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On a two-step algorithm for hierarchical fixed point problems and variational inequalities. (English) Zbl 1180.47040

The paper is concerned with the variational inequality problem of finding \(x^* \in \text{Fix}(T)\) with \(\langle (I-S)x^*, x-x^* \rangle \geq 0\) for all \(x\in \text{Fix}(T)\), where \(T,S: C\to C\) are nonexpansive mappings such that Fix\((T)\), the set of fixed points set of \(T\), is nonempty, and \(C\) is a closed convex subset of a Hilbert space \(H\). Let \(f: C \to C\) be a contraction. The authors study convergence properties of the iterative process \(x_{n+1}=\alpha_n f(x_n)+(1-\alpha_n) T y_n\), \(y_n=\beta_n S x_n+(1-\beta_n) x_n\), where \(\alpha_n,\beta_n \in [0,1]\).

MSC:

47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J40 Variational inequalities
47J25 Iterative procedures involving nonlinear operators
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References:

[1] Byrne, C, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20, 103-120, (2004) · Zbl 1051.65067
[2] Yang, Q; Zhao, J, Generalized KM theorems and their applications, Inverse Problems, 22, 833-844, (2006) · Zbl 1117.65081
[3] Moudafi, A, Krasnoselski-Mann iteration for hierarchical fixed-point problems, Inverse Problems, 23, 1635-1640, (2007) · Zbl 1128.47060
[4] Yao Y, Liou Y-C: Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed point problems.Inverse Problems 2008,24(1):-8. · Zbl 1154.47055
[5] Yamada, I, 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), Amsterdam, The Netherlands · Zbl 1013.49005
[6] Solodov, M, An explicit descent method for bilevel convex optimization, Journal of Convex Analysis, 14, 227-237, (2007) · Zbl 1145.90081
[7] Marino, G; Xu, H-K, A general iterative method for nonexpansive mappings in Hilbert spaces, Journal of Mathematical Analysis and Applications, 318, 43-52, (2006) · Zbl 1095.47038
[8] Moudafi, A, Viscosity approximation methods for fixed-points problems, Journal of Mathematical Analysis and Applications, 241, 46-55, (2000) · Zbl 0957.47039
[9] Xu, H-K, Iterative algorithms for nonlinear operators, Journal of the London Mathematical Society, 66, 240-256, (2002) · Zbl 1013.47032
[10] Xu, H-K, Viscosity approximation methods for nonexpansive mappings, Journal of Mathematical Analysis and Applications, 298, 279-291, (2004) · Zbl 1061.47060
[11] MaingĂ©, 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
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