Existence Of Best Proximity Points Research Articles

A Note on Some Best Proximity Point Theorems Proved underP-Property

We show that some recent results concerning the existence of best proximity points can be obtained from the same results in fixed point theory.

Fixed points and best proximity points in contractive cyclic self-maps satisfying constraints in closed integral form with some applications

This paper investigates the existence of best proximity points and the potential fixed points of p-cyclic self-maps which are defined on the union of a finite set of subsets of a certain uniform metric space under several integral-type contractive conditions. The corresponding properties of the associated restricted composed maps from any of the subsets to itself are also dealt with in a formal context. The above subsets of the uniform metric space are not assumed to necessarily intersect but the case when they intersect leads to particular results of the general case in the sense that best proximity points become fixed points. Common fixed points and best proximity points of pairs of self-maps on the above subsets of the uniform metric space are also investigated.

Best proximity point results in geodesic metric spaces

Abstract In this paper, the existence of a best proximity point for relatively u-continuous mappings is proved in geodesic metric spaces. As an application, we discuss the existence of common best proximity points for a family of not necessarily commuting relatively u-continuous mappings.

Nonlinear conditions for the existence of best proximity points

In this paper, we first introduce the new notion of MT-cyclic contraction and establish some new existence and convergence theorems of iterates of best proximity points for MT-cyclic contractions. Some nontrivial examples illustrating our results are also given.MSC:41A17, 47H09.

The existence of best proximity points for multivalued non-self-mappings

Let A, B be nonempty subsets of a metric space (X, d) and T : A → 2B be a multivalued non-self-mapping. The purpose of this paper is to establish some theorems on the existence of a point \({x^*\in A}\) , called best proximity point, which satisfies \({{\rm inf}\{d(x^*,y):y\in Tx^*\}=dist(A,B).}\) This will be done for contraction multivalued non-self-mappings in metric spaces, as well as for nonexpansive multivalued non-self-mappings in Banach spaces having appropriate geometric property.

Best Proximity Point Theorems for Generalized Cyclic Contractions in Ordered Metric Spaces

In this paper, we generalized a cyclic contraction on a partially ordered complete metric space. We prove some fixed point theorems as well as some theorems on the existence of best proximity points. Our results improve and extend some recent results in the previous work.

Best proximity points for asymptotic cyclic contraction mappings

In this paper, we prove the existence and convergence of best proximity points for asymptotic cyclic contractions in metric spaces with the property UC, as well as for asymptotic proximal pointwise contractions in uniformly convex Banach spaces. Moreover, we consider a generalized cyclic contraction mapping and prove the existence of best proximity points for such a mapping in Banach spaces which do not necessarily satisfy the geometric property UC.

Best proximity results in regular cone metric spaces

First, we define the notion of distance between two subsets in regular cone metric spaces. Then, we establish some conditions which guarantee the existence of best proximity points for cyclic contraction mappings on regular cone metric spaces.

On best proximity pair theorems for relatively [formula omitted]-continuous mappings

A new class of mappings, called relatively u -continuous, is introduced and used to investigate the existence of best proximity points. As an application of the existence theorem, we obtain a generalized version of the Markov–Kakutani theorem for best proximity points in the setting of a strictly convex Banach space.

On the Extensions of Krasnoselskii‐Type Theorems to p‐Cyclic Self‐Mappings in Banach Spaces

A set of np(≥2)‐cyclic and either continuous or contractive self‐mappings, with at least one of them being contractive, which are defined on a set of subsets of a Banach space, are considered to build a composed self‐mapping of interest. The existence and uniqueness of fixed points and the existence of best proximity points, in the case that the subsets do not intersect, of such composed mappings are investigated by stating and proving ad hoc extensions of several Krasnoselskii‐type theorems.

Best proximity points: Convergence and existence theorems for [formula omitted]-cyclic mappings

We introduce a new class of mappings, called p -cyclic φ -contractions, which contains the p -cyclic contraction mappings as a subclass. Then, convergence and existence results of best proximity points for p -cyclic φ -contraction mappings are obtained. Moreover, we prove results of the existence of best proximity points in a reflexive Banach space. These results are generalizations of the results of Al-Thagafi and Shahzad (2009) [8].

Best proximity points: global optimal approximate solutions

Let A and B be non-empty subsets of a metric space. As a non-self mapping $${T:A\longrightarrow B}$$ does not necessarily have a fixed point, it is of considerable interest to find an element x in A that is as close to Tx in B as possible. In other words, if the fixed point equation Tx = x has no exact solution, then it is contemplated to find an approximate solution x in A such that the error d(x, Tx) is minimum, where d is the distance function. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, to the fixed point equation Tx = x when there is no exact solution. As the distance between any element x in A and its image Tx in B is at least the distance between the sets A and B, a best proximity pair theorem achieves global minimum of d(x, Tx) by stipulating an approximate solution x of the fixed point equation Tx = x to satisfy the condition that d(x, Tx) = d(A, B). The purpose of this article is to establish best proximity point theorems for contractive non-self mappings, yielding global optimal approximate solutions of certain fixed point equations. Besides establishing the existence of best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.

Results on the Existence and Convergence of Best Proximity Points

AbstractWe first consider a cyclic "Equation missing"-contraction map on a reflexive Banach space "Equation missing" and provide a positive answer to a question raised by Al-Thagafi and Shahzad on the existence of best proximity points for cyclic "Equation missing"-contraction maps in reflexive Banach spaces in one of their works (2009). In the second part of the paper, we will discuss the existence of best proximity points in the framework of more general metric spaces. We obtain some new results on the existence of best proximity points in hyperconvex metric spaces as well as in ultrametric spaces.

The existence of best proximity points in metric spaces with the property UC

Eldred and Veeramani in [A.A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), 1001–1006. MR2260159] proved a theorem which ensures the existence of a best proximity point of cyclic contractions in the framework of uniformly convex Banach spaces. In this paper we introduce a notion of the property UC and extend the Eldred and Veeramani theorem to metric spaces with the property UC.

Convergence and existence results for best proximity points

We provide a positive answer to a question raised by Eldred and Veeramani [A.A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006) 1001–1006] about the existence of a best proximity point for a cyclic contraction map in a reflexive Banach space. Moreover, we introduce a new class of maps, called cyclic φ -contractions, which contains the cyclic contraction maps as a subclass. Convergence and existence results of best proximity points for cyclic φ -contraction maps are also obtained.

Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces

Given a uniform space X and nonempty subsets A and B of X , we introduce the concepts of some families V of generalized pseudodistances on X , of set-valued dynamic systems of relatively quasi-asymptotic contractions T : A ∪ B → 2 A ∪ B with respect to V and best proximity points for T in A ∪ B , and we describe the methods which we use to establish the conditions guaranteeing the existence of best proximity points for T when T is cyclic (i.e. T : A → 2 B and T : B → 2 A ) or when T is noncyclic (i.e. T : A → 2 A and T : B → 2 B ). Moreover, we establish conditions guaranteeing that for each starting point each generalized sequence of iterations of these contractions (in particular, each dynamic process) converges and the limit is a best proximity point for T in A ∪ B . These best proximity points for T are determined by unique endpoints in A ∪ B for a map T [ 2 ] when T is cyclic and for a map T when T is noncyclic. The results and the methods are new for set-valued and single-valued dynamic systems in uniform, locally convex, metric and Banach spaces. Various examples illustrating the ideas of our definitions and results, and fundamental differences between our results and the well-known ones are given.