Équipe Combinatoire et Optimisation, CNRS FRE3232, Université Pierre et Marie Curie, Paris 6, UFR 929, 175 rue du Chevaleret, 75013 Paris, France. email@example.com
We consider some discrete and continuous dynamics in a Banach space involving a non expansive operator J and a corresponding family of strictly contracting operators Φ (λ, x): = λ J( x) for λ ∈ ] 0,1] . Our motivation comes from the study of two-player zero-sum repeated games, where the value of the n-stage game (resp. the value of the λ-discounted game) satisfies the relation v n = Φ( , ) (resp. = Φ(λ, )) where J is the Shapley operator of the game. We study the evolution equation u'(t) = J(u(t))- u(t) as well as associated Eulerian schemes, establishing a new exponential formula and a Kobayashi-like inequality for such trajectories. We prove that the solution of the non-autonomous evolution equation u'(t) = Φ(λ(t), u(t))- u(t) has the same asymptotic behavior (even when it diverges) as the sequence v n (resp. as the family ) when λ(t) = 1/t (resp. when λ(t) converges slowly enough to 0).
(Received November 4 2008)
(Revised March 18 2009)
(Online publication July 31 2009)
Mathematics Subject Classification: