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( 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(, ) (resp. = Φ(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 ) when λ(t) = 1/t (resp. when λ(t) converges slowly enough to 0).
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}, pages = {809--832}, publisher = {EDP-Sciences}, volume = {16}, number = {4}, year = {2010}, doi = {10.1051/cocv/2009026}, mrnumber = {2744152}, zbl = {1204.47091}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2009026/} }
TY - JOUR AU - Vigeral, Guillaume TI - Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 809 EP - 832 VL - 16 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2009026/ DO - 10.1051/cocv/2009026 LA - en ID - COCV_2010__16_4_809_0 ER -
%0 Journal Article %A Vigeral, Guillaume %T Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 809-832 %V 16 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2009026/ %R 10.1051/cocv/2009026 %G en %F 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://archive.numdam.org/articles/10.1051/cocv/2009026/
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