The generic stability and existence of essentially connected components ofsolutions for nonlinear complementarity problems

The aim of this paper is to develop the general generic stability theory fo r nonlinear complementarity problems in the setting of infinite dimensional Banach spaces. We first show that each nonlinear complementarity problem c an be approximated arbitrarily by a nonlinear complementarity problem which is stable in the sense that the small change of the objective function res ults in the small change of its solution set; and thus we say that almost a ll complementarity problems are stable from viewpoint of Baire category. Se condly, we show that each nonlinear complementarity problem has, at least, one connected component of its solutions which is stable, though in general its solution set may not have good behaviour (i.e., not stable). Our resul ts show that if a complementarity problem has only one connected solution s et, it is then always stable without the assumption that the functions are either Lipschitz or differentiable.