an optimal method for finding common solutions to variational inequality problems via simulation function

In this paper, we study the existence and uniqueness of common best proximity points for new types of generalized z-contraction pairs, generalized proximal contraction pairs, and generalized z- proximal contraction pairs of non-self mappings defined on a complete metric spaces. our results improve and generalize some recent defending in the literature. we provide several examples to illustrate the generality of our main results. as an application, we establish sufficient conditions for the existence of unique common solutions to variational inequality problems in hilbert spaces.