An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings

Let H be a Hilbert space. Consider on H a sequence of nonexpansive mappings {T(n)} with common fixed points, a finite family of equilibrium functions {G(i)}(i=1,...,K), a contraction f with coefficient 0 < alpha < 1 and a strongly positive linear bounded operator A with coefficient (gamma) over bar > 0. Let 0 < gamma < (gamma) over bar/alpha. Assuming there are common equilibrium points of the family {G(i)}i=(1,...,K) which are also fixed points for {T(n)}, we define a suitable sequence which strongly converges to the unique such point which also satisfies the variational inequality <(A - gamma f)x*, x - x*> >= 0 for all the x in the intersection of the equilibrium points and the common fixed points of the sequence {T(n)}. (C) 2009 Elsevier Ltd. All rights reserved.