# 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

## Access Full Article

top## Abstract

top## How to cite

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 -

## References

top- H. Attouch and R. Cominetti, A dynamical approach to convex minimization coupling approximation with the steepest descent method. J. Differ. Equ.128 (1996) 269–275. Zbl0886.49024
- R.J. Aumann and M. Maschler with the collaboration of R.E. Stearns, Repeated Games with Incomplete Information. MIT Press (1995).
- V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff International Publishing (1976). Zbl0328.47035
- T. Bewley and E. Kohlberg, The asymptotic theory of stochastic games. Math. Oper. Res.1 (1976) 197–208. Zbl0364.93031
- T. Bewley and E. Kohlberg, The asymptotic solution of a recursion equation occurring in stochastic games. Math. Oper. Res.1 (1976) 321–336. Zbl0364.93032
- H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Mathematical Studies5. North Holland (1973). Zbl0252.47055
- M.G. Crandall and T.M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces. Amer. J. Math.93 (1971) 265–298. Zbl0226.47038
- H. Everett, Recursive Games, in Contributions to the Theory of Games3, H.W. Kuhn and A.W. Tucker Eds., Princeton University Press (1957) 47–78.
- S. Gaubert and J. Gunawardena, The Perron-Frobenius Theorem for homogeneous, monotone functions. T. Am. Math. Soc.356 (2004) 4931–4950. Zbl1067.47064
- J. Gunawardena, From max-plus algebra to nonexpansive maps: a nonlinear theory for discrete event systems. Theor. Comput. Sci.293 (2003) 141–167. Zbl1036.93045
- J. Gunawardena and M. Keane, On the existence of cycle times for some nonexpansive maps. Technical Report HPL-BRIMS-95-003 Ed., Hewlett-Packard Labs (1995).
- T. Kato, Nonlinear semi-groups and evolution equations. J. Math. Soc. Japan19 (1967) 508–520. Zbl0163.38303
- Y. Kobayashi, Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups. J. Math Soc. Japan27 (1975) 640–665. Zbl0313.34068
- E. Kohlberg, Repeated games with absorbing states. Ann. Stat.2 (1974) 724–738. Zbl0297.90114
- E. Kohlberg and A. Neyman, Asymptotic behavior of nonexpansive mappings in normed linear spaces. Israel J. Math.38 (1981) 269–275. Zbl0476.47045
- E. Lehrer and S. Sorin, A uniform Tauberian theorem in dynamic programming. Math. Oper. Res.17 (1992) 303–307. Zbl0771.90099
- I. Miyadera and S. Oharu, Approximation of semi-groups of nonlinear operators. Tôhoku Math. J.22 (1970) 24–47. Zbl0195.15001
- J.-J. Moreau, Propriétés des applications “prox”. C. R. Acad. Sci. Paris256 (1963) 1069–1071. Zbl0115.10802
- A. Neyman, Stochastic games and nonexpansive maps, in Stochastic Games and Applications, A. Neyman and S. Sorin Eds., Kluwer Academic Publishers (2003) 397–415. Zbl1093.91005
- A. Neyman and S. Sorin, Repeated games with public uncertain duration process. (Submitted). Zbl1211.91063
- S. Reich, Asymptotic behavior of semigroups of nonlinear contractions in Banach spaces. J. Math. Anal. Appl.53 (1976) 277–290. Zbl0337.47027
- J. Renault, The Value of Markov Chain Games with Lack of Information on One Side. Math. Oper. Res.31 (2006) 490–512. Zbl1276.91023
- R. Rockafellar, Convex Analysis. Princeton University Press (1970). Zbl0193.18401
- D. Rosenberg and S. Sorin, An operator approach to zero-sum repeated games. Israel J. Math.121 (2001) 221–246. Zbl1054.91014
- S. Sorin, A First Course on Zero-Sum Repeated Games. Springer (2002). Zbl1005.91019
- S. Sorin, Asymptotic properties of monotonic nonexpansive mappings. Discrete Events Dynamical Systems14 (2004) 109–122. Zbl1035.93047
- W. Walter, Differential and Integral Inequalities. Springer-Verlag (1970).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.