Existence and Convergence Theorem for Fixed Point Problem of Various Nonlinear Mappings and Variational Inequality Problems without Some Assumptions

Abstract

The purpose of this article, we give a necessary and sufficient condition for the modified Mann iterative process in order to obtain strong convergence theorem for finding a common element of the set of fixed point of a finite family of nonexpansive mappings and variational inequality problem in Hilbert space without the conditions $\bigcap_{i=1}^{N}Fix(T_i)\cap VI(C,A)\neq\emptyset$. Moreover, we utilize our main result to fixed point problems of strictly pseudocontractive mappings and the set of solutions of variational inequality problem.