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19 pages, 300 KB

Open AccessArticle

Certain Novel Best Proximity Theorems with Applications to Complex Function Theory and Integral Equations

by Moosa Gabeleh

Axioms 2025, 14(9), 657; https://doi.org/10.3390/axioms14090657 - 27 Aug 2025

Viewed by 544

Let

E

and

F

be nonempty disjoint subsets of a metric space

(M, d)

. For a non-self-mapping

φ : E \to F

, which is fixed-point free, a point

ϰ \in E

is said to be a best proximity [...] Read more.

Let

E

and

F

be nonempty disjoint subsets of a metric space

(M, d)

. For a non-self-mapping

φ : E \to F

, which is fixed-point free, a point

ϰ \in E

is said to be a best proximity point for the mapping

φ

whenever the distance of the point

ϰ

to its image under

φ

is equal to the distance between the sets,

E

and

F

. In this article, we establish new best proximity point theorems and obtain real extensions of Edelstein’s fixed point theorem in metric spaces, Krasnoselskii’s fixed point theorem in strictly convex Banach spaces, Dhage’s fixed point theorem in strictly convex Banach algebras, and Sadovskii’s fixed point problem in strictly convex Banach spaces. We then present applications of these best proximity point results to complex function theory, as well as the existence of a solution of a nonlinear functional integral equation and the existence of a mutually nearest solution for a system of integral equations. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization, 2nd Edition)

19 pages, 365 KB

Open AccessArticle

An Outlook on Hybrid Fractional Modeling of a Heat Controller with Multi-Valued Feedback Control

by Shorouk M. Al-Issa, Ahmed M. A. El-Sayed and Hind H. G. Hashem

Fractal Fract. 2023, 7(10), 759; https://doi.org/10.3390/fractalfract7100759 - 15 Oct 2023

Cited by 14 | Viewed by 1789

Abstract

In this study, we extend the investigations of fractional-order models of thermostats and guarantee the solvability of hybrid Caputo fractional models for heat controllers, satisfying some nonlocal hybrid multi-valued conditions with multi-valued feedback control, which involves the Chandrasekhar kernel, by using hybrid Dhage’s [...] Read more.

In this study, we extend the investigations of fractional-order models of thermostats and guarantee the solvability of hybrid Caputo fractional models for heat controllers, satisfying some nonlocal hybrid multi-valued conditions with multi-valued feedback control, which involves the Chandrasekhar kernel, by using hybrid Dhage’s fixed point theorem. A part of this study is dedicated to transforming this problem into an equivalent integral representation and then proving some existence results to achieve our aims. Furthermore, the continuous dependence of the unique solution on the control variable and on the set of selections will be discussed. Moreover, we provide an illustration to support our results. Full article

(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)

15 pages, 835 KB

Open AccessArticle

A Novel Implementation of Dhage’s Fixed Point Theorem to Nonlinear Sequential Hybrid Fractional Differential Equation

by Muath Awadalla, Mohamed Hannabou, Kinda Abuasbeh and Khalid Hilal

Fractal Fract. 2023, 7(2), 144; https://doi.org/10.3390/fractalfract7020144 - 2 Feb 2023

Cited by 7 | Viewed by 1540

Abstract

In this work, the existence and uniqueness of solutions to a sequential fractional (Hybrid) differential equation with hybrid boundary conditions were investigated by the generalization of Dhage’s fixed point theorem and Banach contraction mapping, respectively. In addition, the U-H technique is employed to [...] Read more.

In this work, the existence and uniqueness of solutions to a sequential fractional (Hybrid) differential equation with hybrid boundary conditions were investigated by the generalization of Dhage’s fixed point theorem and Banach contraction mapping, respectively. In addition, the U-H technique is employed to verify the stability of this solution. This study ends with two examples illustrating the theoretical findings. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Stability Analysis and Applications)

16 pages, 340 KB

Open AccessArticle

New Outcomes Regarding the Existence of Hilfer Fractional Stochastic Differential Systems via Almost Sectorial Operators

by Sivajiganesan Sivasankar and Ramalingam Udhayakumar

Fractal Fract. 2022, 6(9), 522; https://doi.org/10.3390/fractalfract6090522 - 16 Sep 2022

Cited by 13 | Viewed by 1697

Abstract

In this paper, we focus on the existence of Hilfer fractional stochastic differential systems via almost sectorial operators. The main results are obtained by using the concepts and ideas from fractional calculus, multivalued maps, semigroup theory, sectorial operators, and the fixed-point technique. We [...] Read more.

In this paper, we focus on the existence of Hilfer fractional stochastic differential systems via almost sectorial operators. The main results are obtained by using the concepts and ideas from fractional calculus, multivalued maps, semigroup theory, sectorial operators, and the fixed-point technique. We start by confirming the existence of the mild solution by using Dhage’s fixed-point theorem. Finally, an example is provided to demonstrate the considered Hilferr fractional stochastic differential systems theory. Full article

(This article belongs to the Special Issue Finite Difference Methods for Fractional and Stochastic Differential Equations)

18 pages, 419 KB

Open AccessArticle

Existence Results for the Solution of the Hybrid Caputo–Hadamard Fractional Differential Problems Using Dhage’s Approach

by Muhammad Yaseen, Sadia Mumtaz, Reny George and Azhar Hussain

Fractal Fract. 2022, 6(1), 17; https://doi.org/10.3390/fractalfract6010017 - 30 Dec 2021

Cited by 2 | Viewed by 1752

Abstract

In this work, we explore the existence results for the hybrid Caputo–Hadamard fractional boundary value problem (CH-FBVP). The inclusion version of the proposed BVP with a three-point hybrid Caputo–Hadamard terminal conditions is also considered and the related existence results are provided. To achieve [...] Read more.

In this work, we explore the existence results for the hybrid Caputo–Hadamard fractional boundary value problem (CH-FBVP). The inclusion version of the proposed BVP with a three-point hybrid Caputo–Hadamard terminal conditions is also considered and the related existence results are provided. To achieve these goals, we utilize the well-known fixed point theorems attributed to Dhage for both BVPs. Moreover, we present two numerical examples to validate our analytical findings. Full article

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21 pages, 372 KB

Open AccessArticle

Analytical Study of Two Nonlinear Coupled Hybrid Systems Involving Generalized Hilfer Fractional Operators

by Mohammed A. Almalahi, Omar Bazighifan, Satish K. Panchal, S. S. Askar and Georgia Irina Oros

Fractal Fract. 2021, 5(4), 178; https://doi.org/10.3390/fractalfract5040178 - 22 Oct 2021

Cited by 28 | Viewed by 1971

Abstract

In this research paper, we dedicate our interest to an investigation of the sufficient conditions for the existence of solutions of two new types of a coupled systems of hybrid fractional differential equations involving

ϕ

-Hilfer fractional derivatives. The existence results are established [...] Read more.

In this research paper, we dedicate our interest to an investigation of the sufficient conditions for the existence of solutions of two new types of a coupled systems of hybrid fractional differential equations involving

ϕ

-Hilfer fractional derivatives. The existence results are established in the weighted space of functions using Dhage’s hybrid fixed point theorem for three operators in a Banach algebra and Dhage’s helpful generalization of Krasnoselskii fixed- point theorem. Finally, simulated examples are provided to demonstrate the obtained results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

16 pages, 805 KB

Open AccessArticle

Implicit Hybrid Fractional Boundary Value Problem via Generalized Hilfer Derivative

by Abdellatif ‬Boutiara, Mohammed S. ‬Abdo, Mohammed A. ‬Almalahi, Hijaz Ahmad and Amira Ishan

Symmetry 2021, 13(10), 1937; https://doi.org/10.3390/sym13101937 - 15 Oct 2021

Cited by 5 | Viewed by 1673

Abstract

This research paper is dedicated to the study of a class of boundary value problems for a nonlinear, implicit, hybrid, fractional, differential equation, supplemented with boundary conditions involving general fractional derivatives, known as the

ϑ

-Hilfer and

ϑ

-Riemann–Liouville fractional operators. The existence [...] Read more.

This research paper is dedicated to the study of a class of boundary value problems for a nonlinear, implicit, hybrid, fractional, differential equation, supplemented with boundary conditions involving general fractional derivatives, known as the

ϑ

-Hilfer and

ϑ

-Riemann–Liouville fractional operators. The existence of solutions to the mentioned problem is obtained by some auxiliary conditions and applied Dhage’s fixed point theorem on Banach algebras. The considered problem covers some symmetry cases, with respect to a

ϑ

function. Moreover, we present a pertinent example to corroborate the reported results. Full article

(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications II)

26 pages, 342 KB

Open AccessArticle

Fractional Coupled Hybrid Sturm–Liouville Differential Equation with Multi-Point Boundary Coupled Hybrid Condition

by Mohadeseh Paknazar and Manuel De La Sen

Axioms 2021, 10(2), 65; https://doi.org/10.3390/axioms10020065 - 16 Apr 2021

Cited by 4 | Viewed by 1996

Abstract

The Sturm–Liouville differential equation is an important tool for physics, applied mathematics, and other fields of engineering and science and has wide applications in quantum mechanics, classical mechanics, and wave phenomena. In this paper, we investigate the coupled hybrid version of the Sturm–Liouville [...] Read more.

The Sturm–Liouville differential equation is an important tool for physics, applied mathematics, and other fields of engineering and science and has wide applications in quantum mechanics, classical mechanics, and wave phenomena. In this paper, we investigate the coupled hybrid version of the Sturm–Liouville differential equation. Indeed, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with multi-point boundary coupled hybrid condition. Furthermore, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with an integral boundary coupled hybrid condition. We give an application and some examples to illustrate our results. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)

14 pages, 300 KB

Open AccessArticle

On Hybrid Type Nonlinear Fractional Integrodifferential Equations

by Faten H. Damag, Adem Kılıçman and Awsan T. Al-Arioi

Mathematics 2020, 8(6), 984; https://doi.org/10.3390/math8060984 - 16 Jun 2020

Cited by 5 | Viewed by 2385

Abstract

In this paper, we introduce and investigate a hybrid type of nonlinear Riemann Liouville fractional integro-differential equations. We develop and extend previous work on such non-fractional equations, using operator theoretical techniques, and find the approximate solutions. We prove the existence as well as [...] Read more.

In this paper, we introduce and investigate a hybrid type of nonlinear Riemann Liouville fractional integro-differential equations. We develop and extend previous work on such non-fractional equations, using operator theoretical techniques, and find the approximate solutions. We prove the existence as well as the uniqueness of the corresponding approximate solutions by using hybrid fixed point theorems and provide upper and lower bounds to these solutions. Furthermore, some examples are provided, in which the general claims in the main theorems are demonstrated. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)

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