Advertisement

Théorèmes d’existence de solutions d’inclusions différentielles

  • Marlène Frigon
Chapter
Part of the NATO ASI Series book series (ASIC, volume 472)

Résumé

Dans ce texte, on présente quelques applications de méthodes topologiques permettant d’obtenir l’existence de solutions d’inclusions differentielles ordinaires. Trois types de fonctions multivoques sont distingués et un principe general d’existence de solutions est etabli pour chacun d’eux. Des résultats sont obtenus pour des systèmes d’inclusions différentielles du second ordre et pour des inclusions differentielles dans des espaces de Banach. Les principaux théorèmes obtenus découlent soit de théorèmes de point fixe, soit de la théorie de la transversalité topologique pour des operateurs compacts ou con-tractants, univoques ou multivoques.

Abstract

In this text, we present applications of topological methods to ordinary differential inclusions. Three types of multivalued functions are considered and a general existence principle is established for each of them. Results are obtained for second order systems of differential inclusions, and for differential inclusions in Banach spaces. Main theorems rely either on fixed point theorems or on topological transversality theories for compact or contractive, univalued or multivalued operators.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Références

  1. [1]
    Aubin, J.-P. et Cellina, A., Differential Inclusions, Springer-Verlag, Berlin, 1984.zbMATHCrossRefGoogle Scholar
  2. [2]
    Aubin, J.-P. et Frankowska, H., Set-Valued Analysis, Birkhäuser, Boston, 1990.zbMATHGoogle Scholar
  3. [3]
    Benedetto, J. J., Real Variable and Integration, Stuttgart, Teubner, 1976.zbMATHGoogle Scholar
  4. [4]
    Blagodatskikh, V. I. et Filippov, A. F., Differential inclusions and optimal control, Proc. Steklov Inst. Math. 4 (1986), 199–259.Google Scholar
  5. [5]
    Borisovich, Y. G., Gel’man, B. D., Myshkis, A. D. et Obukhovskii, V. V., Multivalued mappings, Itogi Nauki i Tekhniki, Ser. Mat. Anal. 19 (1982), 127–230 (Russian); translation: J. Soviet Math. 24 (1982), 719–791.MathSciNetGoogle Scholar
  6. [6]
    Bressan, A. et Colombo, G., Extensions and selections of maps with decomposable values, Studia Math. 90 (1988), 70–85.MathSciNetGoogle Scholar
  7. [7]
    Castaing, C., Sur les équations differentielles multivoques, C. R. Acad. Sci. Paris Sér. A 263 (1966), 63-66.MathSciNetzbMATHGoogle Scholar
  8. [8]
    Castaing, C. et Valadier, M., Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag, New York, 1977.zbMATHGoogle Scholar
  9. [9]
    Datko, R., On the integration of set-valued mappings in a Banach space, Fund. Math. 78 (1973), 205–208.MathSciNetzbMATHGoogle Scholar
  10. [10]
    Deimling, K., Multivalued Differential Equations, Walter de Gruyter, Berlin, 1992.zbMATHCrossRefGoogle Scholar
  11. [11]
    Dugundji, J. et Granas, A., Fixed Point Theory, vol. 1, PWN, Warszawa, 1982.Google Scholar
  12. [12]
    Erbe, L. H. et Krawcewicz, W., Existence of solutions to boundary value problems for impulsive second order differential inclusions, Rocky Mountain J. Math. 22 (1992), 1–20.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Erbe, L. H. et Krawcewicz, W., Nonlinear boundary value problems for differential inclusions y″F(t,y,y ,), Ann. Polon. Math. 54 (1991), 195–226.MathSciNetzbMATHGoogle Scholar
  14. [14]
    Erbe, L. H. et Palamides, P. K., Boundary value problems for second order differential systems, J. Math. Anal. Appl. 127 (1987), 80–92.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Frigon, M., Boundary et periodic value problems for systems of nonlinear second order differential equations, Topol. Methods Nonlinear Anal. 1 (1993), 259-274.MathSciNetzbMATHGoogle Scholar
  16. [16]
    Frigon, M., Boundary et periodic value problems for systems of differential equations under Bernstein-Nagumo growth condition, Differential Integral Equations (à paraitre).Google Scholar
  17. [17]
    Frigon, M., et Granas, A., Résultats du type de Leray-Schauder pour des contractions multivoques, Topol. Methods Nonlinear Anal. 4 (1994), 197–208.MathSciNetzbMATHGoogle Scholar
  18. [18]
    Frigon, M., et Granas, A., Problèmes aux limites pour des inclusions différentielles semi-continues inférieurement, Riv. Mat. Univ. Parma 17 (1991), 87–97.MathSciNetzbMATHGoogle Scholar
  19. [19]
    Frigon, M., et Granas, A., Theoremes d’existence pour des inclusions differentielles sans convexité, C. R. Acad. Sci. Paris Ser. I Math. 306 (1988), 747-750.MathSciNetzbMATHGoogle Scholar
  20. [20]
    Frigon, M., Granas, A., et Guennoun, Z., Sur l’intervalle maximal d’existence de solutions pour des inclusions differentielles, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), 819–822.zbMATHGoogle Scholar
  21. [21]
    Frigon, M., Granas, A., et Guennoun, Z., Alternative non-linéaire pour les applications contractantes, Ann. Sci. Math. Quebéc (à paraître).Google Scholar
  22. [22]
    Frigon, M. et Lee, J. W., Existence principles for Carathéodory differential equations in Banach spaces, Topol. Methods Nonlinear Anal 1 (1993), 95-111.MathSciNetzbMATHGoogle Scholar
  23. [23]
    Frigon, M. et O’Regan, D., Boundary value problems for second order impulsive differential equations using set-valued maps (soumis).Google Scholar
  24. [24]
    Gaprindashvili, G. D., Solvability of a Dirichlet boundary-value problem for systems of nonlinear ordinary differential equations with singularities, Differentsia’nye Uravneniya 27 (1991), 1521–1525 (Russian); translation: Differential Equations 27 (1991), 1074–1077.MathSciNetzbMATHGoogle Scholar
  25. [25]
    Granas, A., Continuation method for contractive maps, Topol. Methods Nonlinear Anal. 3 (1994), 375–379.MathSciNetzbMATHGoogle Scholar
  26. [26]
    Granas, A., On the Leray-Schauder alternative, Topol. Methods Nonlinear Anal. 2 (1993), 225–231.MathSciNetzbMATHGoogle Scholar
  27. [27]
    Granas, A., Guenther, R. B. et Lee, J. W., Some general existence principles in the Carathéodory theory of nonlinear differential systems, J. Math. Pures Appl. 70 (1991), 153–196.MathSciNetzbMATHGoogle Scholar
  28. [28]
    Granas, A., Guenther, R. B. et Lee, J. W., Nonlinear boundary value problems for ordinary differential equations, Dissertationes Math. (Rozprawy Mat.) 244 (1985).Google Scholar
  29. [29]
    Granas, A., Guenther, R. B. et Lee, J. W., Some existence results for the differential inclusions y (k) F(x, y,…, y (k-1)), y ∈ B, C. R. Acad. Sci. Paris Ser. I Math. 307 (1988), 391–396.MathSciNetzbMATHGoogle Scholar
  30. [30]
    Habets, P. et Schmitt, K., Nonlinear boundary value problems for systems of differential equations, Arch. Math. (Basel) 40 (1983), 441–446.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    Hartman, P., On boundary value problems for systems of ordinary, nonlinear, second order differential equations, Trans. Amer. Math. Soc. 96 (1960), 493–509.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    Hartman, P., Ordinary Differential Equations, Wiley, New York, 1964.zbMATHGoogle Scholar
  33. [33]
    Hiai, F. et Umegaki, H., Integrals, conditional expectations, and martingales of multivalued functions, J. Multivariate Anal. 7 (1977), 149–182.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    Himmelberg, C. J., Measurable relations, Fund. Math. 87 (1975), 53–72.MathSciNetzbMATHGoogle Scholar
  35. [35]
    Kuratowski, K. et Ryll-Nardzewski, C., A general theorem on selectors, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 13 (1965), 397–403.MathSciNetzbMATHGoogle Scholar
  36. [36]
    Lakshmikantham, V., Bainov, D. D. et Simeonov, P. S., Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.zbMATHCrossRefGoogle Scholar
  37. [37]
    Lasota, A. et Yorke, J. A., Existence of solutions of two-point boundary value problems for nonlinear systems, J. Differential Equations 11 (1972), 509–518.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    Lee, J. W. et O’Regan, D., Existence results for differential equations in Banach spaces, Comment. Math. Univ. Carolin. 34 (1993), 239–251.MathSciNetzbMATHGoogle Scholar
  39. [39]
    Lee, J. W. et O’Regan, D., Topological transversality: Applications to initial value problems, Ann. Polon. Math. 48 (1988), 31–36.MathSciNetGoogle Scholar
  40. [40]
    Liz, E. et Nieto, J. J., Periodic boundary value problems for second order impulsive integro-differential equations of Volterra and Fredholm type, pré-publication.Google Scholar
  41. [41]
    Nadler, S. B., Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 415–487.MathSciNetGoogle Scholar
  42. [42]
    Natanson, I. P., The Theory of Functions of Real Variable, Ungar, New York, 1955.Google Scholar
  43. [43]
    Ornelas, A., Approximation of relaxed solutions for lower semi-continuous differential inclusions, Ann. Polon. Math. 56 (1991), 1–10.MathSciNetzbMATHGoogle Scholar
  44. [44]
    Pruszko, T., Some applications of the topological degree theory to the multivalued boundary value problem, Dissertationes Math. (Rozprawy Mat.) 229 (1984).Google Scholar
  45. [45]
    Schmitt, K. et Thompson, R., Boundary value problems for infinite systems of second-order differential equations, J. Differential Equations 18 (1975), 277–295.MathSciNetzbMATHCrossRefGoogle Scholar
  46. [46]
    Wintner, A., The nonlocal existence problem for ordinary differential equations, Amer. J. Math. 67 (1945), 277–284.MathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    Yoshida, K., Functional Analysis, Springer-Verlag, Berlin, 1965.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Marlène Frigon
    • 1
  1. 1.Département de mathématiques et de statistiqueUniversité de MontréalMontréalCanada

Personalised recommendations