Weak convergence of a projection algorithm for variational inequalities in a Banach space

Let C be a nonempty, closed convex subset of a Banach space E. In this paper, motivated by Alber [Ya.I. Alber, Metric and generalized projection operators in Banach spaces: Properties and applications, in: A.G. Kartsatos (Ed.), Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, in: Lecture Notes Pure Appl. Math., vol. 178, Dekker, New York, 1996, pp. 15–50], we introduce the following iterative scheme for finding a solution of the variational inequality problem for an inversestrongly- monotone operator A in a Banach space: x1 = x ∈ C and xn+1 = ΠCJ−1(J xn −λnAxn) for every n = 1, 2, . . ., where ΠC is the generalized projection from E onto C, J is the duality mapping from E into E∗ and {λn} is a sequence of positive real numbers. Then we show a weak convergence theorem (Theorem 3.1). Finally, using this result, we consider the convex minimization problem, the complementarity problem, and the problem of finding a point u ∈ E satisfying 0 = Au. © 2007 Elsevier Inc. All rights reserved