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KRASNOSELSKI–MANN ITERATION FOR HIERARCHICAL FIXED POINTS AND EQUILIBRIUM PROBLEM

Part of: Nonlinear operators and their properties

Published online by Cambridge University Press:  26 February 2009

GIUSEPPE MARINO ,
VITTORIO COLAO ,
LUIGI MUGLIA  and
YONGHONG YAO
GIUSEPPE MARINO*
Affiliation:
Dipartimento di Matematica, Università della Calabria, 87036, Arcavacata di Rende (CS), Italy (email: gmarino@unical.it)
VITTORIO COLAO
Affiliation:
Dipartimento di Matematica, Università della Calabria, 87036, Arcavacata di Rende (CS), Italy (email: colao@mat.unical.it)
LUIGI MUGLIA
Affiliation:
Dipartimento di Matematica, Università della Calabria, 87036, Arcavacata di Rende (CS), Italy (email: muglia@mat.unical.it)
YONGHONG YAO
Affiliation:
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, People’s Republic of China (email: yaoyonghong@yahoo.cn)
*
✉For correspondence; e-mail: gmarino@unical.it
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Abstract

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We give an explicit Krasnoselski–Mann type method for finding common solutions of the following system of equilibrium and hierarchical fixed points: where C is a closed convex subset of a Hilbert space H, G:C×C→ℝ is an equilibrium function, T:C→C is a nonexpansive mapping with Fix(T) its set of fixed points and f:C→C is a ρ-contraction. Our algorithm is constructed and proved using the idea of the paper of [Y. Yao and Y.-C. Liou, ‘Weak and strong convergence of Krasnosel’skiĭ–Mann iteration for hierarchical fixed point problems’, Inverse Problems24 (2008), 501–508], in which only the variational inequality problem of finding hierarchically a fixed point of a nonexpansive mapping T with respect to a ρ-contraction f was considered. The paper follows the lines of research of corresponding results of Moudafi and Théra.

Keywords

hierarchical fixed pointsequilibrium problemnonexpansive mapcontractionvariational inequality problemprojection

MSC classification

Secondary: 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc. 47H10: Fixed-point theorems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 2 , April 2009 , pp. 187 - 200
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

Supported by Ministero dell’Università e della Ricerca of Italy.

References

[1] Blum, E. and Oettli, E., ‘From optimation and variational inequalities to equilibrium problems’, Math. Student 63 (1994), 123–145.Google Scholar
[2] Browder, F. E., ‘Convergence of approximation to fixed points of nonexpansive nonlinear mappings in Hilbert spaces’, Arch. Ration. Mech. Anal. 24 (1967), 82–90.CrossRefGoogle Scholar
[3] Byrne, C., ‘A unified treatment of some iterative algorithms in signal processing and image reconstruction’, Inverse Problems 20 (2004), 103–120.CrossRefGoogle Scholar
[4] Cabot, A., ‘Proximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimization’, SIAM J. Optim. 15 (2005), 555–572.CrossRefGoogle Scholar
[5] Censor, Y., Motova, A. and Segal, A., ‘Perturbed projections and subgradient projections for the multiple-sets split feasibility problem’, J. Math. Anal. Appl. 327 (2007), 1244–1256.CrossRefGoogle Scholar
[6] Combettes, P. L. and Hirstoaga, S. A., ‘Equilibrium programming in Hilbert spaces’, J. Nonlinear Convex Anal. 6(1) (2005), 117–136.Google Scholar
[7] Flåm, S. D. and Antipin, A. S., ‘Equilibrium programming using proximal-like algorithms’, Math. Program. 78 (1997), 29–41.CrossRefGoogle Scholar
[8] Göpfert, A., Riahi, H., Tammer, C. and Zălinescu, C., Variational Methods in Partially Ordered Spaces (Springer, New York, 2003).Google Scholar
[9] Halpern, B., ‘Fixed points of nonexpansive maps’, Bull. Amer. Math. Soc. 73 (1967), 957–961.CrossRefGoogle Scholar
[10] Iusem, A. N. and Sosa, W., ‘Iterative algorithms for equilibrium problems’, Optimization 52 (2003), 301–316.CrossRefGoogle Scholar
[11] Konnov, I. V., ‘Application of the proximal point method to nonmonotone equilibrium problems’, J. Optim. Theory Appl. 119 (2003), 317–333.CrossRefGoogle Scholar
[12] Lions, P. L., ‘Approximation de points fixes de contractions’, C. R. Acad. Sci. Sér. A-B Paris 284 (1977), 1357–1359.Google Scholar
[13] Luo, Z.-Q., Pang, J.-S. and Ralph, D., Mathematical Programs with Equilibrium Constraints (Cambridge University Press, Cambridge, 1996).CrossRefGoogle Scholar
[14] Marino, G. and Xu, H. K., ‘A general iterative method for nonexpansive mappings in Hilbert spaces’, J. Math. Anal. Appl. 318(1) (2006), 43–52.CrossRefGoogle Scholar
[15] Moudafi, A., ‘Viscosity approximation methods for fixed-points problems’, J. Math. Anal. Appl. 241 (2000), 46–55.CrossRefGoogle Scholar
[16] Moudafi, A., ‘Second-order differential proximal methods for equilibrium problems’, J. Inequal. Pure Appl. Math. 4 (2003), art. 18.Google Scholar
[17] Moudafi, A., ‘Krasnoselski–Mann iteration for hierarchical fixed-point problems’, Inverse Problems 23 (2007), 1635–1640.CrossRefGoogle Scholar
[18] Moudafi, A. and Théra, M., ‘Proximal and dynamical approaches to equilibrium problems’, in: Ill-Posed Variational Problems and Regularization Techniques, Lecture Notes in Economics and Math. Sys., 477 (Springer, New York, 1999).Google Scholar
[19] Oettli, W., ‘A remark on vector-valued equilibria and generalized monotonicity’, Acta Math. Vietnamica 22 (1997), 215–221.Google Scholar
[20] Reich, S., ‘Strong convergence theorems for resolvents of accretive operators in Banach spaces’, J. Math. Anal. Appl. 75 (1980), 287–292.CrossRefGoogle Scholar
[21] Reich, S., ‘Approximating fixed points of nonexpansive mappings’, Panamerican. Math. J. 4(2) (1994), 23–28.Google Scholar
[22] Shioji, N. and Takahashi, W., ‘Strong convergence of approximated sequeces for nonexpansive mappings in Banach spaces’, Proc. Amer. Math. Soc. 125 (1997), 3641–3645.CrossRefGoogle Scholar
[23] Solodov, M., ‘An explicit descent method for bilevel convex optimization’, J. Convex Anal. 14 (2007), 227–237.Google Scholar
[24] Suzuki, T., ‘Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals’, J. Math. Anal. Appl. 305(1) (2005), 227–239.CrossRefGoogle Scholar
[25] Wittmann, R., ‘Approximation of fixed points of nonexpansive mappings’, Arch. Math. 58 (1992), 486–491.CrossRefGoogle Scholar
[26] Xu, H. K., ‘Iterative algorithms for nonlinear operators’, J. London Math. Soc. 66 (2002), 240–256.CrossRefGoogle Scholar
[27] Xu, H. K., ‘Another control condition in an iterative method for nonexpansive mappings’, Bull. Austral. Math. Soc. 65 (2002), 109–113.CrossRefGoogle Scholar
[28] Xu, H. K., ‘Remarks on an iterative method for nonexpansive mappings’, Comm. Appl. Nonlinear Anal. 10(1) (2003), 67–75.Google Scholar
[29] Xu, H. K., ‘An iterative approach to quadratic optimization’, J. Optim. Theory Appl. 116(3) (2003), 659–678.CrossRefGoogle Scholar
[30] Xu, H. K., ‘Viscosity approximation methods for nonexpansive mappings’, J. Math. Anal. Appl. 298 (2004), 279–291.CrossRefGoogle Scholar
[31] Xu, H. K., ‘A variable Krasnoselski–Mann algorithm and the multiple-set split feasibility problem’, Inverse Problems 22 (2006), 2021–2034.CrossRefGoogle Scholar
[32] Yamada, I., ‘The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings’, in: Inherently Parallel Algorithm for Feasibility and Optimization and their Applications (eds. D. Butnariu, Y. Censor and S. Reich) (Elsevier, 2001), pp. 473–504.Google Scholar
[33] Yamada, I., Ogura, N. and Shirakawa, N., ‘A numerically robust hybrid steepest descent method for the convexity constrained generalized inverse problems’, Contemp. Math. 313 (2002), 269–305.CrossRefGoogle Scholar
[34] Yang, Q. and Zhao, J., ‘Generalized KM theorems and their applications’, Inverse Problems 22 (2006), 833–844.CrossRefGoogle Scholar
[35] Yao, Y. and Liou, Y.-C., ‘Weak and strong convergence of Krasnosel’skiĭ–Mann iteration for hierarchical fixed point problems’, Inverse Problems 24 (2008), 501–508.CrossRefGoogle Scholar