Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces
| العنوان: | Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces |
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| المؤلفون: | Guillaume Vigeral |
| المساهمون: | CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL) |
| المصدر: | ESAIM: Control, Optimisation and Calculus of Variations ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2010, 16, pp.809-832 |
| بيانات النشر: | arXiv, 2009. |
| سنة النشر: | 2009 |
| مصطلحات موضوعية: | 0209 industrial biotechnology, Computer Science::Computer Science and Game Theory, Control and Optimization, Banach space, 02 engineering and technology, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], Lambda, 01 natural sciences, Combinatorics, symbols.namesake, 020901 industrial engineering & automation, Operator (computer programming), Classical Analysis and ODEs (math.CA), FOS: Mathematics, 0101 mathematics, Mathematics - Optimization and Control, Mathematics, Sequence, 010102 general mathematics, Eulerian path, 47H09, 47J35, 34E10, Computational Mathematics, Exponential formula, Control and Systems Engineering, Mathematics - Classical Analysis and ODEs, Optimization and Control (math.OC), Evolution equation, symbols, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Expansive |
| الوصف: | 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 $\Phi(\lambda,x):=\lambda J(\frac{1-\lambda}{\lambda}x)$ for $\lambda\in]0,1]$. 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 $\lambda$-discounted game) satisfies the relation $v_n=\Phi(\frac{1}{n},v_{n-1})$ (resp. $v_\lambda=\Phi(\lambda,v_\lambda)$) 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 $u'(t)=\Phi(\bm{\lambda}(t),u(t))-u(t)$ has the same asymptotic behavior (even when it diverges) as the sequence $v_n$ (resp. as the family $v_\lambda$) when $\bm{\lambda}(t)=1/t$ (resp. when $\bm{\lambda}(t)$ converges slowly enough to 0). Comment: 28 pages To appear in ESAIM:COCV |
| تدمد: | 1292-8119 1262-3377 |
| DOI: | 10.48550/arxiv.0904.2342 |
| URL الوصول: | https://explore.openaire.eu/search/publication?articleId=doi_dedup___::2e7a2c432272bfec01c8946c299c481b |
| Rights: | OPEN |
| رقم الانضمام: | edsair.doi.dedup.....2e7a2c432272bfec01c8946c299c481b |
| قاعدة البيانات: | OpenAIRE |
| تدمد: | 12928119 12623377 |
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| DOI: | 10.48550/arxiv.0904.2342 |