Abstract:

In this paper, rstly, we review the notion of the SO-complete metric spaces. This notion let us to consider some xed 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: nding x 2 X such that x = Tx, Ax R1 Bx and Cx R2 Dx, where X; d is an incomplete metric space equipped with the two binary relations R1 and R2, A;B;C;D : X ! X are discontinuous mappings and T : X ! X satis es 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 nd that the solutions of a dierential equation.

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