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Title

Nonlinear boundary value problems for differential inclusions y'' ∈ F(t,y,y')

Authors 1, 1

Affiliations

  1. Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada

Abstract

Applying the topological transversality method of Granas and the a priori bounds technique we prove some existence results for systems of differential inclusions of the form y'' ∈ F(t,y,y'), where F is a Carathéodory multifunction and y satisfies some nonlinear boundary conditions.

Keywords

ENG
boundary value problems, differential inclusion, topological transversality

Bibliography

  1. J.-P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin 1984.
  2. J. W. Bebernes and K. Schmitt, Periodic boundary value problems for systems of second order differential equations, J. Differential Equations 13 (1973), 33-47.
  3. J.-M. Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris 265 (1967), 333-336.
  4. H. Brezis, Analyse fonctionnelle, Théorie et applications, Masson, Paris 1982.
  5. C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer, 1977.
  6. M. Cecchi, M. Marini and P. Zecca, Asymptotic properties of the solutions of nonlinear equations with dichotomies and applications, Boll. Un. Mat. Ital. (6) 1-C (1982), 209-234.
  7. M. Cecchi, M. Marini and P. Zecca, Linear boundary value problems for systems of ordinary differential equations on non-compact intervals, I, II, Ann. Mat. Pura Appl. (4) 123 (1980), 267-285; 134 (1980), 367-379.
  8. A. Cellina and A. Lasota, A new approach to the definition of topological degree for multivalued mappings, Rend. Accad. Naz. Lincei 47 (1969), 434-440.
  9. K. C. Chang, The obstacle problems and partial differential equations with discontinuous nonlinearities, Comm. Pure Appl. Math. 33 (1980), 117-146.
  10. J. Dugundji and A. Granas, Fixed Point Theory, Vol. 1, PWN, Warszawa 1982.
  11. L. H. Erbe and H. W. Knobloch, Boundary value problems for systems of second order differential equations, Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 61-76.
  12. L. H. Erbe, W. Krawcewicz and S. Chen, Some existence results for solutions of differential inclusions with retardations, this issue, 227-239.
  13. L. H. Erbe and P. K. Palamides, Boundary value problems for second-order differential systems, J. Math. Anal. Appl. 127 (1) (1987), 80-92.
  14. L. H. Erbe and K. Schmitt, On solvability of boundary value problems for systems of differential equations, J. Appl. Math. Phys. 38 (1987), 184-192.
  15. C. Fabry, Nagumo conditions for systems of second order differential equations, J. Math. Anal. Appl. 107 (1985), 132-143.
  16. C. Fabry and P. Habets, The Picard boundary value problem for nonlinear second order vector differential equations, J. Differential Equations 42 (2) (1987), 186-198.
  17. M. Frigon, Application de la théorie de la transversalité topologique à des problèmes non linéaires pour des équations différentielles ordinaires, Dissertationes Math. 296 (1990).
  18. R. E. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math. 568, Springer, 1977.
  19. K. Gęba, A. Granas, T. Kaczyński et W. Krawcewicz, Homotopie et équations non linéaires dans les espaces de Banach, C. R. Acad. Sci. Paris Sér. I Math. 300 (10) (1985), 303-306.
  20. A. Granas, Homotopy extension theorem in Banach spaces and some of its applications to the theory of non-linear equations, Bull. Acad. Polon. Sci. 7 (1959), 387-394.
  21. A. Granas, Sur la méthode de continuité de Poincaré, C. R. Acad. Sci. Paris 287 (1976), 983-985.
  22. A. Granas et Zine el Abdine Guennoun, Quelques résultats dans la théorie de Bernstein-Carathéodory de l'équation y'' = f(t,y,y'), C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), 703-706.
  23. A. Granas, R. B. Guenther and J. W. Lee, On a theorem of S. Bernstein, Pacific. J. Math. 74 (1978), 78-82.
  24. A. Granas, R. B. Guenther and J. W. Lee, Nonlinear boundary value problems for some classes of ordinary differential equations, Rocky Mountain J. Math. 10 (1980), 35-58.
  25. A. Granas, R. B. Guenther and J. W. Lee, Nonlinear boundary value problems for ordinary differential equations, Dissertationes Math. 244 (1981).
  26. A. Granas, R. B. Guenther and J. W. Lee, Topological transversality II; Applications to the Neumann problem for y'' = f(t,y,y'), Pacific J. Math. 104 (1983), 95-109.
  27. A. Granas, R. B. Guenther, J. W. Lee and D. O'Regan, Topological transversality III; Applications to the boundary value problems on infinite intervals and semiconductor devices, J. Math. Anal. Appl. 116 (1986), 335-348.
  28. J. Haddad and J. M. Lasry, Periodic solutions of functional differential inclusions and fixed points of G-selectionable correspondences, J. Math. Anal. Appl. 110 (1983), 295-312.
  29. P. Hartman, Ordinary Differential Equations, Wiley, New York 1964.
  30. T. Kaczyński, Topological transversality and nonlinear equations in locally convex spaces, 1987, preprint.
  31. H. W. Knobloch, Boundary value problems for systems of nonlinear differential equations, in: Proc. Equadiff IV 1977, Lecture Notes in Math. 703, Springer, 1978, 197-204.
  32. H. W. Knobloch and K. Schmitt, Nonlinear boundary value problems for systems of differential equations, Proc. Roy. Soc. Edinburgh Sect. A 78 (1977), 139-159.
  33. W. Krawcewicz, Contribution à la théorie des équations non linéaires dans les espaces de Banach, Dissertationes Math. 273 (1988).
  34. J. Mawhin and K. Schmitt, Upper and lower solutions and semilinear second order elliptic equations with non-linear boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 199-207.
  35. T. Pruszko, Topological degree methods in multivalued boundary value problems, Nonlinear Anal. 5 (9) (1981), 953-973.
  36. T. Pruszko, Some applications of the topological degree theory to multivalued boundary value problems, Dissertationes Math. 229 (1984).
  37. K. Schmitt, Boundary value problems for quasilinear second order elliptic equations, Nonlinear Anal. 2 (1978), 263-309.
  38. C. A. Stuart, Differential equations with discontinuous nonlinearities, Arch. Rational Mech. Anal. 63 (1976), 59-75.
  39. P. Zecca and P. L. Zecca, Nonlinear boundary value problems in Banach space for multivalued differential equations on a noncompact interval, Nonlinear Anal. 3 (1979), 347-352.
Annales Polonici Mathematici
1991/54/3
Export to PBN, Opens in new tab
Pages:
195-226
Main language of publication
English
Received
1988-08-03
Accepted
1989-02-17
Published
1991
Exact and natural sciences