A new iterative method for equilibrium problems and fixed point problems of infinitely nonexpansive mappings and monotone mappings

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Abstract

In this paper, we introduce a new iterative scheme for finding the common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α-inverse-strongly monotone mapping in Hilbert spaces. We prove that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. This main result improve and extend Plubtieng and Punpaeng’s corresponding result [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Applied Mathematics and Computation 197 (2008), 548–558]. Using this theorem, we obtain three corollaries.

Section snippets

Introduction and preliminaries

Let H be a real Hilbert space with inner product ·,· and norm ·. Let C be a nonempty closed convex subset of H and F:C×CR be a bifunction of C×C into R, where R is the set of real numbers. The equilibrium problem for F:C×CR is to find xC such thatF(x,y)0for all yC. The set of solutions of (1.1) is denoted by EP(F). Given a mapping T:CH, let F(x,y)=Tx,y-x for all x,yC. Then zEP(F) if and only if Tz,y-z0 for all yC, i.e., z is a solution of the variational inequality.

It is well

Main results

Theorem 2.1

Let C be a closed convex subset of a real Hilbert space H. Let F be a bifunction from C×CR satisfying (A1)–(A4), A be an α-inverse-strongly monotone mapping of C into H and {Tn}n=1 be a sequence of nonexpansive self-mappings on C such that n=1F(Tn)EP(F)VI(A,C). Suppose x1=uC, {sn},{αn},{βn},{γn} are sequences in [0,1], {λn} is a sequence in [0,2α] such that λn[a,b] for some a,b with 0<a<b<2α and {rn}(0,) is a real sequence. Suppose the following conditions are satisfied:

  • (i)

    αn+βn+γn=1,

  • (ii)

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This research is supported by Tianjin Natural Science Foundation in China Grant (06YFJMJC12500) and supported by the science research foundation program in Civil Aviation University of China (04-CAUC-15 S) as well.

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