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

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

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

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topVigeral, 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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