Weakly homogeneous variational inequalities and solvability of nonlinear equations over cones

Given a closed convex cone C in a finite dimensional real Hilbert space H, a weakly homogeneous map f : C → H is a sum of two continuous maps h and g, where h is positively homogeneous of degree γ (≥ 0) on C and g(x) = o(||x||) as ||x|| → ∞ in C. Given such a map f , a nonempty closed convex subset K of C, and a q ∈ H, we consider the variational inequality problem, VI(f,K, q), of finding an x∗ ∈ K such that 〈f(x∗)+q, x−x∗〉 ≥ 0 for all x ∈ K. In this paper, we establish some results connecting the variational inequality problem VI(f,K, q) and the cone complementarity problem CP(f∞,K∞, 0), where f∞ := h is the homogeneous part of f and K∞ is the recession cone of K. We show, for example, that VI(f,K, q) has a nonempty compact solution set for every q when zero is the only solution of CP(f∞,K∞, 0) and the local (topological) degree of the map x 7→ x−ΠK∞(x−G(x)) at the origin is nonzero, where G is a continuous extension of f∞ to H. As a consequence, we generalize a complementarity result of Karamardian [13] formulated for homogeneous maps on proper cones to variational inequalities. The results above extend some similar results proved for affine variational inequalities and for polynomial complementarity problems over the nonnegative orthant in R. As an application, we discuss the solvability of nonlinear equations corresponding to weakly homogeneous maps over closed convex cones. In particular, we extend a result of Hillar and Johnson [12] on the solvability of symmetric word equations to Euclidean Jordan algebras. ∗Supported by Fondecyt through grant 3150323