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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: x(c)=ϕ(c)+∫abK(c,r,x(r))dr, where b>a≥0, x∈C[a,b] (the set of continuous real functions defined on [a,b]⊆R), ϕ:[a,b]→R, and K:[a,b]×[a,b]×R→R are given mappings.

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