Best proximity point theorems for generalized rational type contraction conditions involving control functions on complex valued metric spaces

Abstract

Fixed-point theory is being adopted in pure and applied mathematics to a great extent. In scenarios where the fixed point equation lacks a solution, the best approximation theorems and the best proximity pair theorems are used as alternatives. The existence of approximate solutions is guaranteed by the best approximation theorem, but these solutions are not optimal. The best proximity point theorems provide sufficient conditions that guarantee the existence of optimal approximate solutions. In addition, the most effective proximity point theorems serve as generalizations of the fixed point theorems. So we have introduced generalized rational type contraction conditions involving control functions on complex valued metric spaces to prove common best proximity point results for commute proximally non-self mappings under certain assumptions. Many existing results in the literature are extended, generalized, and improvised in the theorems presented in this paper. We have supported our findings with some examples.