Quasi-P-*-maps, P(tau,alpha,beta)-maps, exceptional family of elements, and complementarity problems

Quasi-P*-maps and P(tau, alpha, beta)-maps defined in this paper are two la rge 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 e ncompasses quasimonotone maps (in particular, pseudomonotone maps) as speci al cases. Under a strict feasibility condition, it is shown that the nonlin ear complementarity problem has a solution if the function is a nonlinear q uasi-P*-map or P(tau, alpha, beta)-map. This result generalizes a classical Karamardian existence theorem and a recent result concerning quasimonotone maps established by Hadjisawas and Schaible, but restricted to complementa rity problems. A new existence result under an exceptional regularity condi tion 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.