Nonlinear Problems and Applications of Fixed Point Theory

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (31 July 2021) | Viewed by 23128

Special Issue Editors


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Guest Editor
Graduate School, Mathematics, Gyeongsang National University, Jinju 52828, Republic of Korea
Interests: fixed point theory and applications; stability of functional equations; variational inequality problems; equilibrium problems; optimization problems; inequality theory and applications
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Guest Editor
Department of Mathematics Education, Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju-City, Jeonbuk 54896, Korea
Interests: topology; algebraic topology; digital topology; combinatorial topology; graph theory; discrete mathematics; applied topology; discrete and digital geometry; fixed point theory
Special Issues, Collections and Topics in MDPI journals

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Guest Editor
Department of Mathematics Education, Kyungnam University, Changwon 51767, Republic of Korea
Interests: variational inequality; fixed point theory and applications; fuzzy metric spaces; topology

Special Issue Information

Dear Colleagues,

Recently, fixed point theory (with topological fixed point theory, metric fixed point theory and discrete fixed point theory) is a very important and powerful tool to study nonlinear analysis and applications, especially, nonlinear operator theory and applications, equilibrium problems and applications, variational inequality problems and applications, complementarity problems and applications, saddle point theory and applications, differential and integral equations and applications, optimization problems and applications, approximation theory and applications, numerical analysis and applications, stability of functional equations, game theory and applications, programming problems and applications, engineering, topology, economics, geometry and many others.

The aim of Special Issue of the journal Mathematics is to enhance the new development of fixed point theory and related nonlinear problems with applications. Our Guest Editors will accept very high quality papers containing original results and survey articles of exceptional merits.

Prof. Dr. Yeol Je Cho
Prof. Dr. Sang-Eon Han
Prof. Dr. Jong Kyu Kim
Guest Editors

Manuscript Submission Information

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Keywords

  • fixed point theory and applications
  • best proximity point theory and applications
  • nonlinear operator theory and applications
  • generalized contractive mappings
  • equilibrium problems and applications
  • variational inequality problems and applications
  • optimization problems and applications
  • game theory and applications
  • numerical algorithms for nonlinear problems
  • well-posedness in fixed point theory
  • stability of functional equations related to fixed point theory
  • differential and integral equations by fixed point theory

Published Papers (13 papers)

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Research

21 pages, 380 KiB  
Article
Approximation of Fixed Points of Multivalued Generalized (α,β)-Nonexpansive Mappings in an Ordered CAT(0) Space
by Mujahid Abbas, Hira Iqbal, Manuel De la Sen and Khushdil Ahmad
Mathematics 2021, 9(16), 1945; https://doi.org/10.3390/math9161945 - 15 Aug 2021
Cited by 1 | Viewed by 1551
Abstract
The purpose of this article is to initiate the notion of monotone multivalued generalized (α,β)-nonexpansive mappings and explore the iterative approximation of the fixed points for the mapping in an ordered CAT(0) space. In particular, we employ the S [...] Read more.
The purpose of this article is to initiate the notion of monotone multivalued generalized (α,β)-nonexpansive mappings and explore the iterative approximation of the fixed points for the mapping in an ordered CAT(0) space. In particular, we employ the S-iteration algorithm in CAT(0) space to prove some convergence results. Moreover, some examples and useful results related to the proposed mapping are provided. Numerical experiments are also provided to illustrate and compare the convergence of the iteration scheme. Finally, an application of the iterative scheme has been presented in finding the solutions of integral differential equation. Full article
(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)
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12 pages, 292 KiB  
Article
Nadler’s Theorem on Incomplete Modular Space
by Fatemeh Lael, Naeem Saleem, Liliana Guran and Monica Felicia Bota
Mathematics 2021, 9(16), 1927; https://doi.org/10.3390/math9161927 - 13 Aug 2021
Cited by 3 | Viewed by 2142
Abstract
This manuscript is focused on the role of convexity of the modular, and some fixed point results for contractive correspondence and single-valued mappings are presented. Further, we prove Nadler’s Theorem and some fixed point results on orthogonal modular spaces. We present an application [...] Read more.
This manuscript is focused on the role of convexity of the modular, and some fixed point results for contractive correspondence and single-valued mappings are presented. Further, we prove Nadler’s Theorem and some fixed point results on orthogonal modular spaces. We present an application to a particular form of integral inclusion to support our proposed version of Nadler’s theorem. Full article
(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)
18 pages, 317 KiB  
Article
Inertial Extragradient Methods for Solving Split Equilibrium Problems
by Suthep Suantai, Narin Petrot and Manatchanok Khonchaliew
Mathematics 2021, 9(16), 1884; https://doi.org/10.3390/math9161884 - 8 Aug 2021
Cited by 3 | Viewed by 1456
Abstract
This paper presents two inertial extragradient algorithms for finding a solution of split pseudomonotone equilibrium problems in the setting of real Hilbert spaces. The weak and strong convergence theorems of the introduced algorithms are presented under some constraint qualifications of the scalar sequences. [...] Read more.
This paper presents two inertial extragradient algorithms for finding a solution of split pseudomonotone equilibrium problems in the setting of real Hilbert spaces. The weak and strong convergence theorems of the introduced algorithms are presented under some constraint qualifications of the scalar sequences. The discussions on the numerical experiments are also provided to demonstrate the effectiveness of the proposed algorithms. Full article
(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)
8 pages, 254 KiB  
Article
Approximation of Endpoints for α—Reich–Suzuki Nonexpansive Mappings in Hyperbolic Metric Spaces
by Izhar Uddin, Sajan Aggarwal and Afrah A. N. Abdou
Mathematics 2021, 9(14), 1692; https://doi.org/10.3390/math9141692 - 19 Jul 2021
Cited by 4 | Viewed by 1740
Abstract
The concept of an endpoint is a relatively new concept compared to the concept of a fixed point. The aim of this paper is to perform a convergence analysis of M—iteration involving α—Reich–Suzuki nonexpansive mappings. In this paper, we prove strong [...] Read more.
The concept of an endpoint is a relatively new concept compared to the concept of a fixed point. The aim of this paper is to perform a convergence analysis of M—iteration involving α—Reich–Suzuki nonexpansive mappings. In this paper, we prove strong and Δ—convergence theorems in a hyperbolic metric space. Thus, our results generalize and improve many existing results. Full article
(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)
20 pages, 664 KiB  
Article
A New Forward–Backward Algorithm with Line Searchand Inertial Techniques for Convex Minimization Problems with Applications
by Dawan Chumpungam, Panitarn Sarnmeta and Suthep Suantai
Mathematics 2021, 9(13), 1562; https://doi.org/10.3390/math9131562 - 2 Jul 2021
Cited by 2 | Viewed by 1769
Abstract
For the past few decades, various algorithms have been proposed to solve convex minimization problems in the form of the sum of two lower semicontinuous and convex functions. The convergence of these algorithms was guaranteed under the L-Lipschitz condition on the gradient of [...] Read more.
For the past few decades, various algorithms have been proposed to solve convex minimization problems in the form of the sum of two lower semicontinuous and convex functions. The convergence of these algorithms was guaranteed under the L-Lipschitz condition on the gradient of the objective function. In recent years, an inertial technique has been widely used to accelerate the convergence behavior of an algorithm. In this work, we introduce a new forward–backward splitting algorithm using a new line search and inertial technique to solve convex minimization problems in the form of the sum of two lower semicontinuous and convex functions. A weak convergence of our proposed method is established without assuming the L-Lipschitz continuity of the gradient of the objective function. Moreover, a complexity theorem is also given. As applications, we employed our algorithm to solve data classification and image restoration by conducting some experiments on these problems. The performance of our algorithm was evaluated using various evaluation tools. Furthermore, we compared its performance with other algorithms. Based on the experiments, we found that the proposed algorithm performed better than other algorithms mentioned in the literature. Full article
(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)
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14 pages, 308 KiB  
Article
Some Fixed Point Results of Weak-Fuzzy Graphical Contraction Mappings with Application to Integral Equations
by Shamoona Jabeen, Zhiming Zheng, Mutti-Ur Rehman, Wei Wei and Jehad Alzabut
Mathematics 2021, 9(5), 541; https://doi.org/10.3390/math9050541 - 4 Mar 2021
Cited by 3 | Viewed by 1502
Abstract
The present paper aims to introduce the concept of weak-fuzzy contraction mappings in the graph structure within the context of fuzzy cone metric spaces. We prove some fixed point results endowed with a graph using weak-fuzzy contractions. By relaxing the continuity condition of [...] Read more.
The present paper aims to introduce the concept of weak-fuzzy contraction mappings in the graph structure within the context of fuzzy cone metric spaces. We prove some fixed point results endowed with a graph using weak-fuzzy contractions. By relaxing the continuity condition of mappings involved, our results enrich and generalize some well-known results in fixed point theory. With the help of new lemmas, our proofs are straight forward. We furnish the validity of our findings with appropriate examples. This approach is completely new and will be beneficial for the future aspects of the related study. We provide an application of integral equations to illustrate the usability of our theory. Full article
(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)
25 pages, 455 KiB  
Article
Discrete Group Actions on Digital Objects and Fixed Point Sets by Isok(·)-Actions
by Sang-Eon Han
Mathematics 2021, 9(3), 290; https://doi.org/10.3390/math9030290 - 1 Feb 2021
Viewed by 1994
Abstract
Given a digital image (or digital object) (X,k),XZn, this paper initially establishes a group structure of the set of self-k-isomorphisms of (X,k) with the function composition, denoted [...] Read more.
Given a digital image (or digital object) (X,k),XZn, this paper initially establishes a group structure of the set of self-k-isomorphisms of (X,k) with the function composition, denoted by Isok(X) or Autk(X). In particular, let Ckn,l be a simple closed k-curve with l elements in Zn. Then, the group Isok(Ckn,l) is proved to be isomorphic to the standard dihedral group Dl with order l. The calculation of this quantity Isok(Ckn,l) is a key step for obtaining many new results. Indeed, it is essential for exploring many features of Isok(X). Furthermore, this quantity is proved to be a digital topological invariant. After proceeding with an Isok(X)-action on (X,k), we investigate some properties of fixed point sets by this action. Finally, we explore various features of fixed point sets by this action from the viewpoint of digital k-curve theory. This paper only deals with k-connected digital images (X,k) whose cardinality is equal to or greater than 2. Besides, we discuss some errors that have appeared in the lilterature. Full article
(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)
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10 pages, 1447 KiB  
Article
Approximation of the Constant in a Markov-Type Inequality on a Simplex Using Meta-Heuristics
by Grzegorz Sroka and Mariusz Oszust
Mathematics 2021, 9(3), 264; https://doi.org/10.3390/math9030264 - 29 Jan 2021
Cited by 1 | Viewed by 1630
Abstract
Markov-type inequalities are often used in numerical solutions of differential equations, and their constants improve error bounds. In this paper, the upper approximation of the constant in a Markov-type inequality on a simplex is considered. To determine the constant, the minimal polynomial and [...] Read more.
Markov-type inequalities are often used in numerical solutions of differential equations, and their constants improve error bounds. In this paper, the upper approximation of the constant in a Markov-type inequality on a simplex is considered. To determine the constant, the minimal polynomial and pluripotential theories were employed. They include a complex equilibrium measure that solves the extreme problem by minimizing the energy integral. Consequently, examples of polynomials of the second degree are introduced. Then, a challenging bilevel optimization problem that uses the polynomials for the approximation was formulated. Finally, three popular meta-heuristics were applied to the problem, and their results were investigated. Full article
(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)
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25 pages, 700 KiB  
Article
Fixed Point Sets of Digital Curves and Digital Surfaces
by Sang-Eon Han
Mathematics 2020, 8(11), 1896; https://doi.org/10.3390/math8111896 - 31 Oct 2020
Cited by 2 | Viewed by 1466
Abstract
Given a digital image (or digital object) (X,k), we address some unsolved problems related to the study of fixed point sets of k-continuous self-maps of (X,k) from the viewpoints of digital curve and [...] Read more.
Given a digital image (or digital object) (X,k), we address some unsolved problems related to the study of fixed point sets of k-continuous self-maps of (X,k) from the viewpoints of digital curve and digital surface theory. Consider two simple closed k-curves with li elements in Zn, i{1,2},l1l24. After initially formulating an alignment of fixed point sets of a digital wedge of these curves, we prove that perfectness of it depends on the numbers li,i{1,2}, instead of the k-adjacency. Furthermore, given digital k-surfaces, we also study an alignment of fixed point sets of digital k-surfaces and digital wedges of them. Finally, given a digital image which is not perfect, we explore a certain condition that makes it perfect. In this paper, each digital image (X,k) is assumed to be k-connected and X2 unless stated otherwise. Full article
(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)
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21 pages, 523 KiB  
Article
The Most Refined Axiom for a Digital Covering Space and Its Utilities
by Sang-Eon Han
Mathematics 2020, 8(11), 1868; https://doi.org/10.3390/math8111868 - 27 Oct 2020
Cited by 4 | Viewed by 1824
Abstract
This paper is devoted to establishing the most refined axiom for a digital covering space which remains open. The crucial step in making our approach is to simplify the notions of several types of earlier versions of local [...] Read more.
This paper is devoted to establishing the most refined axiom for a digital covering space which remains open. The crucial step in making our approach is to simplify the notions of several types of earlier versions of local (k0,k1)-isomorphisms and use the most simplified local (k0,k1)-isomorphism. This approach is indeed a key step to make the axioms for a digital covering space very refined. In this paper, the most refined local (k0,k1)-isomorphism is proved to be a (k0,k1)-covering map, which implies that the earlier axioms for a digital covering space are significantly simplified with one axiom. This finding facilitates the calculations of digital fundamental groups of digital images using the unique lifting property and the homotopy lifting theorem. In addition, consider a simple closed k:=k(t,n)-curve with five elements in Zn, denoted by SCkn,5. After introducing the notion of digital topological imbedding, we investigate some properties of SCkn,5, where k:=k(t,n),3tn. Since SCkn,5 is the minimal and simple closed k-curve with odd elements in Zn which is not k-contractible, we strongly study some properties of it associated with generalized digital wedges from the viewpoint of fixed point theory. Finally, after introducing the notion of generalized digital wedge, we further address some issues which remain open. The present paper only deals with k-connected digital images. Full article
(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)
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10 pages, 232 KiB  
Article
Local Sharp Vector Variational Type Inequality and Optimization Problems
by Jong Kyu Kim and Salahuddin
Mathematics 2020, 8(10), 1844; https://doi.org/10.3390/math8101844 - 20 Oct 2020
Cited by 3 | Viewed by 1675
Abstract
In this paper, our goal was to establish the relationship between solutions of local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, also Minty local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, [...] Read more.
In this paper, our goal was to establish the relationship between solutions of local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, also Minty local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, under generalized approximate η-convexity conditions for locally Lipschitzian functions. Full article
(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)
12 pages, 923 KiB  
Article
Linear Convergence of Split Equality Common Null Point Problem with Application to Optimization Problem
by Yaqian Jiang, Rudong Chen and Luoyi Shi
Mathematics 2020, 8(10), 1836; https://doi.org/10.3390/math8101836 - 19 Oct 2020
Viewed by 1398
Abstract
The purpose of this paper is to propose an iterative algorithm for solving the split equality common null point problem (SECNP), which is to find an element of the set of common zero points for a finite family of maximal monotone operators in [...] Read more.
The purpose of this paper is to propose an iterative algorithm for solving the split equality common null point problem (SECNP), which is to find an element of the set of common zero points for a finite family of maximal monotone operators in Hilbert spaces. We introduce the concept of bounded linear regularity for the SECNP and construct several sufficient conditions to ensure the linear convergence of the algorithm. Moreover, some numerical experiments are given to test the validity of our results. Full article
(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)
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26 pages, 473 KiB  
Article
Fixed Point Sets of k-Continuous Self-Maps of m-Iterated Digital Wedges
by Sang-Eon Han
Mathematics 2020, 8(9), 1617; https://doi.org/10.3390/math8091617 - 18 Sep 2020
Cited by 2 | Viewed by 1780
Abstract
Let Ckn,l be a simple closed k-curves with l elements in Zn and W:=Ckn,lCkn,lm-times be an m-iterated [...] Read more.
Let Ckn,l be a simple closed k-curves with l elements in Zn and W:=Ckn,lCkn,lm-times be an m-iterated digital wedges of Ckn,l, and F(Conk(W)) be an alignment of fixed point sets of W. Then, the aim of the paper is devoted to investigating various properties of F(Conk(W)). Furthermore, when proceeding with this work, this paper addresses several unsolved problems. To be specific, we firstly formulate an alignment of fixed point sets of Ckn,l, denoted by F(Conk(Ckn,l)), where l(7) is an odd natural number and k2n. Secondly, given a digital image (X,k) with X=n, we find a certain condition that supports n1,n2F(Conk(X)). Thirdly, after finding some features of F(Conk(W)), we develop a method of making F(Conk(W)) perfect according to the (even or odd) number l of Ckn,l. Finally, we prove that the perfectness of F(Conk(W)) is equivalent to that of F(Conk(Ckn,l)). This can play an important role in studying fixed point theory and digital curve theory. This paper only deals with k-connected digital images (X,k) such that X2. Full article
(This article belongs to the Special Issue Nonlinear Problems and Applications of Fixed Point Theory)
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