Special Issue "Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Dynamical Systems".

Deadline for manuscript submissions: closed (31 December 2019).

Special Issue Editors

Prof. Chi-Ming Chen
Website
Guest Editor
Institute of Computational and Modeling Science, National Tsing Hua University, 521 Nan-Dah Road, Hsinchu City 30014, Taiwan
Interests: mathematical analysis; functional analysis; fixed point theory
Prof. Ing-Jer Lin
Website
Guest Editor
Department of Mathematics, National Kaohsiung Normal University, 82444 No.62, Shenjhong Road, Yanchao District, Kaohsiung City 82446, Taiwan
Interests: multivariate analysis; harmonic analysis; functional analysis; nonlinear analysis; fixed point theory; mathematics education

Special Issue Information

Dear Colleagues,

The main goal of this Special Issue is to gather the recent results and discussion around reseach in nonlinear (functional) analysis, with a particular interest in the improvements to the theoretical side of fixed point theory that may play a crucial role in the solution of real-world problems.

The fixed point $f(x) = x$ equation is equivalent to F(x) = 0 where F(x) = x − f(x). Thus, the concrete solution of such equations takes "fixed point theory" into account. In cases where it is not possible to solve such a problem, any approximative solution is also worth considering and can be identified through the best proximity point theory. Roughly speaking, best proximity means the minimal value of d(x,f(x)) if f(x) is not equal to x.

These approaches help us to overcome the difficulties arising in the study and computational simulation of nonlinear analysis. For the solution of the problems and/or to identify better solutions, researcher refine the existing axioms and requirements. In this Special Issue, we aim to explore the trends in the solutions of real-world problem, in particular by using the fixed/best-proximity point theory.

Topics for this mini symposium include, but are not limited to:

  • Fixed point of continuous and discontinuous mapping;
  • Existence and uniqueness of the solution of the equation F(x) = 0 = x − f(x);
  • Characterization of completeness for abstract spaces;
  • Iterative methods for the nonexpansive-type mappings;
  • Examination of different types of contractions.

Prof. Chi-Ming Chen
Prof. Ing-Jer Lin
Guest Editors

Manuscript Submission Information

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Keywords

  • Fixed point of single and multivalued mappings
  • Best proximity point of single and multivalued mappings
  • Distinct contractions
  • Applications of fixed point results

Published Papers (13 papers)

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Research

Open AccessArticle
Fundamental Questions and New Counterexamples for b-Metric Spaces and Fatou Property
Mathematics 2019, 7(11), 1107; https://doi.org/10.3390/math7111107 - 14 Nov 2019
Abstract
In this paper, we give new examples to show that the continuity actually strictly stronger than the Fatou property in b-metric spaces. We establish a new fixed point theorem for new essential and fundamental sufficient conditions such that a Ćirić type contraction [...] Read more.
In this paper, we give new examples to show that the continuity actually strictly stronger than the Fatou property in b-metric spaces. We establish a new fixed point theorem for new essential and fundamental sufficient conditions such that a Ćirić type contraction with contraction constant λ [ 1 s , 1 ) in a complete b-metric space with s > 1 have a unique fixed point. Many new examples illustrating our results are also given. Our new results extend and improve many recent results and they are completely original and quite different from the well known results on the topic in the literature. Full article
Open AccessArticle
Best Proximity Results with Applications to Nonlinear Dynamical Systems
Mathematics 2019, 7(10), 900; https://doi.org/10.3390/math7100900 - 26 Sep 2019
Abstract
Best proximity point theorem furnishes sufficient conditions for the existence and computation of an approximate solution ω that is optimal in the sense that the error σ ( ω , J ω ) assumes the global minimum value σ ( θ , ϑ [...] Read more.
Best proximity point theorem furnishes sufficient conditions for the existence and computation of an approximate solution ω that is optimal in the sense that the error σ ( ω , J ω ) assumes the global minimum value σ ( θ , ϑ ) . The aim of this paper is to define the notion of Suzuki α - Θ -proximal multivalued contraction and prove the existence of best proximity points ω satisfying σ ( ω , J ω ) = σ ( θ , ϑ ) , where J is assumed to be continuous or the space M is regular. We derive some best proximity results on a metric space with graphs and ordered metric spaces as consequences. We also provide a non trivial example to support our main results. As applications of our main results, we discuss some variational inequality problems and dynamical programming problems. Full article
Open AccessArticle
Existence and Uniqueness of Zeros for Vector-Valued Functions with K-Adjustability Convexity and Their Applications
Mathematics 2019, 7(9), 809; https://doi.org/10.3390/math7090809 - 02 Sep 2019
Abstract
In this paper, we introduce the new concepts of K-adjustability convexity and strictly K-adjustability convexity which respectively generalize and extend the concepts of K-convexity and strictly K-convexity. We establish some new existence and uniqueness theorems of zeros for vector-valued [...] Read more.
In this paper, we introduce the new concepts of K-adjustability convexity and strictly K-adjustability convexity which respectively generalize and extend the concepts of K-convexity and strictly K-convexity. We establish some new existence and uniqueness theorems of zeros for vector-valued functions with K-adjustability convexity. As their applications, we obtain existence theorems for the minimization problem and fixed point problem which are original and quite different from the known results in the existing literature. Full article
Open AccessArticle
A Fixed Point Result with a Contractive Iterate at a Point
Mathematics 2019, 7(7), 606; https://doi.org/10.3390/math7070606 - 07 Jul 2019
Cited by 2
Abstract
In this manuscript, we define generalized Kincses-Totik type contractions within the context of metric space and consider the existence of a fixed point for such operators. Kincses-Totik type contractions extends the renowned Banach contraction mapping principle in different aspects. First, the continuity condition [...] Read more.
In this manuscript, we define generalized Kincses-Totik type contractions within the context of metric space and consider the existence of a fixed point for such operators. Kincses-Totik type contractions extends the renowned Banach contraction mapping principle in different aspects. First, the continuity condition for the considered mapping is not required. Second, the contraction inequality contains all possible geometrical distances. Third, the contraction inequality is formulated for some iteration of the considered operator, instead of the dealing with the given operator. Fourth and last, the iteration number may vary for each point in the domain of the operator for which we look for a fixed point. Consequently, the proved results generalize the acknowledged results in the field, including the well-known theorems of Seghal, Kincses-Totik, and Banach-Caccioppoli. We present two illustrative examples to support our results. As an application, we consider an Ulam-stability of one of our results. Full article
Open AccessArticle
On Bilateral Contractions
Mathematics 2019, 7(6), 538; https://doi.org/10.3390/math7060538 - 12 Jun 2019
Abstract
In this manuscript, we introduce a new type of contraction, bilateral contraction which merges two significant approaches in the fixed point theory: Caristi type contractions and Jaggi type contractions. The principal aim of the main result is to enrich the literature by combining [...] Read more.
In this manuscript, we introduce a new type of contraction, bilateral contraction which merges two significant approaches in the fixed point theory: Caristi type contractions and Jaggi type contractions. The principal aim of the main result is to enrich the literature by combining the techniques of the mentioned two celebrated results that belong to Jaggi and Caristi. We consider an example to indicate the validity and genuine nature of the main result. Full article
Open AccessArticle
On (G, G′)-Prešić–Ćirić Operators in Graphical Metric Spaces
Mathematics 2019, 7(5), 445; https://doi.org/10.3390/math7050445 - 18 May 2019
Abstract
The purpose of this paper is to introduce a new type of operators in graphical metric spaces and to prove some fixed point results for these operators. Several known results are generalized and extended in this new setting of graphical metric spaces. The [...] Read more.
The purpose of this paper is to introduce a new type of operators in graphical metric spaces and to prove some fixed point results for these operators. Several known results are generalized and extended in this new setting of graphical metric spaces. The results are illustrated and justified with examples. Full article
Open AccessArticle
Best Proximity Point Theorems for Two Weak Cyclic Contractions on Metric-Like Spaces
Mathematics 2019, 7(4), 349; https://doi.org/10.3390/math7040349 - 14 Apr 2019
Cited by 1
Abstract
In this paper, we establish two best proximity point theorems in the setting of metric-like spaces that are based on cyclic contraction: Meir–Keeler–Kannan type cyclic contractions and a generalized Ćirić type cyclic φ -contraction via the MT -function. We express some examples to [...] Read more.
In this paper, we establish two best proximity point theorems in the setting of metric-like spaces that are based on cyclic contraction: Meir–Keeler–Kannan type cyclic contractions and a generalized Ćirić type cyclic φ -contraction via the MT -function. We express some examples to indicate the validity of the presented results. Our results unify and generalize a number of best proximity point results on the topic in the corresponding recent literature. Full article
Open AccessArticle
Some Remarks and New Results in Ordered Partial b-Metric Spaces
Mathematics 2019, 7(4), 334; https://doi.org/10.3390/math7040334 - 06 Apr 2019
Cited by 5
Abstract
We improve and generalize some new results in fixed point theory in the context of partial b-metric spaces. Full article
Open AccessArticle
Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities, and Fixed Point Problems
Mathematics 2019, 7(2), 187; https://doi.org/10.3390/math7020187 - 16 Feb 2019
Abstract
In this paper, we introduce a multiple hybrid implicit iteration method for finding a solution for a monotone variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities, and a common fixed point problem [...] Read more.
In this paper, we introduce a multiple hybrid implicit iteration method for finding a solution for a monotone variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities, and a common fixed point problem of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping in Hilbert spaces. Strong convergence of the proposed method to the unique solution of the problem is established under some suitable assumptions. Full article
Open AccessArticle
Strong Convergence of a New Iterative Algorithm for Split Monotone Variational Inclusion Problems
Mathematics 2019, 7(2), 123; https://doi.org/10.3390/math7020123 - 24 Jan 2019
Abstract
The main aim of this work is to introduce an implicit general iterative method for approximating a solution of a split variational inclusion problem with a hierarchical optimization problem constraint for a countable family of mappings, which are nonexpansive, in the setting of [...] Read more.
The main aim of this work is to introduce an implicit general iterative method for approximating a solution of a split variational inclusion problem with a hierarchical optimization problem constraint for a countable family of mappings, which are nonexpansive, in the setting of infinite dimensional Hilbert spaces. Convergence theorem of the sequences generated in our proposed implicit algorithm is obtained under some weak assumptions. Full article
Open AccessArticle
Interpolative Ćirić-Reich-Rus Type Contractions via the Branciari Distance
Mathematics 2019, 7(1), 84; https://doi.org/10.3390/math7010084 - 15 Jan 2019
Cited by 11
Abstract
In this paper, we initiate the concept of interpolative Ćirić-Reich-Rus type contractions via the Branciari distance and prove some related fixed points results for such mappings. Moreover, an example is provided to show the useability of our obtained results. Full article
Open AccessArticle
Global Optimization for Quasi-Noncyclic Relatively Nonexpansive Mappings with Application to Analytic Complex Functions
Mathematics 2019, 7(1), 46; https://doi.org/10.3390/math7010046 - 04 Jan 2019
Cited by 1
Abstract
The purpose of this article is to resolve a global optimization problem for quasi-noncyclic relatively nonexpansive mappings by giving an algorithm that determines an optimal approximate solution of the following minimization problem, min x A d ( x , T x ) [...] Read more.
The purpose of this article is to resolve a global optimization problem for quasi-noncyclic relatively nonexpansive mappings by giving an algorithm that determines an optimal approximate solution of the following minimization problem, min x A d ( x , T x ) , min y B d ( y , T y ) and min ( x , y ) A × B d ( x , y ) ; also, we provide some illustrative examples to support our results. As an application, the existence of a solution of the analytic complex function is discussed. Full article
Open AccessArticle
Double Controlled Metric Type Spaces and Some Fixed Point Results
Mathematics 2018, 6(12), 320; https://doi.org/10.3390/math6120320 - 12 Dec 2018
Cited by 22
Abstract
In this article, in the sequel of extending b-metric spaces, we modify controlled metric type spaces via two control functions α ( x , y ) and μ ( x , y ) on the right-hand side of the b - triangle [...] Read more.
In this article, in the sequel of extending b-metric spaces, we modify controlled metric type spaces via two control functions α ( x , y ) and μ ( x , y ) on the right-hand side of the b - triangle inequality, that is, d ( x , y ) α ( x , z ) d ( x , z ) + μ ( z , y ) d ( z , y ) , for all x , y , z X . Some examples of a double controlled metric type space by two incomparable functions, which is not a controlled metric type by one of the given functions, are presented. We also provide some fixed point results involving Banach type, Kannan type and ϕ -nonlinear type contractions in the setting of double controlled metric type spaces. Full article
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