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

Let H be a Hilbert space and C be a nonempty closed convex subset of H, {T(i)}(i is an element of N) be a family of nonexpansive mappings from C into H, G(i) : C x C -> R be a finite family of equilibrium functions (i is an element of {1, 2, ..., K}), A be a strongly positive bounded linear operator with a coefficient (gamma) over bar and B lambda-Lipschitzian, relaxed (mu, nu)-cocoercive map of C into H. Moreover, let {r(k,n)}(k=1)(K), {alpha(n)} satisfy appropriate conditions and F := (boolean AND(K)(k=1) EP(G(k))) boolean AND VI(c, B) boolean AND (boolean AND(n is an element of N) Fix(T(n))) not equal empty set; we introduce an explicit scheme which defines a suitable sequence as follows: x(n+1) = alpha(n)gamma f(x(n)) + (1 - alpha(n)A)W(n)P(C)(I - s(n)B)S(r1,n)(1) S(r2,n)(2) ... S(rK,n)(K) x(n) for all n is an element of N and {x(n)} strongly converges to x* is an element of F which satisfies the variational inequality <(A - gamma f)x*, x - x*> >= 0 for all x epsilon F. The results presented in this paper mainly extend and improve a recent result of Colao [V. Colao, An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings, Nonlinear Analysis (2009), doi: 10.1016/j.na.2009.01.115] and Qin [X. Qin, M. Shang, Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Analysis 69(2008)3897-3909]. (C) 2009 Elsevier Ltd. All rights reserved.