Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces
| Autor: | Vigeral, Guillaume |
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| Rok vydání: | 2009 |
| Předmět: | |
| Zdroj: | ESAIM: COCV 16 (2010) 809-832 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1051/cocv/2009026 |
| Popis: | 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 $u'(t)=J(u(t))-u(t)$ 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 $u'(t)=\Phi(\bm{\lambda}(t),u(t))-u(t)$ has the same asymptotic behavior (even when it diverges) as the sequence $v_n$ (resp. as the family $v_\lambda$) when $\bm{\lambda}(t)=1/t$ (resp. when $\bm{\lambda}(t)$ converges slowly enough to 0). Comment: 28 pages To appear in ESAIM:COCV |
| Databáze: | arXiv |
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