Mathematics – Classical Analysis and ODEs
Scientific paper
2009-04-15
ESAIM: COCV 16 (2010) 809-832
Mathematics
Classical Analysis and ODEs
28 pages To appear in ESAIM:COCV
Scientific paper
10.1051/cocv/2009026
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).
No associations
LandOfFree
Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-592340