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

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 $\Phi (\lambda ,x):=\lambda J(\frac{1-\lambda }{\lambda }x)$ for $\lambda \in ]0,1]$. 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 $\lambda$-discounted game) satisfies the relation $v_n=\Phi (\frac{1}{n},v_{n-1})$ (resp. $v_{\lambda }=\Phi (\lambda ,v_{\lambda })$) where $J$ is the Shapley operator of the game. We study the evolution equation $u^{\prime }(t)=J(u(t))-u(t)$ 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 $u^{\prime }(t)=\Phi (\mathbf{\lambda }(t),u(t))-u(t)$ has the same asymptotic behavior (even when it diverges) as the sequence $v_n$ (resp. as the family $v_{\lambda }$) when $\mathbf{\lambda }(t)=1/t$ (resp. when $\mathbf{\lambda }(t)$ converges slowly enough to 0).

28 pages To appear in ESAIM:COCV

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