Krasnoselski-Mann iteration for hierarchical fixed points and equilibrium problem

We give an explicit Krasnoselski-Mann type method for finding common solutions of the following system of equilibrium and hierarchical fixed points: {G(x*, y) >= 0, for all y is an element of C, find x* is an element of Fix(T) such that < x* - f(x*), x - x*> >= 0, for all x is an element of Fix (T), where C is a closed convex subset of a Hilbert space H, G : C x 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 rho-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'skn-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 rho-contraction f was considered. The paper follows the lines of research of corresponding results of Moudafi and Thera.