Equilibria of pairs of nonlinear maps associated with cones

doi:10.1007/s00020-003-1259-3

淡江大學機構典藏 > 理學院 > 應用數學與數據科學學系 > 期刊論文 > Item 987654321/41237

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Title:	Equilibria of pairs of nonlinear maps associated with cones
Authors:	Barker, George P.;Neumann-coto, Max;Schneider, Hans;Takane, Martha;譚必信;Tam, Bit-shun
Contributors:	淡江大學數學學系
Keywords:	Proper cone;convex set;nonlinear map;equilibrium point;Ky Fan;Borsuk-Ulam
Date:	2005-03
Issue Date:	2010-01-28 07:02:46 (UTC+8)
Publisher:	Springer
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(K1∖{0})∩(−K2)=∅F(K1∖{0})∩(−K2)=∅ and K2⊆FK1.K2⊆FK1. 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.
Relation:	Integral Equations and Operator Theory 51(3), pp.357-373
DOI:	10.1007/s00020-003-1259-3
Appears in Collections:	[應用數學與數據科學學系] 期刊論文

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