Best proximity points and extension of Mizoguchi-Takahashi’s fixed point theorems

Poom Kumam,Hassen Aydi,Wutiphol Sintunavarat,Erdal Karapınar

doi:10.1186/1687-1812-2013-242

Abstract

Abstract In this paper, we introduce a multi-valued cyclic generalized contraction by extending the Mizoguchi and Takahashi’s contraction for non-self mappings. We also establish a best proximity point for such type contraction mappings in the context of metric spaces. Later, we characterize this result to investigate the existence of best proximity point theorems in uniformly convex Banach spaces. We state some illustrative examples to support our main theorems. Our results extend, improve and enrich some celebrated results in the literature, such as Nadler’s fixed point theorem, Mizoguchi and Takahashi’s fixed point theorem. MSC:41A65, 46B20, 47H09, 47H10.

Highlights

1 Introduction It is evident that the fixed point theory is one of the fundamental tools in nonlinear functional analysis
The celebrated Banach contraction mapping principle [ ] is the most known and crucial result in fixed point theory. It says that each contraction in a complete metric space has a unique fixed point
It is clear that a fixed point coincides with a best proximity point if d(A, B) =

Summary

Open Access

Best proximity points and extension of Mizoguchi-Takahashi’s fixed point theorems. Poom Kumam[1], Hassen Aydi[2], Erdal Karapınar[3] and Wutiphol Sintunavarat4* Dedicated to Prof. W Takahashi on the occasion of his 70th birthday

Introduction

Let α