Advertisement

Annali di Matematica Pura ed Applicata

, Volume 115, Issue 1, pp 99–117 | Cite as

Monotone trajectories of multivalued dynamical systems

  • Jean-Pierre Aubin
  • Arrigo Cellina
  • John Nohel
Article

Summary

We prove existence of « monotone trajectories » for a class of discrete and continuous systems sufficiently general to include problems of some interest in economic and biological theory. We prove existence of critical points which are Pareto minima. We study stability properties of Pareto minima.

Keywords

Dynamical System Stability Property Continuous System Biological Theory Multivalued Dynamical System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Antosiewicz H. A. -Cellina A.,Continuous selections and differential relations, J. Diff. Eq.,19 (1975), pp. 386–398.CrossRefMathSciNetGoogle Scholar
  2. Arrow K. - Hahn F. H.,General competitive analysis, Holden-Day (1971).Google Scholar
  3. Aubin J.-P.,Mathematical methods of game and economic theory, North-Holland (to appear).Google Scholar
  4. Aubin J.-P. - Clarke F. H.,Monotone invariant solutions to differential inclusion, J. London Math. Soc., to appear.Google Scholar
  5. Brauer F. - Nohel J.,Qualitative theory of ordinary differential equations, Benjamin (1969).Google Scholar
  6. Brözis H.,Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland (1973).Google Scholar
  7. Browder F.,The fixed point theory of multivalued mappings in topological vector spaces, Math. Annalen,177 (1968), pp. 283–301.CrossRefzbMATHMathSciNetGoogle Scholar
  8. Castaing C. -Valadier M.,Equations différentielles multivoques dans les espaces localement convexes, Revue Française Informatique Recherche Opérationnelle,16 (1969), pp. 3–16.MathSciNetGoogle Scholar
  9. Cellina A.,Approximation of set valued functions, Ann. di Mat. pura e appl.,82 (1969), pp. 17–24.CrossRefzbMATHMathSciNetGoogle Scholar
  10. Champsaur P.,Neutrality of planning procedures in an economy with public goods, Core discussion paper 7410.Google Scholar
  11. Champsaur P. - Dreze J. - Henry C.,Dynamic processes in economic theory, Core discussion paper 7417.Google Scholar
  12. Clarke F. H.,Generalized gradients and applications, T.A.M.S.,205 (1975), pp. 247–262.CrossRefzbMATHGoogle Scholar
  13. Cornet B. [1],Fixed point and surjectivity theorems for correspondences; applications, Cahiers de Mathématiques de la Décision no. 75-21.Google Scholar
  14. Cornet B. [2],Paris avec handicaps et théorèmes de surjectivité de correspondances, C. R. Ac. Sc.,281 (1975), pp. 479–482.zbMATHMathSciNetGoogle Scholar
  15. Crandall M. G.,A generalization of Peano's existence theorem and flow invariance, Proc. A.M.S.,36 (1972), pp. 151–155.CrossRefzbMATHMathSciNetGoogle Scholar
  16. Day R. H.,Adaptative processes and economic theory, SSRI 7514, Univ. of Wisconsin.Google Scholar
  17. Dreze J. -De la Vallée Poussin,A tâtonnement process for public goods, Revie of Economic Studies,38 (1971), pp. 133–150.Google Scholar
  18. Henry C. [1],An existence theory for a class of differential equations with multivalued right-hand side, J. Math. Anal. Appl.,41 (1973), pp. 178–186.CrossRefMathSciNetGoogle Scholar
  19. Henry C. [2],Problèmes d'existence et de stabilité pour des processus dynamiques considérés en économie mathématique, C. R. Ac. Sc.,278 (1974), pp. 97–100.zbMATHMathSciNetGoogle Scholar
  20. Kalai G. - Maschler M. - Owen G.,Asymptotic stability and other properties of trajectories and transfer sequences leading to bargaining sets, Int. J. of Game theory (to appear).Google Scholar
  21. Ky Fan,A minimax inequality and applications, inInequalities III, Shishia Ed., Academic Press (1972), pp. 103–113.Google Scholar
  22. La Salle J. P.,Vector Liapunov functions (to appear).Google Scholar
  23. Lasry J. M. - Robert R. (to appear).Google Scholar
  24. Malinvaud E.,Procedures for the determination of a program of collective consumption, European Economic Review, 1970–1971 (Winter), pp. 187–217.Google Scholar
  25. Maschler M. -Peleg B.,Stable sets and stable points of set valued dynamic systems, SIAM J. Control and Optimization,14 (1976), pp. 985–995.CrossRefMathSciNetGoogle Scholar
  26. Moreau J. J.,Râfle par un convexe variable, Séminaire Analyse Convexe, Montpellier., 1971, exp. no. 15 et 1972, exp. no. 3.Google Scholar
  27. Nagumo N.,Über die Laga der Integralkurven gewöhnlichen Differentialgleichungen, Proc. Phys. Math. Soc. Japan,24 (1942), pp. 551–559.zbMATHMathSciNetGoogle Scholar
  28. Smale S.,An approach to the analysis of dynamic processes in economic systems (to appear).Google Scholar
  29. Uzawa H.,The stability of dynamic processes, Econometrica,29 (1961), pp. 617–631.zbMATHMathSciNetGoogle Scholar
  30. Valadier M.,Existence globale pour les équations différentielles multivoques, C. R. Acad. Sc.,272 (1971), pp. 474–477.zbMATHMathSciNetGoogle Scholar
  31. Yoshizawa T.,Stability theory and the existence of periodic solutions and almost periodic solutions, Springer-Verlag (1975), Appl. Math. Sc.,14.Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1977

Authors and Affiliations

  • Jean-Pierre Aubin
    • 1
  • Arrigo Cellina
    • 1
  • John Nohel
    • 1
  1. 1.ParisFrance

Personalised recommendations