On modified hybrid steepest-descent methods for general variational inequalities

We consider the general variational inequality GVI(F,g,C)GVI(F,g,C), where F and g are mappings from a Hilbert space into itself and C is intersection of the fixed point sets of a finite family of nonexpansive mappings. We suggest and analyze an iterative algorithm with variable parameters as follows:un+1=(1−αn+1+θn+1)T[n+1]un+αn+1un−θn+1g(T[n+1]un)−λn+1μn+1F(T[n+1]un),n⩾0. The sequence {un}{un} is shown to converge in norm to the solutions of the general variational inequality GVI(F,g,C)GVI(F,g,C) under some mild conditions. Application to constrained generalized pseudo-inverse is included. Since the general variational inequalities include variational inequalities, quasi-variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. Results obtained in this paper may be viewed as a refinement and improvement of the previously known results.