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, p. 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($\frac{1-\lambda }{\lambda }$ 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).

DOI : https://doi.org/10.1051/cocv/2009026
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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