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Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces

Published online by Cambridge University Press:  31 July 2009

Guillaume Vigeral
Guillaume Vigeral*
Affiliation:
Équipe Combinatoire et Optimisation, CNRS FRE3232, Université Pierre et Marie Curie, Paris 6, UFR 929, 175 rue du Chevaleret, 75013 Paris, France. guillaumevigeral@gmail.com
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Abstract

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).

Keywords

Banach spacesnonexpansive mappingsevolution equationsasymptotic behaviorShapley operator
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 4 , October 2010 , pp. 809 - 832
Copyright
© EDP Sciences, SMAI, 2009

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