Equilibria of pairs of nonlinear maps associated with cones
Por:
Barker G.P., Neumann-Coto M., Schneider H., Takane M., Tam B.-S.
Publicada:
1 ene 2005
Resumen:
Let K 1, K 2 be closed, full, pointed convex cones in finite-dimensional real vector spaces of the same dimension, and let F : K 1 ? span K 2 be a homogeneous, continuous, K 2-convex map that satisfies F(?K 1) ?int K 2= ø and FK 1 ?int K 2 ? ø. Using an equivalent formulation of the Borsuk-Ulam theorem in algebraic topology, we show that we have F(K1 {0}) ? (-K 2)= ø and K2 ? FK1. We also prove that if, in addition, G : K 1 ? span K 2 is any homogeneous, continuous map which is (K 1, K 2)-positive and K 2-concave, then there exist a unique real scalar ?0 and a (up to scalar multiples) unique nonzero vector x 0 ? K 1 such that Gx 0 = ?0 Fx 0, and moreover we have ?0 > 0 and x 0 ? int K 1 and we also have a characterization of the scalar ?0. Then, we reformulate the above result in the setting when K 1 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. © 2005 Birkhäuser Verlag Basel/Switzerland.
Filiaciones:
Barker G.P.:
Department of Mathematics, University of Missouri-Kansas City, Kansas City, MO 64110-2499, United States
Neumann-Coto M.:
Instituto de Matemáticas, UNAM, Ciudad Universitaria, 04510 México D.F., Mexico
Schneider H.:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, United States
Takane M.:
Instituto de Matemáticas, UNAM, Ciudad Universitaria, 04510 México D.F., Mexico
Tam B.-S.:
Department of Mathematics, Tamkang University, Tamsui, 251, Taiwan
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