Quasi-P‚-Maps, P(τ, α, β)-Maps, Exceptional Family of Elements,

Abstract. Quasi-P‚-maps and P(τ, α, β)-maps defined 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 quasimonotone maps (in particular, pseudo-monotone 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(τ, α, β)-map. This result generalizes a classical Karamardian existence theorem and a recent result concerning quasimonotone maps 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. Key Words. Nonlinear complementarity problems, exceptional family of elements, quasi-P‚-maps, P(τ, α, β)-maps, exceptional regularity. 1