Positive definite and Gram tensor complementarity problems

Given a continuous function [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.], the non-linear complementarity problem NCP(g,q) is to find a vector [InlineEquation not available: see fulltext.] such that x≥0,y:=g(x)+q≥0andxTy=0.We say that g has the Globally Uniquely Solvable (GUS)-property if NCP(g,q) has a unique solution for all [InlineEquation not available: see fulltext.] and C-property if NCP (g, q) has a convex solution set for all [InlineEquation not available: see fulltext.]. In this paper, we find a class of non-linear functions that have the GUS-property and C-property. These functions are constructed by some special tensors which are positive semidefinite. We call these tensors as Gram tensors.