Coincidence and fixed points of nonlinear Hybrid contractions

Shyam Lal Singh,Yeol Je Cho,Ki Sik Ha

doi:10.1155/s0161171289000281

Abstract

In this Der, we show the existence of solutions of functional equationsfx∈sx∩txandx=fx∈sx∩Txunder certain contraction and asymptotic regularity conditions, where f, S and T are single-valued and multl-valued mappings on a metric space, respectively. We also observe that MukherJee's fixed point theorem for a single-valued mapping commuting with a multl-valued mapping admits of a counterexample and suggest some modifications. While doing so, we also answer an open question raised in [I] and [2]. Moreover, our results extend and unify a multitude of fixed point theorems for multi-valued mappings.

Highlights

The study of fixed points of multl-valued mappings using the Hausdorff metric was initiated by Markin [3] and Nadler [4]
A number of generalizations of the multl-valued contraction principle were obtained, among others, by Ciric [5], Khan [6], Kubiak [7], Reich [8], Smithson [9] and .egrzyk [I0]
In [14], Sessa introduced the concept of weak commutativity for single-valued mappings on a metric space. We extend this concept to the setting of a singlev.alued mapping and a multl-valued mapping on a metric space as follows: DEFINITION 1.5. f and S are said to be commute weakly at z X if H(fSz, Sfz) D(fz, Sz). f and S are said to commute weakly on X if they commute weakly at every point in X

Summary

Introduction

The study of fixed points of multl-valued mappings using the Hausdorff metric was initiated by Markin [3] and Nadler [4]. Let (X,d) be a complete metric space, f a continuous mapping from X The following thereom is an interesting result for the existence of coincidence points of hybrid contractions, that is, contractive conditions involving slngle-valued and multi-valued mappings.

Results

Conclusion