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

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. ©2013 Sintunavarat et al.; licensee Springer.