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 Original Research Article

Let HH be a Hilbert space and CC be a nonempty closed convex subset of HH, {Ti}i∈N{Ti}i∈N be a family of nonexpansive mappings from CC into HH, Gi:C×C→RGi:C×C→R be a finite family of equilibrium functions (i∈{1,2,…,K})(i∈{1,2,…,K}), AA be a strongly positive bounded linear operator with a coefficient View the MathML sourceγ̄ and View the MathML sourceBλ-Lipschitzian, relaxed (μ,ν)(μ,ν)-cocoercive map of CC into HH. Moreover, let View the MathML source{rk,n}k=1K, {αn}{αn} satisfy appropriate conditions and View the MathML sourceF≔(∩k=1KEP(Gk))∩VI(C,B)∩(∩n∈NFix(Tn))≠0̸; we introduce an explicit scheme which defines a suitable sequence as follows: View the MathML sourcexn+1=αnγf(xn)+(1−αnA)WnPC(I−snB)Sr1,n1Sr2,n2⋯SrK,nKxn∀n∈N Turn MathJax on and {xn}{xn} strongly converges to x∗∈Fx∗∈F which satisfies the variational inequality 〈(A−γf)x∗,x−x∗〉≥0〈(A−γf)x∗,x−x∗〉≥0 for all x∈Fx∈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].