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

Vigeral, Guillaume

ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 4, pp. 809-832.

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($\frac{1-\lambda }{\lambda }$ 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($\frac{1}{n}$, $v_{n-1}$) (resp. $v_{\lambda }$ = Φ(λ, $v_{\lambda }$)) 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 $v_{\lambda }$) when λ(t) = 1/t (resp. when λ(t) converges slowly enough to 0).

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