Quasi-P*-Maps, P(τ, α, β)-Maps, Exceptional Family of Elements, and Complementarity Problems

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, 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(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.