which was extensively studied in the last several decades. In the case when K = R n + (the nonnegative orthant), problems (2) and (3) are further reduced to the following generalized complementarity problem (GCP):
and complementarity problem (NCP):
respectively. Given functions f; g and h, the existence of a solution to NPE is not always assured. In this paper, we study the existence of a solution to NPE by using new concepts, that is, the exceptional families for NPE. Based on these concepts, two important alternative theorems for NPE are established, which state that there exists either a solution or an exceptional family for NPE. Thus, the condition "there exists no exceptional family for NPE" is su cient for the existence of a solution to NPE. Due to the importance of complementarity problems in practical applications (see ), in Section 3 of this paper, we use one of the alternative theorems to develop several new existence conditions for GCP.
So far, a large number of existence results have been proved for complementarity problems with di erent classes of the functions. Most of these results assume monotonicity or certain generalized monotonicity of the functions . These concepts of generalized monotonicity were emerged from the pseudomonotonicity introduced by Karamardian [10,. He showed that the complementarity problem over a pointed, solid closed convex cone K in R n has a solution if f is a continuous pseudo-monotone function and satisÿes the strictly feasible condition, i.e., there exists an x ∈ K such that f(x) is an interior point of K * , the dual cone of K. Particularly, when K = R n + , this strictly feasible condition reduces to the following: There exists a point u ≥ 0 such that f(u) ¿ 0. Later, several generalizations of Karamardian result have been developed. Under the same pseudo-monotonicity assumption, Cottle and Yao [3] generalized the Karamardian result to the case of a solid, closed and convex cone in Hilbert space. Harker and Pang [6] and Yao extended the Karamardian result to variational inequality problems. The class of quasi-monotone maps is larger than the pseudo-monotone maps. Hadjisavvas and Schaible [5] established an existence result for quasi-monotone VIP in re exive Banach space. When restricted to NCP, their result states that if the strictly feasible condition holds, there exists a solution to the complementarity problem with a quasi-monotone map. The concept of monotonicity is also generalized in other directions, for instance, the class of nonlinear P * -maps. It is easy to give examples to show that a P * -map need not to be a quasi-monotone map, and the vice versa. Zhao and Han showed that if the strictly feasible condition is satisÿed then there exists a solution to NCP with a nonlinear P * -map.
In this paper, we introduce several new classes of nonlinear functions including so-called quasi-P * , quasi-P M * and P( ; ; ÿ)-maps. Each of these classes can be viewed as the generalization of P * -maps. For these maps, our main results state that the complementarity problem has a solution if it is strictly feasible. Since quasi-P * -maps and