ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - October 2010 - Volume 16, Issue 04  

Research Article

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

Vigeral, Guillaume

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

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 v n = 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 v n (resp. as the family $v_\lambda$ ) when λ(t) = 1/t (resp. when λ(t) converges slowly enough to 0).

(Received November 4 2008)

(Revised March 18 2009)

(Online publication July 31 2009)

Key Words:

  • Banach spaces;
  • nonexpansive mappings;
  • evolution equations;
  • asymptotic behavior;
  • Shapley operator

Mathematics Subject Classification:

  • 47H09;
  • 47J35;
  • 34E10