Proximal Contraction Research Articles

Common Best Proximity Points and Completeness of ℱ−Metric Spaces

In this paper, we introduce three classes of proximal contractions that are called the proximally λ−ψ−dominated contractions, generalized ηβγ−proximal contractions and Berinde-type weak proximal contractions, and obtain common best proximity points for these proximal contractions in the setting of F−metric spaces. Further, we obtain the best proximity point result for generalized α−φ−proximal contractions in F−metric spaces. As an application, fixed point and coincidence point results for these contractions are obtained. Some examples are provided to support the validity of our main results. Moreover, we obtain a completeness characterization of the F−metric spaces via best proximity points.

A new approach to generalized interpolative proximal contractions in non archimedean fuzzy metric spaces

<abstract><p>We introduce a new type of interpolative proximal contractive condition that ensures the existence of the best proximity points of fuzzy mappings in the complete non-archimedean fuzzy metric spaces. We establish certain best proximity point theorems for such proximal contractions. We improve and generalize the fuzzy proximal contractions by introducing fuzzy proximal interpolative contractions. The obtained results improve and generalize the best proximity point theorems published in Fuzzy Information and Engineering, 5 (2013), 417–429. Moreover, we provide many nontrivial examples to validate our best proximity point theorem.</p></abstract>

Best proximity point for generalized proximal contraction in a complete metric space

In this article, we have proved some best proximity point theorems for a non-self mapping by using generalized proximal contraction in a complete metric space. An example is also given in the support of our result.

Best Proximity Point Results for Generalization of α ̌–η ̌ Proximal Contractive Mapping in Fuzzy Banach Spaces

<p>The best proximity point is a generalization of a fixed point that is beneficial when the contraction map is not a self-map. On other hand, best approximation theorems provide an approximate solution to the fixed-point equation Tҳ = ҳ. It is used to solve the problem to determine an approximate solution that is optimum. The main goal of this paper is to present new types of proximal contraction for nonself mappings in a fuzzy Banach space. At first, the notion of the best proximity point is presented. We introduce the notion of α ̌–η ̌-β ̌ proximal contractive. After that, the best proximity point theorem for such type of mappings in a fuzzy Banach space is proved. In addition, the concept of α ̌–η ̌-φ ̌ proximal contractive mapping is presented in a fuzzy Banach space and under specific conditions, the best proximity point theorem for such type of mapping is proved. Additionally, some examples are supplied to show the results' applicability.</p>

Best Proximity Point Theorems without Fuzzy P-Property for Several (ψ − ϕ)-Weak Contractions in Non-Archimedean Fuzzy Metric Spaces

This paper addresses a problem of global optimization in a non-Archimedean fuzzy metric space context without fuzzy P-property. Specifically, it concerns the determination of the fuzzy distance between two subsets of a non-Archimedean fuzzy metric space. Our approach to solving this problem is to find an optimal approximate solution to a fixed point equation. This approach has been well studied within a category of problems called proximity point problems. We explore some new types of (ψ−ϕ)-weak proximal contractions and investigate the existence of the unique best proximity point for such kinds of mappings. Subsequently, some fixed point results for corresponding contractions are proved, and some illustrative examples are presented to support the validity of the main results. Moreover, an interesting application in computer science, particularly in the domain of words has been provided. Our work is a fuzzy generalization of the proximity point problem by means of fuzzy fixed point method.

Equivalence between the Existence of Best Proximity Points and Fixed Points for Some Classes of Proximal Contractions

In the year 2014, Almeida et al. introduced a new class of mappings, namely, contractions of Geraghty type. Additionally, in the year 2021, Beg et al. introduced the concept of generalized F-proximal contraction of the first kind and generalized F-proximal contraction of the second kind, respectively. After developing these concepts, authors mainly studied the best proximity points for these classes of mappings. In this short note, we prove that the problem of the existence of the best proximity points for the said classes of proximal contractions is equivalent to the corresponding fixed points problems.

Best Proximity Point Theorems for the Generalized Fuzzy Interpolative Proximal Contractions

The idea of best proximity points of the fuzzy mappings in fuzzy metric space was intorduced by Vetro and Salimi. We introduce a new type of proximal contractive condition that ensures the existence of best proximity points of fuzzy mappings in the fuzzy complete metric spaces. We establish certain best proximity point theorems for such proximal contractions. We improve and generalize the fuzzy proximal contractions by introducing Ψ,Φ-fuzzy proximal contractions and Ψ,Φ-fuzzy proximal interpolative contractions. The obtained results improve and generalize many best proximity point theorems published earlier. Moreover, we provide many nontrivial examples to validate our best proximity point theorem.

Interpolative Prešić Type Contractions and Related Results

In this article, we will extend the notion of interpolative Kannan contraction by introducing the notions of interpolative Prešić type contractions and interpolative Prešić type proximal contractions for mappings defined on product spaces. Through these notions, we will derive some results to ensure the existence of fixed points and best proximity points for such mappings.

Best proximity point theorems for $ \alpha^{+} F, (\theta-\phi )$-proximal contraction

In this paper, inspired by the idea of Suzuki type $ \alpha^{+} F$-proximal contraction in metric spaces, we prove a new existence of best proximity point for Suzuki type $ \alpha^{+} F$-proximal contraction and $ \alpha^{+} (\theta-\phi )$-proximal contraction defined on a closed subset of a complete metric space. Our theorems extend, generalize, and improve many existing results.

Interpolative Ćirić-Reich-Rus-type best proximity point results with applications

<abstract><p>In this paper, we introduce the notion of $ \omega $-interpolative Ćirić-Reich-Rus-type proximal contraction. We obtain some best proximity point results for these mappings using the concept of $ \omega $-admissibility in complete metric spaces. Some best proximity results are extended to partial ordered metric spaces and graphical metric spaces. Several new definitions are presented by considering the special cases of aforementioned results. The application of these results in fixed point theory is also discussed. The acquired results extend $ \omega $-interpolative Ćirić-Reich-Rus-type contraction for obtaining fixed points.</p></abstract>

SOME COMMENTS ON BEST PROXIMITY POINTS FOR ORDERED PROXIMAL CONTRACTIONS

In this article, we focus on the existence of an optimal approximate solution, designated as a best proximity point for non-self mappings which are ordered proximal contractions in the setting of partially ordered metric spaces and prove that these results are particular cases of existing fixed point theorems in the literature.

Completeness of metric spaces and existence of best proximity points

<abstract><p>In this paper, we discuss the existence of best proximity points of new generalized proximal contractions of metric spaces. Moreover, we obtain a completeness characterization of underlying metric space via the best proximity points. Some new best proximity point theorems have been derived as consequences of main results in (partially ordered) metric spaces.</p></abstract>

On Some Coincidence Best Proximity Point Results

In this paper, we discuss some (coincidence) best proximity point results for generalized proximal contractions and λ − μ -proximal Geraghty contractions in controlled metric type spaces. To clarify our study, various examples are given and some conclusions are drawn.

Best Proximity Point Theorems for Single and Multivalued Mappings in Fuzzy Multiplicative Metric Space

In this paper, we introduce fuzzy multiplicative metric space and prove some best proximity point theorems for single-valued and multivalued proximal contractions on the newly introduced space. As corollaries of our results, we prove some fixed-point theorems. Also, we present best proximity point theorems for Feng-Liu-type multivalued proximal contraction in fuzzy metric space. Moreover, we illustrate our results with some interesting examples.

Relaxed inertial methods for solving the split monotone variational inclusion problem beyond co-coerciveness

We study the split monotone variational inclusion problem in two real Hilbert spaces. Combining the inertial and relaxation techniques with the proximal contraction algorithm, we propose two new methods for solving this problem without the usual co-coerciveness assumption on the associated operators.

Existence of best proximity points satisfying two constraint inequalities

In this paper, we prove the existence of best proximity point and coupled best proximity point on metric spaces with partial order for weak proximal contraction mappings such that these critical points satisfy some constraint inequalities.

Some notes on the paper “On best proximity points of interpolative proximal contractions”

In a recent paper [I. Altun and A. Tasdemir, On best proximity points of interpolative proximal contractions, Quaestiones Math. (2020), DOI: 10.2989/16073606.2020.1785576] the authors proved two best proximity point theorems for a new class of non-self mappings, called interpolative Reich-Rus-Ćirić type proximal contractions of the first and second kinds. Our purpose is to show that their results follow from a fixed point theorem for interpolative Reich-Rus-Ćirić type contraction mappings [E. Karapinar, R. Agarwal and H. Aydi, Interpolative Reich-Rus-Ćirić contractions on partial metric spaces, Mathematics 6 (2018), 7 pages].

Best proximity point theory on vector metric spaces

In this paper, we first give a new definition of Ω-Dedekind complete Riesz space (E,≤) in the frame of vector metric space (Ω,ρ,E) and we investigate the relation between Dedekind complete Riesz space and our new concept. Moreover, we introduce a new contraction so called α-vector proximal contraction mapping. Then, we prove certain best proximity point theorems for such mappings in vector metric spaces (Ω,ρ,E) where (E,≤) is Ω-Dedekind complete Riesz space. Thus, for the first time, we acquire best proximity point results on vector metric spaces. As a result, we generalize some fixed point results proved in both vector metric spaces and partially ordered vector metric spaces such as main results of V4 . Further, we provide nontrivial and comparative examples to show the effectiveness of our main results.

Existence and Iterative Approximation of Hybrid Best Proximity Points for Set-Valued Mappings*

In this paper, hybrid best proximity points for a class of proximally mixed monotone set-valued mappings are discussed. In order to get rid of the fetter of continuity, we introduce the proximally ordered contraction instead of various proximal distance contractions widely applied in recent literatures and present some new existence and iterative approximation theorems of best proximity points for such mappings. Some examples are given to illustrate the advantages of the obtained results.

A note on the paper ``Best proximity point results for $$p$$-proximal contractions"

A note on the paper ``Best proximity point results for $$p$$-proximal contractions"