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Zbl 1129.49012
Iiduka, Hideaki; Takahashi, Wataru
Weak convergence of a projection algorithm for variational inequalities in a Banach space.
(English)
[J] J. Math. Anal. Appl. 339, No. 1, 668-679 (2008). ISSN 0022-247X

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 $0=Au$.
MSC 2000:
*49J40 Variational methods including variational inequalities
47J20 Inequalities involving nonlinear operators
49M15 Methods of Newton-Raphson, Galerkin and Ritz types
49J45 Optimal control problems inv. semicontinuity and convergence

Keywords: generalized projections; inverse-strongly-monotone operators; variational inequalities; $p$-uniformly convex; weak convergence

Citations: Zbl 0883.47083

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