A fixed point problem via simulation functions in incomplete metric spaces with its application

Abstract

In this paper, firstly, we review the notion of the SO-complete metric spaces. This notion let us to consider some fixed point theorems for single-valued mappings in incomplete metric spaces. Secondly, as motivated by the recent work of A.H. Ansari et al. [J. Fixed Point Theory Appl. (2017), 1145–1163], we obtain that an existence and uniqueness result for the following problem: finding x ? X such that x = T x, Ax R? Bx and Cx R? Dx, where (X, d) is an incomplete metric space equipped with the two binary relations R? and R?, A, B, C, D : X ? X are discontinuous mappings and T : X ? X satisfies in a new contractive condition. This result is a real generalization of main theorem of A.H. Ansari’s. Finally, we provide some examples for our results and as an application, we find that the solutions of a differential equation.