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).

Classification: 47H09, 47J35, 34E10

Keywords: Banach spaces, nonexpansive mappings, evolution equations, asymptotic behavior, Shapley operator

@article{COCV_2010__16_4_809_0, author = {Vigeral, Guillaume}, title = {Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, publisher = {EDP-Sciences}, volume = {16}, number = {4}, year = {2010}, pages = {809-832}, doi = {10.1051/cocv/2009026}, zbl = {1204.47091}, mrnumber = {2744152}, language = {en}, url = {http://www.numdam.org/item/COCV_2010__16_4_809_0} }

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://www.numdam.org/item/COCV_2010__16_4_809_0/

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