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Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems. (English) Zbl 1222.47091

The authors consider an iterative process for finding a common element of the set of common fixed points of $N$ nonexpansive mappings and the set of solutions of the variational inequality for a pseudomonotone, Lipschitz-continuous mapping $T$ in a real Hilbert space. To overcome the difficulty of dealing with an operator $T$ which is not maximal monotone, the idea proposed in this paper is to investigate an iterative scheme based on the combination of the extragradient method and the approximate proximal method introduced in [R. T. Rockafellar, SIAM J. Control Optim. 14, 877–898 (1976; Zbl 0358.90053)]. A necessary and sufficient condition for weak convergence of the sequences produced by this scheme is established. The scheme proposed in this paper is related to the scheme in [N. Nadezhkina and W. Takahashi, SIAM J. Optim. 16, 1230–1241 (2006; Zbl 1143.47047)] utilized for the approximation of a common element of the solution set of a monotone variational inequality problem and the fixed point set of a single nonexpansive mapping.

The present paper considers the more general case of several nonexpansive mappings. More importantly, it drops the maximal monotonicity, using instead the weaker assumption of (algebraic) pseudomonotonicity. The price for this is the assumption that the nonlinear pseudomonotone mapping $T$, in addition to being $k$-Lipschitz, must be weak-strong sequentially continuous or completely continuous. In the finite-dimensional case, the assumption of Lipschitz continuity is enough, and the result presented in this paper extends the result of Nadezhkina and Takahashi.

MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47J20 Inequalities involving nonlinear operators 47H05 Monotone operators (with respect to duality) and generalizations 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking" properties 49J40 Variational methods including variational inequalities 49J45 Optimal control problems involving semicontinuity and convergence; relaxation 49J53 Set-valued and variational analysis
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