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Very recently, P. Saha et al. [A global optimality result in probabilistic spaces using control function. Optimization. 2021;70:2387–2400] studied the existence of best proximity points for Banach type ϕ-proximal contractions in the framework of Menger probabilistic metric spaces. In this paper, we present a simple proof of their main existence result which is based on a fixed point theorem for ϕ-contractive self-mappings in complete Menger PM-spaces due to B.S. Choudhury and K. Das [A new contraction principle in Menger spaces. Acta Math Sin. 2008;24:13791386]. [ABSTRACT FROM AUTHOR]

Recently, Altun et al. [1] introduced the notion of p-proximal contractions and p-proximal contractive mappings and discussed about best proximity point results for this two classes of mappings. After that, Gabeleh and Markin [3] showed that the best proximity point theorem for p-proximal contractions proved in [1] follows from the same conclusion fixed point theory. In this short note, we show that the best proximity point theorem for p-proximal contractive mappings follows from the corresponding fixed point result for metric spaces. [ABSTRACT FROM AUTHOR]

In a recent paper [I. Altun and A. Tasdemir, On best proximity points of interpolative proximal contractions, Quaestiones Math. (2020), DOI: ] 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]. [ABSTRACT FROM AUTHOR]

Very recently Jain et al. (J. Inequal. Appl. 2022:73, 2022) introduced the concept of multivalued F-contraction with altering distance function and investigated the existence of best proximity points for this class of mappings. In this article we prove that the existence of best proximity points for multivalued F-contraction non-self mappings can be obtained from the corresponding fixed point result for multivalued F-contraction self mappings, and so the main conclusion due to Jain et al. is not a real generalization of fixed point theory. [ABSTRACT FROM AUTHOR]

Recently, Altun et al. [1] introduced the notion of p-proximal contractions and discussed about best proximity point results for this class of mappings. Then Gabeleh and Markin [4] showed that the best proximity point theorem proved by Altun et al. in [1] follows from the fixed point theory. In this short note, we show that if the p-proximal contraction constant k < 1 3 then the existence of best proximity point for p-proximal contractions follows from the celebrated Banach contraction principle. [ABSTRACT FROM AUTHOR]

Very recently, I. Altun, M. Aslantas and H. Sahin [1] introduced the notion of p -proximal contractions and surveyed the existence of best proximity points for such class of non-self mappings in metric spaces. In this note we show that this existence result is a straightforward consequence of the same conclusion in fixed point theory. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of proximal (α , F) -contractions in F -metric spaces. We establish novel results concerning the existence and uniqueness of best proximity points for such mappings. The validity of our findings is corroborated through a non-trivial example. Furthermore, we demonstrate the applicability of these results by proving the existence of solutions for Volterra integral equations related to population growth models. This approach not only extends best proximity theory, but also paves the way for further research in applied mathematics and beyond. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the novel concept of generalized distance denoted as J θ and call it an extended b-generalized pseudo-distance. With the help of this generalized distance, we define a generalized point to set distance J θ (u , H ★) , a generalized Hausdorff type distance and a P J θ -property of a pair (H ★ , K ★) of nonempty subsets of extended b-metric space (U ★ , ρ θ). Additionally, we establish several best proximity point theorems for multi-valued contraction mappings of Nadler type defined on b-metric spaces and extended b-metric spaces. Our findings generalize numerous existing results found in the literature. To substantiate the introduced notion and validate our main results, we provide some concrete examples. [ABSTRACT FROM AUTHOR]

This paper explores the concept of normal structure by introducing a generalized form known as P-proximal normal structure. Focusing on the framework of Hausdorff locally convex spaces, the study establishes optimal proximity outcomes for both cyclic and noncyclic relatively P-nonexpansive mappings. Furthermore, the paper presents in the realm of probabilistic normed spaces, considered as instances of Hausdorff locally convex spaces, some theorems that address the existence of best proximity points within this context. [ABSTRACT FROM AUTHOR]

Cyclic mappings describe fixed paths for which each point is sequentially transmitted from one set to another. Cyclic mappings satisfying certain cyclic contraction conditions have been used to obtain the best proximity points, which constitute a suitable framework for the mirror reflection model. Alternative contraction mappings introduced by Chen (Symmetry 11:750, 2019) built a new model containing several mirrors in which the light reflected from a mirror does not go to the next mirror sequentially, and its path may diverge to any other mirror. The aim of this paper is to present a new variant of alternative contraction called alternative p-contraction and study its properties. The best proximity point result for such contractions under the alternative UC property is proved. An example to support the result proved herein is provided. [ABSTRACT FROM AUTHOR]

In this paper, we initiate a concept of graph-proximal functions. Furthermore, we give a notion of being generalized Geraghty dominating for a pair of mappings. This permits us to establish the existence of and unique results for a common best proximity point of complete metric space. Additionally, we give a concrete example and corollaries related to the main theorem. In particular, we apply our main results to the case of metric spaces equipped with a reflexive binary relation. Finally, we demonstrate the existence of a solution to boundary value problems of particular second-order differential equations. [ABSTRACT FROM AUTHOR]

Let X be an arbitrary topological space and g : X x X → R be a real valued continuous function defined on X x X. In this article, we introduce two notions like topologically Berinde weak proximal contraction and topologically proximal weakly contractive mapping with respect to g. We explore sufficient conditions for the existence and uniqueness of best proximity points for these classes of mappings. Moreover, in the last part of the paper, we show that the best proximity point theorem for topologically proximal weakly contractive mapping can be deduced from some fixed point theorems in topological spaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the new notion of generalized α-φ-Geraghty proximal contraction mappings and investigate the existence of the best proximity point for such mappings in complete metric spaces. The obtained results extend, generalize, and complement some known fixed and best proximity point results from the literature. [ABSTRACT FROM AUTHOR]

The underlying aim of this paper is first to state the cyclic version of J -integral Banach type contractive mappings introduced by Fallahi, Ghahramani and Soleimani Rad [Integral type contractions in partially ordered metric spaces and best proximity point, Iran. J. Sci. Technol. Trans. Sci. 44 (2020), 177-183] and second to show the existence of best proximity points for such contractive mappings in a metric space with a graph, which can entail a large number of former best proximity point results. One fundamental issue that can be distinguished between this work and previous researches is that it can also involve all of results stated by taking comparable and ϑ-close elements. [ABSTRACT FROM AUTHOR]

This paper introduces new kind of algorithms for multivalued non-self mapping to obtain the best proximity point without assuming the continuity of involved mapping. Some non-trivial examples are presented to illustrate the facts. Consequently, an application to finding an optimal approximate solution for the homotopy theory is also discussed. [ABSTRACT FROM AUTHOR]

The paper presents existence results of (φ, φ) best proximity points for operators that fulfill implicit type inequalities. Classes of mappings endowed with continuity, monotone or monotone-type properties, and which additionally satisfy some adequate inequalities are studied from this point of view. Applications of our results are given with regard to fixed point theory. [ABSTRACT FROM AUTHOR]

In this paper, we first review some important definitions and notions related to best proximity point theory by considering the lack of symmetry property of quasi metric. Then, we introduce a new notion of BG-multivalued contraction mappings. Then, a best proximity point result is obtained via Q-function on quasi metric. Also, we present a noteworthy illustrative example to show the effect of our results. Finally, we give an existence and uniqueness result for the solution of nonlinear integral equations via our result. [ABSTRACT FROM AUTHOR]

Let (A1, A2, A3) be a triple of nonempty convex subsets of a metric space Ω. In this paper, we determine optimal problems of the best proximity pair by proximal normal structure between two sets A1 and A2 with the help of a third set A3 and we find some necessary and sufficient conditions for existence these optimal problems. Also, we provide an example to illustrate the convergence behavior of our proposed results. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the existence of the best proximity points in binormed linear spaces. As a consequence, we obtain some fixed point results. We also provide some illustrations to support our claims. As applications, we obtain the existence of a solution to split feasible and variational inequality problems. [ABSTRACT FROM AUTHOR]

In this paper we list some geometric properties of Banach spaces and some fixed point theory results which we use in the subsequent topic . Certain geometric properties of a Banach space X are helpful to study the existence of fixed points for non expansive mappings and asymptotically non expansive mappings on closed bounded convex subsets of X. In this paper first we give a brief introduction to Geometry of Banach spaces, a brief introduction to fixed point theory, a brief introduction to best proximity point theorems, and finally we give contents of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we aim to present a new and unified way, including the previously mentioned solution methods, to overcome the problem in [7] for closed and bounded valued F -contraction mappings. We also want to obtain a real generalization of fixed point results existing in the literature by using best proximity point theory. Further, considering the strong relationship between homotopy theory and various branches of mathematics such as category theory, topological spaces, and Hamiltonian manifolds in quantum mechanics, our objective is to present an application to homotopy theory of our best proximity point results obtained in the paper. In this sense, we first introduce a new family, which is larger than F* that has often been used to give a positive answer to the problem. Then, we prove some best proximity point results for the new kind of F -contractions on quasi metric spaces via the new family. Additionally, we show that the note given by Almeida et al. [4] is not valid for our results. Therefore, our results are real generalizations of fixed point results in the literature. Moreover, we give comparative examples to demonstrate that our results unify and generalize some well-known results in the literature. As an application, we show that each homotopic mapping to f satisfying all the hypotheses of our best proximity point result has also a best proximity point. [ABSTRACT FROM AUTHOR]

The objective of this research paper is to establish the existence and uniqueness of the best proximity and coincidence with best proximity point results, specifically focusing on Geraghty–Pata–Suzuki-type proximal mappings. To achieve this, we introduce three types of mappings, all within the context of a complete metric space: an α - θ - Geraghty–Pata–Suzuki-type proximal contraction; an α - θ - generalized Geraghty–Pata–Suzuki-type proximal contraction; and an α - θ - modified Geraghty–Pata–Suzuki-type proximal contraction. These new results generalize, extend, and unify various results from the existing literature. Symmetry plays a crucial role in solving nonlinear problems in operator theory, and the variables involved in the metric space are symmetric. Several illustrative examples are provided to showcase the superiority of our results over existing approaches. [ABSTRACT FROM AUTHOR]

In this paper a new definition for the fuzzy inner product spaces of the type Kramosil-Michálek and the type George-Veeramani is presented. In the new setting a fuzzy inner product space can naturally become a fuzzy normed space, and a classical inner product space can be considered as a special case of fuzzy inner product spaces. Several examples are given to illustrate that, the new definition is a nontrivial generalization for classical inner product spaces, and so it has rich contents in fuzziness. By virtue of this definition, some elementary properties are described in terms of the families of semi-inner products and a fuzzy version of Pythagorean theorem is given. As applications, a fixed point theorem for nonlinear contractions is established and the existence of solution of global optimization problem is obtained. [ABSTRACT FROM AUTHOR]

There are situations when we have to resort to the approximate optimal solution of equations of the type g(t) = t when g is not a self-map, because exact solution of that equation does not exist. The existence of such optimal solutions are ensured by best proximity point theorems. In this paper, we define multivalued Geraghty contraction (MVGC) in a complete metric space and establish the corresponding best proximity point (BPP) result. Our result extends the famous result due to Geraghty on fixed points. [ABSTRACT FROM AUTHOR]

In this paper, we extend the result of Romaguera [21] with the aid of best proximity point theory on partial metric spaces by considering the approach of Haghi et al. [9], and so celebrated Boyd-Wong fixed point theorem [7]. We first introduce two concepts called generalized proximal BW-contraction and generalized best BW-contraction. Then, we obtain some best proximity point theorems for such mappings. To illustrate the effectiveness of our results, we provide some nontrivial and interesting examples. Finally, unlike homotopy applications existing in the literature, we present for the first time an application of the best proximity result to the homotopy theory. [ABSTRACT FROM AUTHOR]

In this paper, we point out an error in proving famous Achari type nonunique fixed point results. Also, we prove some best proximity point results in b -metric spaces by introducing new concepts. Hence, we develop some results existing in the literature. Finally, we give a result for the existence of the solution of nonlinear fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, using Bianchini-Grandolfi gauge functions, we define a new notion of BG-contraction mapping and proximal BG-contraction mapping. After that, we prove some best proximity point results. Hence, we generalize and improve results existing in the literature such as [12]. Also, we provide nontrivial example to demonstrate the effectiveness of our result. Finally, we give an existence and uniqueness result for the solution of nonlinear Freldhom and Volterra integral equations via our result. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we extend the fixed point (respectively, best proximity point) theorems for Reich type and Hardy-Rogers type contraction mappings in partial metric space and validate our results with non-trivial examples. In addition, we apply our results to Fredholm integral equation. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of a common fixed point for generalized weakly Zamfirescu-type mappings. Finally, we give an application to our main result in the fractal space which proves the existence of a common attractor for a new kind of deterministic fractal. [ABSTRACT FROM AUTHOR]

In this paper, in the setting of metric spaces we introduce the notions of generalized cyclic quasi-contractions and the ultrametric property as an applied geometric concept. Then we study the existence and uniqueness of best proximity points for such mappings by using this property and the UC property. Also, iterative algorithms are furnished to determine such best proximity points. As a result, we establish a fixed point result and a common fixed point theorem. The presented results extend and improve some recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some best proximity point results on 0-complete partial metric spaces by introducing a new concept of mixed multivalued contraction mapping. Thus, we generalize and extend some important and famous results existing in the literature. To support our results, we present a noteworthy illustrative and comparative example. Finally, we give some applications of our new best proximity point theorems to homotopy theory as directly unlike homotopy applications existing in the literature. Hence, we prove some best proximity point results for homotopic mappings. [ABSTRACT FROM AUTHOR]

In the present paper, we aim to extend and unify the results obtained for the multivalued F -contractions, which have been frequently studied recently, in a different way from the results in the literature without using the Pompeiu-Hausdorff metric. Hence, we first introduce a new class of multivalued mappings that includes multivalued F -contractions. Then, we obtain some best proximity point results for new kind of F -contraction mappings. Thus, we unify and improve many results in the literature. To see this fact, we give some nontrivial and interesting examples. Also, considering the strong relationship between homotopy theory and various branches of mathematics, we obtain an application to homotopy theory of our main result. [ABSTRACT FROM AUTHOR]

In the present paper, by introducing a new notion named as nonunique cyclic contractions, we give some best proximity point results for such mappings. Then, we indicate the shortcoming of the concept of best periodic proximity point which is defined for cyclic mapping by giving a simple example. To overcome this deficiency, we give a more suitable definition named as best cyclic periodic point. Finally, we obtain some best cyclic periodic point theorems, including the famous periodic point result of Ćirić [8], for nonunique cyclic contractions. We also provide some illustrative and comparative examples to support our results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a weak MT-cyclic Reich type contractions and obtain the existence theorems for best proximity point for self-mappings defined on the complete metric spaces. Our results improve and generalize some results in literature. Also, we give some applications of our results to solving some classes of non-linear integral and differential equations. [ABSTRACT FROM AUTHOR]

Let Ŕ, Ś be subsets of a partial metric space and: Ŕ Ś be a mapping. If Ŕ n Ś = 1, it cannot have a solution of equation for some Ŕ. Hence, it is sensible to investigate if there is a point satisfying which is called a best proximity point. In this paper, we first introduce a concept of Hausdorff cyclic mapping pair. Then, we revise the definition of 0-boundedly compact on partial metric spaces. After that, we give some best proximity point results for these mappings. Hene, our results combine, generalize and extend many fixed point and best proximity point theorems in the literature as properly. Moreover, a comparative and illustrative example to demonstrate the effectiveness of our results has been presented. [ABSTRACT FROM AUTHOR]

In the current paper, we discuss sufficient and necessary conditions for the existence of best proximity points for cyclic f- contractive mappings in the setting of strictly convex Banach spaces. Extensions of Edelstein's theorem are considered as well as an extension of a main result in Park [Park, S. (1978). Fixed points of f-contractive maps. Rocky Mountain J. Math. 8:743–750]. Another existence result of best proximity points will be obtained for asymptotically relatively nonexpansive mappings under different conditions with respect to the recent paper of Rajesh and Veeramani [Rajesh, S., Veeramani, P. (2016). Best Proximity point theorems for asymptotically relatively nonexpansive mappings. Numer. Funct. Anal. Optim. 37:80–91]. [ABSTRACT FROM AUTHOR]

Best proximity point theorems for self-mappings were investigated with different conditions on spaces for contraction mappings. In this paper, we prove best proximity point theorems for proximal F∗-weak contraction mappings. [ABSTRACT FROM AUTHOR]

In this paper we solve the global optimality problem of determining probabilistic distance between two sets. We treat this problem as that of finding an optimal approximate solution of a fixed point equation. It is well studied in the context of metric spaces under the name of best proximity point problem. Our work is in the domain of the extension of this line of research to probabilistic metric spaces. For our purpose we define a probabilistic contraction mapping with the help of a control function. This control function plays a central role in the proof of our main theorem. We also use probabilistic P-property for pairs of subsets which is essentially a geometric property. A well known probabilistic extension of the Banach's contraction mapping principle follows as corollary of our main results. We illustrate our results with an example. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some kinds of interpolative proximal contractions named as Reich-Rus-Ćirić and Kannan types. Then taking into account the aforementioned mappings, we give a few best proximity point results. As special cases we obtain some fixed point results for interpolative contractions. To support the theory, some examples are provided. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of graph contraction using a special class of functions and give a best proximity point theorem for this contraction in complete metric spaces endowed with a graph. Then we support our main theorem by a non-trivial example and give some consequences of it for usual graphs. [ABSTRACT FROM AUTHOR]

In this paper, we employ two types of implicit relations to define some new kind of proximal contractions and study about their best proximity points. More precisely, we use two class of functions A and A' to explore proximal A, A'-contractions of first and second kind and strong proximal A, A'-contractions. We investigate the existence of best proximity points results of the same. It is worth mentioning that the well-known results of Sadiq Basha [J. Approx. Theory, 2011] on proximal contractions are the special cases of our obtained results. We authenticate our results by suitable examples. Finally, we point out some areas where our obtained results can be applied. [ABSTRACT FROM AUTHOR]

In this paper, we present two main results about best proximity points for multivalued mappings. First, taking into account Klim and Wardowski's approach in fixed-point theory, we obtain a new result for multivalued mappings which includes the main theorem of Kamran [Kamranm, T. (2009). Mizoguchi–Takahashi's type fixed point theorem. Comput. Math. Appl. 57(3):507–511]. Second, combining this approach with cyclic contraction, we give a general best proximity point result, and so we obtain many well-known fixed-point theorems such as Mizoguchi–Takahashi, Feng–Liu and Klim–Wardowski [Mizoguchi, N., Takahashi, W. (1989). Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 141(1):177–188; Feng, Y., Liu, S. (2006). Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. J. Math. Anal. Appl. 317(1):103–112; Klim, D., Wardowski, D. (2007). Fixed point theorems for set-valued contractions in complete metric spaces. J. Math. Anal. Appl. 334(1):132–139]. Also, we support our results by providing some nontrivial, illustrative and comparative examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of proximal Berinde g-cyclic contractions of two non-self-mappings and proximal Berinde g-contractions, called proximal Berinde g-cyclic contraction of the first and second kind. Coincidence best proximity point theorems for these types of mappings in a metric space are presented. Some examples illustrating our main results are also given. Our main results extend and generalize many existing results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a new notion of the property (MC) to improve and generalize the property (GC). After that, we present two new concepts, proximal b-cyclic contraction of the first type and second type, on b-metric spaces. Then, we obtain two best proximity point theorems for such mappings in the framework of best proximally complete b-metric spaces by using the property (MC). Hence, we generalize some results existing in the literature. Finally, we present some illustrative and interesting examples. [ABSTRACT FROM AUTHOR]

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 on 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 on both vector metric spaces and partially ordered vector metric spaces. Further, we provide nontrivial and comparative examples to show the effectiveness of our main results. [ABSTRACT FROM AUTHOR]

Let A and B be nonempty subsets of a Banach space X and T : A → B be a non-self mapping. An approximate sequence of best proximity points for the mapping T is a sequence {xn} in A such that limn→∞ || xn − T xn || → dist(A, B). In the current paper, we survey the existence of approximate best proximity point sequences for single and multivalued non-self mappings in strictly convex Banach spaces. We also introduce a geometric notion on a nonempty and convex pair of subsets of a Banach space, called semi-Opial condition, and establish some new best proximity point theorems. [ABSTRACT FROM AUTHOR]

Recall from [2], that a mapping T : X 7→ X is called a Reich mapping if it satisfies for all x; yϵ X, d(Tx; Ty) ≤ ad(x; Tx) + bd(y; Ty) + cd(x; y), where a,b,c are nonnegative and satisfy a + b + c < 1. Alternatively, one could define a Reich mapping as follows: T : X 7→ X is called a Reich mapping if there exists a nonnegative constant k with k < 1 3 such that d(Tx; Ty) ≤ k[d(x; Tx)+d(y; Ty)+d(x; y)]. In the present paper, we address the following: How do we characterize Theorem 3 [2], when T is a non-self map? We show such a characterization is given by Theorem 3.1 or Corollary 3.2 in this paper. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce new class of proximal contractions in non-Archimedean modular metric spaces and to prove some best proximity point theorems for such kind of mappings. As application we deduce best proximity results in fuzzy metric spaces. Consequently, some basic fixed point results in both modular and fuzzy metric spaces are obtained as corollaries of our work. Finally, an example is provided to illustrate the usability of our obtained results. [ABSTRACT FROM AUTHOR]

In this paper the existence and uniqueness of best proximity point for Reich-type contraction on b-metric spaces endowed with a graph is established. These results are significant, since we replace the condition of continuity of mapping with the condition of G-continuity of mapping and we consider b-metric spaces endowed with a graph instead of metric spaces, under which can be unified some theorems of the existing literature. [ABSTRACT FROM AUTHOR]