Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, pp. 809-832.

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(1-λ λ 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(1 n, v n-1 ) (resp. v λ = Φ(λ, v λ )) 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 λ ) when λ(t) = 1/t (resp. when λ(t) converges slowly enough to 0).

DOI: 10.1051/cocv/2009026
Classification: 47H09, 47J35, 34E10
Keywords: Banach spaces, nonexpansive mappings, evolution equations, asymptotic behavior, Shapley operator
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     title = {Evolution equations in discrete and continuous time for nonexpansive operators in {Banach} spaces},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {809--832},
     publisher = {EDP-Sciences},
     volume = {16},
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Vigeral, Guillaume. Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, pp. 809-832. doi : 10.1051/cocv/2009026. http://archive.numdam.org/articles/10.1051/cocv/2009026/

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