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 v_{n} = 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 v_{n} (resp. as the family ${v}_{\lambda}$) when λ(t) = 1/t (resp. when λ(t) converges slowly enough to 0).

Keywords: Banach spaces, nonexpansive mappings, evolution equations, asymptotic behavior, Shapley operator

@article{COCV_2010__16_4_809_0, author = {Vigeral, Guillaume}, title = {Evolution equations in discrete and continuous time for nonexpansive operators in {Banach} spaces}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {809--832}, publisher = {EDP-Sciences}, volume = {16}, number = {4}, year = {2010}, doi = {10.1051/cocv/2009026}, mrnumber = {2744152}, zbl = {1204.47091}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2009026/} }

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 PY - 2010 SP - 809 EP - 832 VL - 16 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2009026/ DO - 10.1051/cocv/2009026 LA - en ID - COCV_2010__16_4_809_0 ER -

%0 Journal Article %A Vigeral, Guillaume %T Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 809-832 %V 16 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2009026/ %R 10.1051/cocv/2009026 %G en %F COCV_2010__16_4_809_0

Vigeral, Guillaume. Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, pp. 809-832. doi : 10.1051/cocv/2009026. http://archive.numdam.org/articles/10.1051/cocv/2009026/

[1] A dynamical approach to convex minimization coupling approximation with the steepest descent method. J. Differ. Equ. 128 (1996) 269-275. | Zbl

and ,[2] Zbl

and with the collaboration of R.E. Stearns, Repeated Games with Incomplete Information. MIT Press (1995). |[3] Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff International Publishing (1976). | Zbl

,[4] The asymptotic theory of stochastic games. Math. Oper. Res. 1 (1976) 197-208. | Zbl

and ,[5] The asymptotic solution of a recursion equation occurring in stochastic games. Math. Oper. Res. 1 (1976) 321-336. | Zbl

and ,[6] Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Mathematical Studies 5. North Holland (1973). | Zbl

,[7] Generation of semi-groups of nonlinear transformations on general Banach spaces. Amer. J. Math. 93 (1971) 265-298. | Zbl

and ,[8] Recursive Games, in Contributions to the Theory of Games 3, H.W. Kuhn and A.W. Tucker Eds., Princeton University Press (1957) 47-78. | Zbl

,[9] The Perron-Frobenius Theorem for homogeneous, monotone functions. T. Am. Math. Soc. 356 (2004) 4931-4950. | Zbl

and ,[10] From max-plus algebra to nonexpansive maps: a nonlinear theory for discrete event systems. Theor. Comput. Sci. 293 (2003) 141-167. | Zbl

,[11] On the existence of cycle times for some nonexpansive maps. Technical Report HPL-BRIMS-95-003 Ed., Hewlett-Packard Labs (1995).

and ,[12] Nonlinear semi-groups and evolution equations. J. Math. Soc. Japan 19 (1967) 508-520. | Zbl

,[13] Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups. J. Math Soc. Japan 27 (1975) 640-665. | Zbl

,[14] Repeated games with absorbing states. Ann. Stat. 2 (1974) 724-738. | Zbl

,[15] Asymptotic behavior of nonexpansive mappings in normed linear spaces. Israel J. Math. 38 (1981) 269-275. | Zbl

and ,[16] A uniform Tauberian theorem in dynamic programming. Math. Oper. Res. 17 (1992) 303-307. | Zbl

and ,[17] Approximation of semi-groups of nonlinear operators. Tôhoku Math. J. 22 (1970) 24-47. | Zbl

and ,[18] Propriétés des applications “prox”. C. R. Acad. Sci. Paris 256 (1963) 1069-1071. | Zbl

,[19] Stochastic games and nonexpansive maps, in Stochastic Games and Applications, A. Neyman and S. Sorin Eds., Kluwer Academic Publishers (2003) 397-415. | Zbl

,[20] Repeated games with public uncertain duration process. (Submitted).

and ,[21] Asymptotic behavior of semigroups of nonlinear contractions in Banach spaces. J. Math. Anal. Appl. 53 (1976) 277-290. | Zbl

,[22] The Value of Markov Chain Games with Lack of Information on One Side. Math. Oper. Res. 31 (2006) 490-512.

,[23] Convex Analysis. Princeton University Press (1970). | Zbl

,[24] An operator approach to zero-sum repeated games. Israel J. Math. 121 (2001) 221-246. | Zbl

and ,[25] A First Course on Zero-Sum Repeated Games. Springer (2002). | Zbl

,[26] Asymptotic properties of monotonic nonexpansive mappings. Discrete Events Dynamical Systems 14 (2004) 109-122. | Zbl

,[27] Differential and Integral Inequalities. Springer-Verlag (1970). | Zbl

,*Cited by Sources: *