Communicated by F.giannessi

1 We thank the anonymous referees and Professor F.Giannessi for helpful comments and suggestions which lead to improvement of the paper. Abstract. Quasi-P-maps and P(; ;)-maps deened in this paper are two large classes of nonlinear mappings which are broad enough to include P-maps as special cases. It is of interest that the class of quasi-P-maps also encompasses quasimono-tone maps (in particular, pseudomonotone maps) as special cases. Under a strict feasibility condition, it is shown that the nonlinear complementarity problem has a solution if the function is a nonlinear quasi-P-map or P(; ;)-map. This result generalizes a classical Karamardian existence theorem and a recent result concerning quasimonotone map established by Hadjisawas and Schaible (but restricted to complementarity problems). A new existence result under an exceptional regularity condition is also established. Our method is based on the concept of exceptional family of elements for a continuous function, which is a powerful tool for investigating the solvability of complementarity problems.