The existence of fixed point theorems for partial q-set-valued quasi-contractions in b-metric spaces and related results

Abstract

In this paper, we present a new type of set-valued mappings called partial q-set-valued quasi-contraction mappings and give results as regards fixed points for such mappings in b-metric spaces. By providing some examples, we show that our results are real generalizations of the main results of Aydi et al. (Fixed Point Theory Appl. 2012:88, 2012) and many results in the literature. We also consider fixed point results for single-valued mapping, fixed point results for set-valued mapping in b-metric space endowed with an arbitrary binary relation, and fixed point results in a b-metric space endowed with a graph. By using our result, we establish the existence of solution for the following an integral equations: [InlineEquation not available: see fulltext.], where [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.] (the set of continuous real functions defined on [InlineEquation not available: see fulltext.]), [InlineEquation not available: see fulltext.], and [InlineEquation not available: see fulltext.] are given mappings. MSC:47H10, 54H25.