Proximity Point Theorems Research Articles

$$\varphi $$ φ -Best proximity point theorems and applications to variational inequality problems

The main concern of this study is to introduce the notion of $$\varphi $$ -best proximity points and establish the existence and uniqueness of $$\varphi $$ -best proximity point for non-self mappings satisfying $$(F,\varphi )$$ -proximal and $$(F,\varphi )$$ -weak proximal contraction conditions in the context of complete metric spaces. Some examples are supplied to support the usability of our results. As applications of the obtained results, some new best proximity point results in partial metric spaces are presented. Furthermore, sufficient conditions to ensure the existence of a unique solution for a variational inequality problem are also discussed.

Equilibria of free abstract economies via best proximity point theorems

In this paper, as applications of best proximity point theorems, we obtain new existence theorems of an equilibrium pair in free abstract economies. Also, we state examples which guarantee that the similar results cannot be employed for this type of free abstract economies.

Best proximity point theorems for proximal cyclic contractions

The purpose of this article is to compute a global minimizer of the function $$x\longrightarrow d(x, Tx)$$ , where T is a proximal cyclic contraction in the framework of a best proximally complete space, thereby ensuring the existence of an optimal approximate solution, called a best proximity point, to the equation $$Tx=x$$ when T is not necessarily a self-mapping.

Global optimality results for multivalued non-self mappings in b-metric spaces

In this paper, we introduce a new class of multivalued contractions with respect to b-generalized pseudodistances and prove a best proximity point theorem for this class of non-self mappings. In this way, we improve and extend several existing results in the literature. Examples are given to support our main results. This work is a continuation of studies on the use of a new type of distances in the fixed point theory. The pioneering effort in direction of defining distance is inter alia paper of O. Kada, T. Suzuki and W. Takahashi.

Remarks on Minimal Sets for Cyclic Mappings in Uniformly Convex Banach Spaces

ABSTRACTIn this article, we prove that every nonempty and convex pair of subsets of uniformly convex in every direction Banach spaces has the proximal normal structure and then we present a best proximity point theorem for cyclic relatively nonexpansive mappings in such spaces. We also study the structure of minimal sets of cyclic relatively nonexpansive mappings and obtain the existence results of best proximity points for cyclic mappings using some new geometric notions on minimal sets. Finally, we prove a best proximity point theorem for a new class of cyclic contraction-type mappings in the setting of uniformly convex Banach spaces and so, we improve the main conclusions of Eldred and Veeramani.

New nonlinear conditions for approximate sequences and new best proximity point theorems

New nonlinear conditions for approximate sequences and new best proximity point theorems

New best proximity point theorems with applications to simultaneous generalizations of common fixed point theorems

New best proximity point theorems with applications to simultaneous generalizations of common fixed point theorems

New best proximity point theorems for a pair of nonlinear non-self mappings in metric spaces

New best proximity point theorems for a pair of nonlinear non-self mappings in metric spaces

Common best proximity points in complex valued b-metric spaces

In this paper, we obtain some results on common best proximity points for non-self mappings between two subsets of complex valued b-metric spaces. For this purpose, first we generalize some well-known results that were proved in classic metric spaces on complex valued b-metric space by some new definitions. Second we present a type of contractive condition and develop a common best proximity point theorem for non-self mappings in complex-valued b-metric spaces. Our results are supported by some examples.

Best Proximity Point Theorems in Partially Ordered b-Quasi Metric Spaces

In this paper, we introduce the notion of an ordered rational proximal contraction in partially ordered b-quasi metric spaces. We shall then prove some best proximity point theorems in partially ordered b-quasi metric spaces.

Multivariate best proximity point theorems in metric spaces

Multivariate best proximity point theorems in metric spaces

Best proximity point theorems in the frameworks of fairly and proximally complete spaces

Let us contemplate the problem of solving the linear or non-linear equations of the form \(Tx=gx\) in the framework of metric space. When T is a non-self mapping and g is a self-mapping, it may cause the non-existence of a solution to the preceding equation. At this juncture, one is of course interested in computing an approximate solution \(x^*\) in the space such that \(Tx^*\) is as close to \(gx^*\) as possible. To be precise, if T is from A to B and g is from A to A, where A and B are subsets of a metric space, one is concerned with the computation of a global minimizer of the mapping \(x\longrightarrow d(gx, Tx)\) which serves as a measure of closeness between Tx and gx. This paper is concerned with the resolution of the aforesaid global minimization problem if T is a proximal contraction and g is an isometry in the frameworks of fairly and proximally complete spaces.

BEST PROXIMITY POINT THEOREMS FOR CYCLIC QUASI-CONTRACTION MAPS IN UNIFORMLY CONVEX BANACH SPACES

In this paper we first give a negative answer to a question of Amini-Harandi [‘Best proximity point theorems for cyclic strongly quasi-contraction mappings’, J. Global Optim.56 (2013), 1667–1674] on a best proximity point theorem for cyclic quasi-contraction maps. Then we prove some new results on best proximity point theorems that show that results of Amini-Harandi for cyclic strongly quasi-contractions are true under weaker assumptions.

Best Proximity point for Z-contraction and Suzuki type Z-contraction mappings with an application to fractional calculus

In this article, we introduced the best proximity point theorems for $\mathcal{Z}$-contraction and Suzuki type $\mathcal{Z}$-contraction in the setting of complete metric spaces. Also by the help of weak $P$-property and $P$-property, we proved existence and uniqueness of best proximity point. There is a simple example to show the validity of our results. Our results extended and unify many existing results in the literature. Moreover, an application to fractional order functional differential equation is discussed.<br /><br />

A new class of $\mathcal{S}$-contractions in complete metric spaces and $\mathcal{G_{P}}$-contractions in ordered metric spaces

AbstractThe basic purpose of this article is to define new so-called $\mathcal{S}$ S -contractions and discuss the presence of common best proximity point theorems for such contractions in the setting of Cauchy metric spaces. We also calculate some common optimal approximate solutions of some fixed point equations when there does not exist any common fixed point. We also introduce the notions of $\mathcal{G_{P}}$ GP -functions and $\mathcal {G_{P}}$ GP -contractions with the help of $\mathcal{P}$ P -functions and prove the existence of a unique best proximity point in partially order metric spaces. We give some examples that justify the validity of our results. These results extend and unify many existing results in the literature.

Common best proximity point theorems: Global minimization of some real-valued multi-objective functions

Let us deliberate the question of computing a solution to the problems that can be articulated as the simultaneous equations \({Sx = x}\) and \({Tx = x}\) in the framework of metric spaces. However, when the mappings in context are not necessarily self-mappings, then it may be consequential that the equations do not have a common solution. At this juncture, one contemplates to compute a common approximate solution of such a system with the least possible error. Indeed, for a common approximate solution \({x^*}\) of the equations, the real numbers \({d(x^*, Sx^*)}\) and \({d(x^*,Tx^*)}\) measure the errors due to approximation. Eventually, it is imperative that one pulls off the global minimization of the multiobjective functions \({x \rightarrow d(x, Sx)}\) and \({x \rightarrow d(x, Tx)}\). When S and T are mappings from A to B, it follows that \({d(x, Sx) \geq d(A, B)}\) and \({d(x, Tx) \geq d(A, B)}\) for every \({x \in A}\). As a result, the global minimum of the aforesaid problem shall be actualized if it is ascertained that the functions \({x \rightarrow d(x, Sx)}\) and \({x \rightarrow d(x, Tx)}\) attain the lowest possible value d(A, B). The target of this paper is to resolve the preceding multiobjective global minimization problem when S is a T-cyclic contraction or a generalized cyclic contraction, thereby enabling one to determine a common optimal approximate solution to the aforesaid simultaneous equations.

Best proximity point theorems for cyclic generalized proximal contractions

AbstractIn this paper we introduce the notion of a cyclic generalized proximal contraction, and then prove some best proximity point theorems in metric spaces. We will provide examples to illustrate our results.

Chebyshev center, best proximity point theorems and fixed point theorems

Chebyshev center, best proximity point theorems and fixed point theorems

Some common best proximity point theorems in a complete metric space

In this article, we prove the existence and uniqueness of common best proximity point for a pair of non-self mappings satisfying weak P-property in a complete metric space. Moreover, we prove the same for Kannan type mappings and C-contractive mappings using weak P-property by changing non-self mappings to self mappings in a complete metric space. We take much weaker conditions on the mappings to prove our results. An example is presented here to support our main result.

Best proximity point theorems for multivalued mappings on partially ordered metric spaces

Best proximity point theorems for multivalued mappings on partially ordered metric spaces