Strong Convergence of Iterative Algorithms for Variational Inequalities in Banach Spaces

Abstract

Let C be a nonempty closed convex subset of a Banach space E with the dual E*, let T:C→E* be a Lipschitz continuous mapping and let S:C→C be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator, we study the following variational inequality (for short, VI(T−f,C)): find x∈C such that $$\langle y-x,Tx-f\rangle\geq0,\quad\mbox{for all }y\in C,$$ where f∈E* is a given element. Utilizing the modified Ishikawa iteration and the modified Halpern iteration for relatively nonexpansive mappings, we propose two modified versions of J.L. Li’s (J. Math. Anal. Appl. 295:115–126, 2004) iterative algorithm for finding approximate solutions of VI(T−f,C). Moreover, it is proven that these iterative algorithms converge strongly to the same solution of VI(T−f,C), which is also a fixed point of S.