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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 {\it 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 $$x_{n+1}= \Pi_C J^{-1}(Jx_n- \lambda_nAx_n),$$ 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 $E^*$ 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 $0=Au$.

49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
49M15Newton-type methods in calculus of variations
49J45Optimal control problems involving semicontinuity and convergence; relaxation
Full Text: DOI
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