Equilibria of Pairs of Nonlinear Maps Associated with Cones

Abstract.

Let K₁, K₂ be closed, full, pointed convex cones in finite-dimensional real vector spaces of the same dimension, and let F : K₁ → span K₂ be a homogeneous, continuous, K₂-convex map that satisfies F(∂K₁) ∩ int K₂=∅ and FK₁ ∩ int K₂ ≠ ∅. Using an equivalent formulation of the Borsuk-Ulam theorem in algebraic topology, we show that we have $F(K_1 \setminus\{0\}) \cap (-K_2)=\emptyset$ and $K_2 \subseteq FK_1.$ We also prove that if, in addition, G : K₁ → span K₂ is any homogeneous, continuous map which is (K₁, K₂)-positive and K₂-concave, then there exist a unique real scalar ω₀ and a (up to scalar multiples) unique nonzero vector x₀ ∈ K₁ such that Gx₀ = ω₀Fx₀, and moreover we have ω₀ > 0 and x₀ ∈ int K₁ and we also have a characterization of the scalar ω₀. Then, we reformulate the above result in the setting when K₁ is replaced by a compact convex set and recapture a classical result of Ky Fan on the equilibrium value of a finite system of convex and concave functions.