Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces
Vigeral, Guillaume (2010), Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces, ESAIM. COCV, 16, p. 809-832. http://dx.doi.org/10.1051/cocv/2009026
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Article accepté pour publication ou publiéExternal document link
http://hal.archives-ouvertes.fr/hal-00549765/fr/Date
2010Journal name
ESAIM. COCVVolume
16Publisher
EDP Sciences
Pages
809-832
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Vigeral, GuillaumeAbstract (EN)
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) := λJ (1−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 , vn−1) (resp. vλ = Φ(λ, vλ)) where J is the Shapley operator of the game. We study the evolution n 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 vλ) when λ(t) = 1/t (resp. when λ(t) converges slowly enough to 0).Subjects / Keywords
Kobayashi inequality; Banach spaces; Evolution equations; discrete and continuous time; games; Shapley valueRelated items
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