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| - Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces
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| - 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 vn = 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 vn (resp. as the family $v_\lambda$) when λ( t) = 1/ t (resp. when λ( t) converges slowly enough to 0).
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| - © EDP Sciences, SMAI, 2009
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