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An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings. (English) Zbl 1175.47058
Let H be a Hilbert space and C a closed convex subset of H. An equilibrium function is a mapping G:H×H such that (A 1 ) G(x,x)=0 for all xH. A strongly positive operator is a bounded linear operator A:HH such that for all xH, Ax,xγ ¯x 2 for some γ ¯>0. Supposing that the equilibrium function G satisfies further the conditions (A 2 ) for all x,yC, G(x,y)+G(y,x)0 (i.e., G is monotone); (A 3 ) for all x,y,zC lim sup t0 G(tz+(1-t)x,y)G(x,y), and (A 4 ) for all xC, the mapping G(x,·) is convex and lsc. S. Plubtieng and R. Punpaeng [J. Math. Anal. Appl. 336, No. 1, 455–469 (2007; Zbl 1127.47053)] proposed an iteration procedure to find the unique solution z Fix(T)SEP(G) of the variational inequality: (1) (A-γf)z,z-x0, for all xFix(T)SEP(G). Here, T is a nonexpansive mapping on H, A is a strongly positive operator on H, f an α-contraction on H and γ>0 an appropriate constant. In this paper, the authors extend the above result by considering a family G i ,i=1,,K, of equilibrium functions satisfying (A 2 )–(A 4 ) and a family (T n ) n of nonexpansive mappings. Supposing that D:= i=1 K SEP(G i ) n Fix(T n ). They propose an implicit iteration procedure for finding the unique solution of the variational inequality (1) on the set D and prove the strong convergence of this procedure.

MSC:
47J20Inequalities involving nonlinear operators
47J25Iterative procedures (nonlinear operator equations)
41A65Abstract approximation theory
47H09Mappings defined by “shrinking” properties