Proximally Compatible Mappings and Common Best Proximity Points

The purpose of this paper is to introduce and analyze a new idea of proximally compatible mappings and we extend some results of Jungck via proximally compatible mappings. Furthermore, we obtain common best proximity point theorems for proximally compatible mappings through two different ways of construction of sequences. In addition, we provide an example to support our main result.This work has been partially funded by Basque Government through Grant IT1207-19

Metric fixed point theory on hyperconvex spaces: recent progress

In this survey we present an exposition of the development during the last decade of metric fixed point theory on hyperconvex metric spaces. Therefore we mainly cover results where the conditions on the mappings are metric. We will recall results about proximinal nonexpansive retractions and their impact into the theory of best approximation and best proximity pairs. A central role in this survey will be also played by some recent developments on R-trees. Finally, some considerations and new results on the extension of compact mappings will be shown

Best proximity points for asymptotic proximal pointwise weaker Meir–Keeler-type ψ-contraction mappings

AbstractIn this paper, we study the new class of an asymptotic proximal pointwise weaker Meir–Keeler-type ψ-contraction and prove the existence of solutions for the minimization problem in a uniformly convex Banach space. Also, we give some an example for support our main result

EXISTENCE OF BEST PROXIMITY POINTS: GLOBAL OPTIMAL APPROXIMATE SOLUTION

Abstract. Given non-empty subsets A and B of a metric space, let S : A → B and T : A → B be non-self mappings. Taking into account the fact that, given any element x in A, the distance between x and Sx, and the distance between x and T x are at least d(A, B), a common best proximity point theorem affirms global minimum of both functions x → d(x, Sx) and x → d(x, T x) by imposing a common approximate solution of the equations Sx = x and T x = x to satisfy the constraint that d(x, Sx) = d(x, T x) = d(A, B). In this work we introduce a new notion of proximally dominating type mappings and derive a common best proximity point theorem for proximally commuting non-self mappings, thereby producing common optimal approximate solutions of certain simultaneous fixed point equations when there is no common solution. We furnish suitable examples to demonstrate the validity of the hypotheses of our results

Best proximity point theorems for rational proximal contractions

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Generalized Proximal ψ

We generalized the notion of proximal contractions of the first and the second kinds and established the best proximity point theorems for these classes. Our results improve and extend recent result of Sadiq Basha (2011) and some authors

Some Results on Fixed and Best Proximity Points of Multivalued

This paper is devoted to investigate the fixed points and best proximity points of multivalued cyclic self-mappings on a set of subsets of complete metric spaces endowed with a partial order under a generalized contractive condition involving a Hausdorff distance. The existence and uniqueness of fixed points of both the cyclic self-mapping and its associate composite self-mappings on each of the subsets are investigated, if the subsets in the cyclic disposal are nonempty, bounded and of nonempty convex intersection. The obtained results are extended to the existence of unique best proximity points in uniformly convex Banach spaces.Spanish Government DPI2012-30651; Basque Government IT378-10 and SAIOTEK S-PE12UN015; UPV/EHU UFI 2011/0