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 \(\alpha_n,\beta_n \in [0,1]\).


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


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.