An iterative method for solving equilibrium problem fixed point problem and generalized variational inequalities problem

Abstract

Abstract. In this paper, we introduce a new iterative scheme for ndinga common element of the set of an equilibrium problem, the set of xedpoints of nonexpansive mapping and the set of solutions of the generalizedvariational inequality for -inverse strongly g-monotone mapping in aHilbert space. Under suitable conditions, strong convergence theoremsfor approximating a common element of the above three sets are obtained. 1. IntroductionLet Cbe a closed convex subset of a real Hilbert space H. Recall that aself-mapping f : C !C is Lipschitz continuous on C if there is a constantk>0 such that kf(x) f(y)kkkx yk;x;y2C; if k= 1, the mapping iscalled nonexpansive. We denoted by F(f) the set of xed pints of f.Let A : C !H, g : C !C two nonlinear operators. We consider ageneralized variational inequality (GVI) problem as follows: to nd u2C,g(u) 2Csuch thathg(v) g(u);Aui0: (1:1)The set of solutions of above variational inequality (which is called Noor [1]variational inequality) problem is denoted by GVI(C;A;g). Such a problem isconnected with the convex minimization problem, the complementarity prob-lem, the problem of nding a point u2Hsatisfying 0 = Auand so on. Anoperator Aof Cinto His said -inverse-strongly g-monotone if there exists apositive real number such thathg(u) g(v);Au Avi kAu Avk