28 Apr 2024

Cited by 1 | Viewed by 952

Abstract

The author considers a nonlinear Caputo fractional-order delay differential equation (CFrDDE) with multiple variable delays. First, we study the existence and uniqueness of the solutions of the CFrDDE with multiple variable delays. Second, we obtain two new results on the Ulam–Hyers–Mittag–Leffler (UHML) stability [...] Read more.

The author considers a nonlinear Caputo fractional-order delay differential equation (CFrDDE) with multiple variable delays. First, we study the existence and uniqueness of the solutions of the CFrDDE with multiple variable delays. Second, we obtain two new results on the Ulam–Hyers–Mittag–Leffler (UHML) stability of the same equation in a closed interval using the Picard operator, Chebyshev norm, Bielecki norm and the Banach contraction principle. Finally, we present three examples to show the applications of our results. Although there is an extensive literature on the Lyapunov, Ulam and Mittag–Leffler stability of fractional differential equations (FrDEs) with and without delays, to the best of our knowledge, there are very few works on the UHML stability of FrDEs containing a delay. Thereby, considering a CFrDDE containing multiple variable delays and obtaining new results on the existence and uniqueness of the solutions and UHML stability of this kind of CFrDDE are the important aims of this work. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

24 pages, 365 KiB

Open AccessFeature PaperArticle

Existence of Solutions to a System of Fractional

q

-Difference Boundary Value Problems

by Alexandru Tudorache and Rodica Luca

Mathematics 2024, 12(9), 1335; https://doi.org/10.3390/math12091335 - 27 Apr 2024

Viewed by 825

Abstract

We are investigating the existence of solutions to a system of two fractional

q

-difference equations containing fractional

q

-integral terms, subject to multi-point boundary conditions that encompass

q

-derivatives and fractional

q

-derivatives of different orders. In our main results, we rely [...] Read more.

We are investigating the existence of solutions to a system of two fractional

q

-difference equations containing fractional

q

-integral terms, subject to multi-point boundary conditions that encompass

q

-derivatives and fractional

q

-derivatives of different orders. In our main results, we rely on various fixed point theorems, such as the Leray–Schauder nonlinear alternative, the Schaefer fixed point theorem, the Krasnosel’skii fixed point theorem for the sum of two operators, and the Banach contraction mapping principle. Finally, several examples are provided to illustrate our findings. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

11 pages, 253 KiB

Open AccessArticle

A Comprehensive Study of the Langevin Boundary Value Problems with Variable Order Fractional Derivatives

by John R. Graef, Kadda Maazouz and Moussa Daif Allah Zaak

Axioms 2024, 13(4), 277; https://doi.org/10.3390/axioms13040277 - 21 Apr 2024

Viewed by 816

Abstract

The authors investigate Langevin boundary value problems containing a variable order Caputo fractional derivative. After presenting the background for the study, the authors provide the definitions, theorems, and lemmas that are required for comprehending the manuscript. The existence of solutions is proved using [...] Read more.

The authors investigate Langevin boundary value problems containing a variable order Caputo fractional derivative. After presenting the background for the study, the authors provide the definitions, theorems, and lemmas that are required for comprehending the manuscript. The existence of solutions is proved using Schauder’s fixed point theorem; the uniqueness of solutions is obtained by adding an additional hypothesis and applying Banach’s contraction principle. An example is provided to demonstrate the results. Full article

(This article belongs to the Special Issue Fractional Calculus and Its Applications: Historical and Recent Developments)

17 pages, 323 KiB

Open AccessArticle

Hybrid Newton-like Inverse Free Algorithms for Solving Nonlinear Equations

by Ioannis K. Argyros, Santhosh George, Samundra Regmi and Christopher I. Argyros

Algorithms 2024, 17(4), 154; https://doi.org/10.3390/a17040154 - 10 Apr 2024

Cited by 1 | Viewed by 1161

Abstract

Iterative algorithms requiring the computationally expensive in general inversion of linear operators are difficult to implement. This is the reason why hybrid Newton-like algorithms without inverses are developed in this paper to solve Banach space-valued nonlinear equations. The inverses of the linear operator [...] Read more.

Iterative algorithms requiring the computationally expensive in general inversion of linear operators are difficult to implement. This is the reason why hybrid Newton-like algorithms without inverses are developed in this paper to solve Banach space-valued nonlinear equations. The inverses of the linear operator are exchanged by a finite sum of fixed linear operators. Two types of convergence analysis are presented for these algorithms: the semilocal and the local. The Fréchet derivative of the operator on the equation is controlled by a majorant function. The semi-local analysis also relies on majorizing sequences. The celebrated contraction mapping principle is utilized to study the convergence of the Krasnoselskij-like algorithm. The numerical experimentation demonstrates that the new algorithms are essentially as effective but less expensive to implement. Although the new approach is demonstrated for Newton-like algorithms, it can be applied to other single-step, multistep, or multipoint algorithms using inverses of linear operators along the same lines. Full article

(This article belongs to the Special Issue Numerical Optimization and Algorithms: 2nd Edition)

22 pages, 360 KiB

Open AccessArticle

Boundary Value Problem for a Coupled System of Nonlinear Fractional q-Difference Equations with Caputo Fractional Derivatives

by Saleh S. Redhwan, Maoan Han, Mohammed A. Almalahi, Mona Alsulami and Maryam Ahmed Alyami

Fractal Fract. 2024, 8(1), 73; https://doi.org/10.3390/fractalfract8010073 - 22 Jan 2024

Cited by 2 | Viewed by 1518

Abstract

This paper focuses on the analysis of a coupled system governed by a Caputo-fractional derivative with q-integral-coupled boundary conditions. This system is particularly relevant in modeling multi-atomic systems, including scenarios involving adsorbed atoms or clusters on crystalline surfaces, surface–atom scattering, and atomic [...] Read more.

This paper focuses on the analysis of a coupled system governed by a Caputo-fractional derivative with q-integral-coupled boundary conditions. This system is particularly relevant in modeling multi-atomic systems, including scenarios involving adsorbed atoms or clusters on crystalline surfaces, surface–atom scattering, and atomic friction. To investigate this system, we introduce an operator that exhibits fixed points corresponding to the solutions of the problem, effectively transforming the system into an equivalent fixed-point problem. We established the necessary conditions for the existence and uniqueness of solutions using the Leray–Schauder nonlinear alternative and the Banach contraction mapping principle, respectively. Stability results in the Ulam sense for the coupled system are also discussed, along with a sensitivity analysis of the range parameters. To support the validity of their findings, we provide illustrative examples. Overall, the paper offers a thorough examination and analysis of the considered coupled system, making important contributions to the understanding of multi-atomic systems and their mathematical modeling. Full article

(This article belongs to the Special Issue Advances in Fractional Integral and Derivative Operators with Applications)

28 pages, 412 KiB

Open AccessArticle

Positive Solutions for a System of Fractional q-Difference Equations with Multi-Point Boundary Conditions

by Rodica Luca

Fractal Fract. 2024, 8(1), 70; https://doi.org/10.3390/fractalfract8010070 - 21 Jan 2024

Cited by 1 | Viewed by 1127

Abstract

We explore the existence, uniqueness, and multiplicity of positive solutions to a system of fractional q-difference equations that include fractional q-integrals. This investigation is carried out under coupled multi-point boundary conditions featuring q-derivatives and fractional q-derivatives of various orders. [...] Read more.

We explore the existence, uniqueness, and multiplicity of positive solutions to a system of fractional q-difference equations that include fractional q-integrals. This investigation is carried out under coupled multi-point boundary conditions featuring q-derivatives and fractional q-derivatives of various orders. The proofs of our principal findings employ a range of fixed-point theorems, including the Guo–Krasnosel’skii fixed-point theorem, the Leggett–Williams fixed-point theorem, the Schauder fixed-point theorem, and the Banach contraction mapping principle. Full article

(This article belongs to the Special Issue Advances in Fractional Integral and Derivative Operators with Applications)

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