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Weak convergence of a projection algorithm for variational inequalities in a Banach space. (English) Zbl 1129.49012

Summary: Let $C$ be a nonempty, closed convex subset of a Banach space $E$. In this paper, motivated by 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: Lect. Notes Pure Appl. Math. 178, Dekker, New York, 15–50 (1996; Zbl 0883.47083)], 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

${x}_{n+1}={{\Pi }}_{C}{J}^{-1}\left(J{x}_{n}-{\lambda }_{n}A{x}_{n}\right),$

for every $n=1,2,\cdots$, where ${{\Pi }}_{C}$ is the generalized projection from $E$ onto $C$, $J$ is the duality mapping from $E$ into ${E}^{*}$ and $\left\{{\lambda }_{n}\right\}$ 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 $0=Au$.

##### MSC:
 49J40 Variational methods including variational inequalities 47J20 Inequalities involving nonlinear operators 49M15 Newton-type methods in calculus of variations 49J45 Optimal control problems involving semicontinuity and convergence; relaxation
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