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

Abstract

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: $x_1=x\in C$ and

$$x_{n+1}={\Pi }_C{J}^{\mathrm{-1}}(J{x}_n-{\lambda }_{n}A{x}_n)$$

for every $n=1,2,\dots$, where ${\Pi }_C$ is the generalized projection from E onto C, J is the duality mapping from E into $E^{\ast }$ and $\{{\lambda }_n\}$ 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 $u\in E$ satisfying $0=Au$.