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In this paper, we first review some important definitions and notions related to best proximity point theory by considering the lack of symmetry property of quasi metric. Then, we introduce a new notion of  BG -multivalued contraction mappings. Then, a best proximity point result is obtained via  Q -function on quasi metric. Also, we present a noteworthy illustrative example to show the effect of our results. Finally, we give an existence and uniqueness result for the solution of nonlinear integral equations via our result.

A new contribution to best proximity point theory on quasi metric spaces and an application to nonlinear integral equations