KRASNOSELSKI–MANN ITERATION FOR HIERARCHICAL FIXED POINTS AND EQUILIBRIUM PROBLEM

Publisher： Cambridge University Press

E-ISSN： 1755-1633|79|2|187-200

ISSN： 0004-9727

Source： Bulletin of the Australian Mathematical Society, Vol.79, Iss.2, 2009-04, pp. : 187-200

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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: \[ \begin {cases} G(x^*,y)\geq 0, \forall y \in C, \cr \mbox {find } x^*\in \mathrm {Fix}(T) \mbox { such that } \langle x^*-f(x^*),x-x^* \rangle \geq 0 , \forall x\in \mathrm {Fix}(T), \end {cases} \] where C is a closed convex subset of a Hilbert space H, G:C×C→ 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 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.

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