Fixed point result and applications on a b-metric space endowed with an arbitrary binary relation

Wutiphol Sintunavarat,Somyot Plubtieng,Phayap Katchang

doi:10.1186/1687-1812-2013-296

Abstract

In this paper, we introduce the concept of q-set-valued α-quasi-contraction mapping and establish the existence of a fixed point theorem for this mapping in b-metric spaces. Our results are generalizations and extensions of the result of Aydi et al. (Fixed Point Theory Appl. 2012:88, 2012) and some recent results. We also state some illustrative examples to claim that our results properly generalize some results in the literature. Further, by applying the main results, we investigate a fixed point theorem in a b-metric space endowed with an arbitrary binary relation. At the end of this paper, we give open problems for further investigation.

Highlights

1 Introduction Fixed point theory is one of the cornerstones in the development of mathematics since it plays a basic role in applications of many branches of mathematics
Aydi et al [ ] established the q-set-valued quasi-contraction mapping which is a generalization of the q-set-valued quasi-contraction mapping due to Amini-Harandi [ ] in. They established a fixed point theorem for such a mapping in b-metric spaces
Inspired and motivated by Aydi et al [ ] and Mohammadi et al [ ], we introduce the class of q-set-valued α-quasi-contraction mappings and give a fixed point theorem for such mappings via the idea of α-admissible mapping

Summary

Introduction

Fixed point theory is one of the cornerstones in the development of mathematics since it plays a basic role in applications of many branches of mathematics. Aydi et al [ ] established the q-set-valued quasi-contraction mapping which is a generalization of the q-set-valued quasi-contraction mapping due to Amini-Harandi [ ] in They established a fixed point theorem for such a mapping in b-metric spaces. We introduce the q-set-valued α-quasi-contraction mapping and obtain the existence of a fixed point theorem for such a mapping in b-metric spaces. Let (X, d) be a complete b-metric space (with constant s ≥ ) such that the b-metric is a continuous functional on X × X, let α : X × X → [ , ∞) be a mapping, and let T : X → Pb,cl(X) be a q-set-valued α-quasi-contraction.

Since q

Since s