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Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces
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- Journal:
- ESAIM: Control, Optimisation and Calculus of Variations / Volume 16 / Issue 4 / October 2010
- Published online by Cambridge University Press:
- 31 July 2009, pp. 809-832
- Print publication:
- October 2010
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- Article
- Export citation
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We consider some discrete and continuous dynamics in a Banach spaceinvolving a non expansive operator J and a corresponding family ofstrictly contracting operators Φ (λ, x): = λJ( $\frac{1-\lambda}{\lambda}$
x) for λ ∈ ] 0,1] . Our motivationcomes from the study of two-player zero-sum repeated games, wherethe value of the n-stage game (resp. the value of theλ-discounted game) satisfies the relationv n = Φ( $\frac{1}{n}$ 
, $v_{n-1}$ 
) (resp. $v_\lambda$ 
= Φ(λ, $v_\lambda$ )) where J is the Shapleyoperator of the game. We study the evolution equationu'(t) = J(u(t))- u(t) as well as associated Eulerian schemes,establishing a new exponential formula and a Kobayashi-likeinequality for such trajectories. We prove that the solution of thenon-autonomous evolution equationu'(t) = Φ(λ(t), u(t))- u(t) has the same asymptoticbehavior (even when it diverges) as the sequence v n (resp. as thefamily $v_\lambda$ ) when λ(t) = 1/t (resp. whenλ(t) converges slowly enough to 0).
