Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces

Équipe Combinatoire et Optimisation, CNRS FRE3232, Université Pierre et Marie Curie, Paris 6, UFR 929, 175 rue du Chevaleret, 75013 Paris, France. guillaumevigeral@gmail.com

Received: 4 November 2008
Revised: 18 March 2009

Abstract

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).