The purpose of this paper is to introduce and analyze a new idea of proximally compatible mappings and we extend some results of Jungck via proximally compatible mappings. Furthermore, we obtain common best proximity point theorems for proximally compatible mappings through two different ways of construction of sequences. In addition, we provide an example to support our main result.

In the present paper, we focus our attention on the existence of the fixed point for the sum of the cyclic contraction and the noncyclic accretive operator. Also, we study the best proximity point for the sum of two non-self-mappings. Moreover, we provide the existence of the best proximity point for the cyclic contraction through the notion of the nonlinearD-set contraction. Finally, we give the existence of the best proximity point for the sum of the nonlinearD-set contraction mapping and partially completely continuous mapping in the setting of the partially ordered complete normed linear space.