Alternative theorems for nonlinear projection equations and applications to generalized complementarity problems

Abstract

Let K be a closed convex subset in Rn, and let f; g and h be three functions from Rn into itself. We consider the following nonlinear projection equation (NPE): h(x) = :K (g(x)− f(x)); x ∈ R; (1) where :K (·) is the orthogonal projection operator onto the set K . This equation provides a uni<ed formulation of several interesting and important special cases. In the case when h(x)= g(x) for any x∈Rn; then (1) reduces to the equation studied by Pang and Yao [17] under the name of generalized normal equation which is equivalent to the following generalized variational inequality (GVI): g(x) ∈ K; (y − g(x))f(x) ≥ 0 for all y ∈ K: (2)