Solving variational inequalities involving nonexpansive type mappings

The purpose of this paper is to investigate the asymptotic behavior of algorithms for finding solutions for a certain class of variational inequalities V ID(C,I−f) involving nonexpansive type mappings in smooth Banach spaces. We study the existence of solutions of variational inequalities V ID(C,I−f) when DD is the set of solutions of zeros of accretive operators or the set of fixed points of nonexpansive mappings or the set of fixed points of pseudocontractive mappings. Our convergence analysis covers proximal point algorithm for finding zeros of accretive operators as well as functional Halpern algorithm for finding fixed points of nonexpansive mappings in Banach spaces. Our results improve a number of results concerned with viscosity approximation methods in the context of weakly contraction mappings.