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

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