TWO EXTRAGRADIENT APPROXIMATION METHODS FOR VARIATIONAL INEQUALITIES AND FIXED POINT PROBLEMS OF STRICT PSEUDO-CONTRACTIONS

Abstract

Let $\{S_i\}^N_{i=1}$ be $N$ strict pseudo-contractions defined on a nonempty closed convex subset $C$ of a real Hilbert space $H$. Consider the problem of finding a common element of the set of common fixed points of these mappings $\{S_i\}^N_{i=1}$ and the set of solutions of the variational inequality for a monotone Lipschitz continuous mapping of $C$ into $H$, and consider the parallel-extragradient and cyclic-extragradient algorithms for solving this problem. We will derive the weak convergence of these algorithms. Moreover, these weak convergence results will be applied to finding a common zero point of a finite family of maximal monotone mappings. Further we prove that these algorithms can be modified to have strong convergence by virtue of additional projections. Our results represent the improvement, generalization and development of the previously known results in the literature.