Characterizations of a Homotopy Solution Mapping for Nonlinear Complementarity Problems

We study several properties of a homotopy solution mapping, based on Zhang and Zhang's homotopy formulation, for continuous nonlinear complementarity problems with such non-monotone maps as quasi-monotone, E 0-, and exceptional regular functions. For these classes of complementarity problems, we establish several suucient conditions to assure the nonemptyness and boundedness of the homotopy solution mapping. Under the P 0 property, all these suucient conditions also guarantee the uniqueness and continuity of the homotopy solution mapping, and hence the range of this mapping forms a continuous path passing through an arbitrary point in R n ++ R n ++ to a solution of complementarity problem. Our analysis is very diierent from Zhang and Zhang's. We use homotopy invariance theorem of degree to develop a general suucient condition for the nonemptyness of the homotopy solution mapping instead of using the parameterized Sard's theorem. As a result of this, we only require the continuity of functions instead of continuous diierentiability.