Editorial

3 pages, 175 KiB

Open AccessEditorial

Preface to the Special Issue “Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications”

by Wei-Shih Du, Marko Kostić, Vladimir E. Fedorov and Manuel Pinto

Mathematics 2023, 11(23), 4751; https://doi.org/10.3390/math11234751 - 24 Nov 2023

Viewed by 490

Abstract

Fractional calculus has played a significant role in modeling complex systems in disciplines such as mathematics, physics, biology, and engineering [...] Full article

(This article belongs to the Special Issue Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications)

Research

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11 pages, 231 KiB

Open AccessArticle

Global Convergence of Algorithms Based on Unions of Non-Expansive Maps

by Alexander J. Zaslavski

Mathematics 2023, 11(14), 3213; https://doi.org/10.3390/math11143213 - 21 Jul 2023

Viewed by 492

Abstract

In his recent research, M. K. Tam (2018) considered a framework for the analysis of iterative algorithms which can be described in terms of a structured set-valued operator. At each point in the ambient space, the value of the operator can be expressed [...] Read more.

In his recent research, M. K. Tam (2018) considered a framework for the analysis of iterative algorithms which can be described in terms of a structured set-valued operator. At each point in the ambient space, the value of the operator can be expressed as a finite union of values of single-valued para-contracting operators. He showed that the associated fixed point iteration is locally convergent around strong fixed points. In the present paper we generalize the result of Tam and show the global convergence of his algorithm for an arbitrary starting point. An analogous result is also proven for the Krasnosel’ski–Mann iterations. Full article

(This article belongs to the Special Issue Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications)

12 pages, 276 KiB

Open AccessArticle

Stability of Nonlinear Implicit Differential Equations with Caputo–Katugampola Fractional Derivative

by Qun Dai and Yunying Zhang

Mathematics 2023, 11(14), 3082; https://doi.org/10.3390/math11143082 - 12 Jul 2023

Cited by 1 | Viewed by 666

Abstract

The purpose of this paper is to study nonlinear implicit differential equations with the Caputo–Katugampola fractional derivative. By using Gronwall inequality and Banach fixed-point theorem, the existence of the solution of the implicit equation is proved, and the relevant conclusions about the stability [...] Read more.

The purpose of this paper is to study nonlinear implicit differential equations with the Caputo–Katugampola fractional derivative. By using Gronwall inequality and Banach fixed-point theorem, the existence of the solution of the implicit equation is proved, and the relevant conclusions about the stability of Ulam–Hyers are obtained. Finally, the correctness of the conclusions is verified by an example. Full article

(This article belongs to the Special Issue Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications)

19 pages, 328 KiB

Open AccessArticle

On Two-Point Boundary Value Problems and Fractional Differential Equations via New Quasi-Contractions

by Maha Noorwali and Mohammed Shehu Shagari

Mathematics 2023, 11(11), 2477; https://doi.org/10.3390/math11112477 - 27 May 2023

Viewed by 853

Abstract

The aim of this paper is to introduce new forms of quasi-contractions in metric-like spaces and initiate more general conditions for the existence of invariant points for such operators. The proposed notions are then applied to study novel existence criteria for the existence [...] Read more.

The aim of this paper is to introduce new forms of quasi-contractions in metric-like spaces and initiate more general conditions for the existence of invariant points for such operators. The proposed notions are then applied to study novel existence criteria for the existence of solutions to two-point boundary value problems in the domains of integer and fractional orders. To attract further research in this direction, important consequences are deduced and discussed to indicate the novelty and generality of our proposed concepts. Full article

(This article belongs to the Special Issue Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications)

15 pages, 787 KiB

Open AccessArticle

Existence Theoremsfor Solutions of a Nonlinear Fractional-Order Coupled Delayed System via Fixed Point Theory

by Xin Liu, Lili Chen and Yanfeng Zhao

Mathematics 2023, 11(7), 1634; https://doi.org/10.3390/math11071634 - 28 Mar 2023

Cited by 1 | Viewed by 847

Abstract

In this paper, the problem of the existence and uniqueness of solutions for a nonlinear fractional-order coupled delayed system with a new kind of boundary condition is studied. For this reason, we transform the above problem into an equivalent fixed point problem using [...] Read more.

In this paper, the problem of the existence and uniqueness of solutions for a nonlinear fractional-order coupled delayed system with a new kind of boundary condition is studied. For this reason, we transform the above problem into an equivalent fixed point problem using the integral operator. Moreover, by applying fixed point theorems, a novel set of sufficient conditions that guarantee the existence and uniqueness of solutions of the coupled system is derived. Eventually, an example is presented to illustrate the effectiveness of the obtained results. Full article

(This article belongs to the Special Issue Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications)

16 pages, 1092 KiB

Open AccessArticle

On New Generalized Viscosity Implicit Double Midpoint Rule for Hierarchical Problem

by Thanyarat Jitpeera, Anantachai Padcharoen and Wiyada Kumam

Mathematics 2022, 10(24), 4755; https://doi.org/10.3390/math10244755 - 14 Dec 2022

Viewed by 940

Abstract

The implicit midpoint rules are employed as a powerful numerical technique, and in this article we attend a class of viscosity iteration approximations on hierarchical problems for the implicit double midpoint rules. We prove the strong convergence theorem to the unique solution on [...] Read more.

The implicit midpoint rules are employed as a powerful numerical technique, and in this article we attend a class of viscosity iteration approximations on hierarchical problems for the implicit double midpoint rules. We prove the strong convergence theorem to the unique solution on hierarchical problem of this technique is established under some favorable conditions imposed on the control parameters in Hilbert spaces. Furthermore, we propose some applications to the constrained convex minimization problem, nonlinear Fredholm integral equation and variational inequality on fixed point problem. Moreover, some numerical examples are also presented to illustrate the different proposed methods and convergence results. Our results modified the implicit double midpoint rules with the hierarchical problem. Full article

(This article belongs to the Special Issue Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications)

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18 pages, 339 KiB

Open AccessArticle

The Gao-Type Constant of Absolute Normalized Norms on ℝ²

by Zhanfei Zuo, Yimin Huang, Huaping Huang and Jing Wang

Mathematics 2022, 10(23), 4591; https://doi.org/10.3390/math10234591 - 04 Dec 2022

Viewed by 881

Abstract

In this paper, we present an extraordinary method to calculate the Gao-type constant of absolute normalized norms on

R^{2}

. By using our method, we not only provide the values of the Gao-type constant in new specific spaces, but also improve well-known [...] Read more.

In this paper, we present an extraordinary method to calculate the Gao-type constant of absolute normalized norms on

R^{2}

. By using our method, we not only provide the values of the Gao-type constant in new specific spaces, but also improve well-known results from the existing literature. Full article

(This article belongs to the Special Issue Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications)

15 pages, 325 KiB

Open AccessArticle

An Inertial Modified S-Algorithm for Convex Minimization Problems with Directed Graphs and Its Applications in Classification Problems

by Kobkoon Janngam and Suthep Suantai

Mathematics 2022, 10(23), 4442; https://doi.org/10.3390/math10234442 - 24 Nov 2022

Cited by 2 | Viewed by 822

Abstract

In this paper, we propose a new accelerated common fixed-point algorithm for two countable families of G-nonexpansive mappings. Weak convergence results are obtained in the context of directed graphs in real Hilbert spaces. As applications, we apply the obtained results to solving [...] Read more.

In this paper, we propose a new accelerated common fixed-point algorithm for two countable families of G-nonexpansive mappings. Weak convergence results are obtained in the context of directed graphs in real Hilbert spaces. As applications, we apply the obtained results to solving some convex minimization problems and employ our proposed algorithm to solve the data classification of Breast Cancer, Heart Diseases and Ionosphere. Moreover, we also compare the performance of our proposed algorithm with other algorithms in the literature and it is shown that our algorithm has a better convergence behavior than the others. Full article

(This article belongs to the Special Issue Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications)

12 pages, 299 KiB

Open AccessArticle

Contraction in Rational Forms in the Framework of Super Metric Spaces

by Erdal Karapinar and Andreea Fulga

Mathematics 2022, 10(17), 3077; https://doi.org/10.3390/math10173077 - 26 Aug 2022

Viewed by 1252

Abstract

In this paper, we investigate contractions in a rational form in the context of the supermetric space, which is a very interesting generalization of the metric space. We consider an illustrative example to support this new result on supermetric space. Full article

(This article belongs to the Special Issue Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications)

11 pages, 312 KiB

Open AccessArticle

Finite-Time Stability Analysis of Fractional Delay Systems

by Ahmed M. Elshenhab, Xingtao Wang, Clemente Cesarano, Barakah Almarri and Osama Moaaz

Mathematics 2022, 10(11), 1883; https://doi.org/10.3390/math10111883 - 31 May 2022

Cited by 8 | Viewed by 1402

Abstract

Nonhomogeneous systems of fractional differential equations with pure delay are considered. As an application, the representation of solutions of these systems and their delayed Mittag-Leffler matrix functions are used to obtain the finite time stability results. Our results improve and extend the previous [...] Read more.

Nonhomogeneous systems of fractional differential equations with pure delay are considered. As an application, the representation of solutions of these systems and their delayed Mittag-Leffler matrix functions are used to obtain the finite time stability results. Our results improve and extend the previous related results. Finally, to illustrate our theoretical results, we give an example. Full article

(This article belongs to the Special Issue Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications)

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8 pages, 259 KiB

Open AccessArticle

Remarks on Recent Results for Generalized F-Contractions

by Huaping Huang, Vesna Todorčević and Stojan Radenović

Mathematics 2022, 10(5), 768; https://doi.org/10.3390/math10050768 - 28 Feb 2022

Cited by 4 | Viewed by 1473

Abstract

In this paper, we give some remarks on the recent papers on generalized F-contractions. Our results unify and generalize the previous results in the existing literature. Moreover, we give an example to support our results. As an application, we give the existence [...] Read more.

In this paper, we give some remarks on the recent papers on generalized F-contractions. Our results unify and generalize the previous results in the existing literature. Moreover, we give an example to support our results. As an application, we give the existence and uniqueness of a solution to a class of differential equations. Full article

(This article belongs to the Special Issue Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications)

19 pages, 338 KiB

Open AccessArticle

Analytic Resolving Families for Equations with Distributed Riemann–Liouville Derivatives

by Vladimir E. Fedorov, Wei-Shih Du, Marko Kostić and Aliya A. Abdrakhmanova

Mathematics 2022, 10(5), 681; https://doi.org/10.3390/math10050681 - 22 Feb 2022

Cited by 5 | Viewed by 1132

Abstract

Some new necessary and sufficient conditions for the existence of analytic resolving families of operators to the linear equation with a distributed Riemann–Liouville derivative in a Banach space are established. We study the unique solvability of a natural initial value problem with distributed [...] Read more.

Some new necessary and sufficient conditions for the existence of analytic resolving families of operators to the linear equation with a distributed Riemann–Liouville derivative in a Banach space are established. We study the unique solvability of a natural initial value problem with distributed fractional derivatives in the initial conditions to corresponding inhomogeneous equations. These abstract results are applied to a class of initial boundary value problems for equations with distributed derivatives in time and polynomials with respect to a self-adjoint elliptic differential operator in spatial variables. Full article

(This article belongs to the Special Issue Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications)

27 pages, 425 KiB

Open AccessArticle

Doss ρ-Almost Periodic Type Functions in

R

ⁿ

by Marko Kostić, Wei-Shih Du and Vladimir E. Fedorov

Mathematics 2021, 9(21), 2825; https://doi.org/10.3390/math9212825 - 07 Nov 2021

Cited by 4 | Viewed by 1152

Abstract

In this paper, we investigate various classes of multi-dimensional Doss

ρ

-almost periodic type functions of the form

F : Λ \times X \to Y,

where

n \in N,

\emptyset \neq Λ \subseteq R^{n},

X and Y are complex [...] Read more.

In this paper, we investigate various classes of multi-dimensional Doss

ρ

-almost periodic type functions of the form

F : Λ \times X \to Y,

where

n \in N,

\emptyset \neq Λ \subseteq R^{n},

X and Y are complex Banach spaces, and

ρ

is a binary relation on

Y .

We work in the general setting of Lebesgue spaces with variable exponents. The main structural properties of multi-dimensional Doss

ρ

-almost periodic type functions, like the translation invariance, the convolution invariance and the invariance under the actions of convolution products, are clarified. We examine connections of Doss

ρ

-almost periodic type functions with

(ω, c)

-periodic functions and Weyl-

ρ

-almost periodic type functions in the multi-dimensional setting. Certain applications of our results to the abstract Volterra integro-differential equations and the partial differential equations are given. Full article

(This article belongs to the Special Issue Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications)

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