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KRASNOSELSKI–MANN ITERATION FOR HIERARCHICAL FIXED POINTS AND EQUILIBRIUM PROBLEM

Published online by Cambridge University Press:  26 February 2009

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)
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Abstract

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:CC is a nonexpansive mapping with Fix(T) its set of fixed points and f:CC 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 Problems 24 (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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

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

References

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KRASNOSELSKI–MANN ITERATION FOR HIERARCHICAL FIXED POINTS AND EQUILIBRIUM PROBLEM
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