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

摘要

Let H be a Hilbert space and C be a nonempty closed convex subset of H, fTigi2N be a family of nonexpansive mappings from C into H, Gi V C C ! R be a finite family of equilibrium functions .i 2 f1; 2; : : : ; Kg/, A be a strongly positive bounded linear operator with a coefficient N and B -Lipschitzian, relaxed .; /-cocoercive map of C into H. Moreover, let frk;ngK k D1, fng satisfy appropriate conditions and F VD .K k D1 EP.Gk// VI.C; B/ .2N Fix.Tn// 6D ;; we introduce an explicit scheme which defines a suitable sequence as follows: xnC1 D n f .xn/ C .1 nA/WnPC .IsnB/S1 r1;n S2 r2;n SK rK;n xn 8n 2 N and fxng strongly converges to x 2 F which satisfies the variational inequality h.A f /x; x xi 0 for all x 2 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) 38973909].