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A hybrid method for monotone variational inequalities involving pseudocontractions. (English) Zbl 1215.49021
Summary: We use strongly pseudocontraction to regularize the following ill-posed monotone variational inequality: finding a point \(x^*\) with the property \(x^*\in \text{Fix}(T)\) such that \(\langle(I-S)x^*,x-x^*\rangle\geq 0\), \(x\in\text{Fix}(T)\) where \(S,T\) are two pseudocontractive self-mappings of a closed convex subset \(C\) of a Hilbert space with the set of fixed points \(\text{Fix}(T)\neq\emptyset\). Assume the solution set \(\Omega\) of (VI) is nonempty. In this paper, we introduce one implicit scheme which can be used to find an element \(x^*\in\Omega\). Our results improve and extend a recent result of X. Lu, H.-K. Xu and X. Yin [Nonlinear Anal., Theory Methods Appl. 71, No. 3–4, A, 1032–1041 (2009; Zbl 1176.90462)].

MSC:
49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
49N60 Regularity of solutions in optimal control
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References:
[1] Yamada, I; Butnariu, D (ed.); Censor, Y (ed.); Reich, S (ed.), The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, No. 8, 473-504, (2001) · Zbl 1013.49005
[2] Moudafi, A; Maingé, P-E, Towards viscosity approximations of hierarchical fixed-point problems, No. 2006, 10, (2006) · Zbl 1143.47305
[3] Lu, X; Xu, H-K; Yin, X, Hybrid methods for a class of monotone variational inequalities, Nonlinear Analysis: Theory, Methods & Applications, 71, 1032-1041, (2009) · Zbl 1176.90462
[4] Chen, R; Su, Y; Xu, H-K, Regularization and iteration methods for a class of monotone variational inequalities, Taiwanese Journal of Mathematics, 13, 739-752, (2009) · Zbl 1179.58008
[5] Cianciaruso, F; Colao, V; Muglia, L; Xu, H-K, On an implicit hierarchical fixed point approach to variational inequalities, Bulletin of the Australian Mathematical Society, 80, 117-124, (2009) · Zbl 1168.49005
[6] Maingé, P-E; Moudafi, A, Strong convergence of an iterative method for hierarchical fixed-point problems, Pacific Journal of Optimization, 3, 529-538, (2007) · Zbl 1158.47057
[7] Moudafi, A, Krasnoselski-Mann iteration for hierarchical fixed-point problems, Inverse Problems, 23, 1635-1640, (2007) · Zbl 1128.47060
[8] Yao, Y; Liou, Y-C, “weak and strong convergence of krasnoselski-Mann iteration for hierarchical fixed point problems, Inverse Problems, 24, 8, (2008) · Zbl 1154.47055
[9] Marino, G; Colao, V; Muglia, L; Yao, Y, Krasnoselski-Mann iteration for hierarchical fixed points and equilibrium problem, Bulletin of the Australian Mathematical Society, 79, 187-200, (2009) · Zbl 1165.47050
[10] Zhou, H, Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, 70, 4039-4046, (2009) · Zbl 1218.47131
[11] Deimling, K, Zeros of accretive operators, Manuscripta Mathematica, 13, 365-374, (1974) · Zbl 0288.47047
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