Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 May 2023) | Viewed by 15696

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Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 82444, Taiwan
Interests: nonlinear analysis and its applications; fixed point theory; variational principles and inequalities; optimization theory; equilibrium problems; fractional calculus theory
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Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21125 Novi Sad, Serbia
Interests: abstract Volterra integro-differential equations; abstract fractional differential equations; topological dynamics of linear operators and abstract PDEs
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Department of Mathematical Analysis, Chelyabinsk State University, Chelyabinsk, Russia
Interests: fractional derivative; distributed derivative; fractional differential equation; initial value problem; boundary value problem; inverse problem; identification problem; existence and uniqueness of solution; group analysis
Special Issues, Collections and Topics in MDPI journals

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Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Santiago de Chile, Chile
Interests: differential equation; approximation; differential systems

Special Issue Information

Dear Colleagues, 

Abstract fractional integro-differential equations arise from approximation theory and operator theory, numerical computational methods, the modeling of nonlinear phenomena, optimal control of complex systems, and other scientific research. During the previous more than eight decades, fixed point theory and its application have made a more important contribution to promote our understanding of the real world around us in various fields, such as nonlinear functional analysis, differential equations, economics, game theory, optimization, dynamic system theory, signal and image processing, and so forth.

This Special Issue will focus more on the originality of the recent results concerning the abstract (degenerate) fractional integro-differential equations in Banach spaces and locally convex spaces, the corresponding semilinear Cauchy problems, and applications of fixed point theory. We are particularly interested in the qualitative analysis of solutions for various classes of the abstract fractional integro-differential equations. We would also like to receive new results concerning the existence and uniqueness of almost periodic solutions (almost automorphic solutions, hypercyclic and topologically mixing solutions) of the abstract fractional integro-differential equations. We cordially and earnestly invite researchers to contribute their original and high-quality research papers which will inspire an advance in abstract fractional integro-differential equations and fixed point theory with applications. Potential topics include, but are not limited to:

  • Initial value problems of fractional integro-differential equations
  • Boundary value problems of fractional integro-differential equations
  • Singular and impulsive fractional integro-differential equations
  • Well-posedness and optimal control
  • Fixed point theory with applications
  • Best proximity point theory with applications
  • Algorithms for fixed points and best proximity points
  • Nonlinear problems via fractional calculus and fixed point theory approaches
  • Optimization

Prof. Dr. Wei-Shih Du
Prof. Dr. Marko Kostić
Prof. Dr. Vladimir E. Fedorov
Prof. Dr. Manuel Pinto
Guest Editors

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Keywords

  • fractional integro-differential equations
  • boundary value problem
  • fixed point theory
  • best proximity point theory
  • algorithms
  • optimization

Published Papers (13 papers)

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Editorial

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3 pages, 175 KiB  
Editorial
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 596
Abstract
Fractional calculus has played a significant role in modeling complex systems in disciplines such as mathematics, physics, biology, and engineering [...] Full article

Research

Jump to: Editorial

11 pages, 231 KiB  
Article
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
Cited by 1 | Viewed by 633
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
12 pages, 276 KiB  
Article
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 823
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
19 pages, 328 KiB  
Article
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 997
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
15 pages, 787 KiB  
Article
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 995
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
16 pages, 1092 KiB  
Article
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 1052
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
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18 pages, 339 KiB  
Article
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 - 4 Dec 2022
Viewed by 1060
Abstract
In this paper, we present an extraordinary method to calculate the Gao-type constant of absolute normalized norms on R2. 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 R2. 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
15 pages, 325 KiB  
Article
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 995
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
12 pages, 299 KiB  
Article
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
Cited by 2 | Viewed by 1405
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
11 pages, 312 KiB  
Article
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 1575
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
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8 pages, 259 KiB  
Article
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 5 | Viewed by 1608
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
19 pages, 338 KiB  
Article
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 1279
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
27 pages, 425 KiB  
Article
Doss ρ-Almost Periodic Type Functions in Rn
by Marko Kostić, Wei-Shih Du and Vladimir E. Fedorov
Mathematics 2021, 9(21), 2825; https://doi.org/10.3390/math9212825 - 7 Nov 2021
Cited by 4 | Viewed by 1247
Abstract
In this paper, we investigate various classes of multi-dimensional Doss ρ-almost periodic type functions of the form F:Λ×XY, where nN,ΛRn, 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×XY, where nN,ΛRn, 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
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