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 inverse-strongly-monotone operator A   in a Banach space: x1=x∈Cx1=x∈C andxn+1=ΠCJ−1(Jxn−λnAxn)xn+1=ΠCJ−1(Jxn−λnAxn) for every n=1,2,…, where ΠCΠC is the generalized projection from E onto C, J is the duality mapping from E   into E∗E∗ and {λn}{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∈Eu∈E satisfying 0=Au0=Au.