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

Guillaume Vigeral

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 4, page 809-832
  • ISSN: 1292-8119

Abstract

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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( 1 - λ λ 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( 1 n , v n - 1 ) (resp.  v λ = Φ(λ, v 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 v λ ) when λ(t) = 1/t (resp. when λ(t) converges slowly enough to 0).

How to cite

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Vigeral, Guillaume. "Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces." ESAIM: Control, Optimisation and Calculus of Variations 16.4 (2010): 809-832. <http://eudml.org/doc/250699>.

@article{Vigeral2010,
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). },
author = {Vigeral, Guillaume},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Banach spaces; nonexpansive mappings; evolution equations; asymptotic behavior; Shapley operator},
language = {eng},
month = {10},
number = {4},
pages = {809-832},
publisher = {EDP Sciences},
title = {Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces},
url = {http://eudml.org/doc/250699},
volume = {16},
year = {2010},
}

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
DA - 2010/10//
PB - EDP Sciences
VL - 16
IS - 4
SP - 809
EP - 832
AB - 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).
LA - eng
KW - Banach spaces; nonexpansive mappings; evolution equations; asymptotic behavior; Shapley operator
UR - http://eudml.org/doc/250699
ER -

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