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
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00549765/fr/
Journal nameESAIM. COCV
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Abstract (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 / KeywordsKobayashi inequality; Banach spaces; Evolution equations; discrete and continuous time; games; Shapley value
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