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

L. H. Erbe; W. Krawcewicz

Annales Polonici Mathematici (1991)

  • Volume: 54, Issue: 3, page 195-226
  • ISSN: 0066-2216

Abstract

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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.

How to cite

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L. H. Erbe, and W. Krawcewicz. "Nonlinear boundary value problems for differential inclusions y'' ∈ F(t,y,y')." Annales Polonici Mathematici 54.3 (1991): 195-226. <http://eudml.org/doc/262452>.

@article{L1991,
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.},
author = {L. H. Erbe, W. Krawcewicz},
journal = {Annales Polonici Mathematici},
keywords = {boundary value problems; differential inclusion; topological transversality; nonlinear boundary conditions; Carathéodory multifunction; topological transversality method; a priori bounds technique},
language = {eng},
number = {3},
pages = {195-226},
title = {Nonlinear boundary value problems for differential inclusions y'' ∈ F(t,y,y')},
url = {http://eudml.org/doc/262452},
volume = {54},
year = {1991},
}

TY - JOUR
AU - L. H. Erbe
AU - W. Krawcewicz
TI - Nonlinear boundary value problems for differential inclusions y'' ∈ F(t,y,y')
JO - Annales Polonici Mathematici
PY - 1991
VL - 54
IS - 3
SP - 195
EP - 226
AB - 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.
LA - eng
KW - boundary value problems; differential inclusion; topological transversality; nonlinear boundary conditions; Carathéodory multifunction; topological transversality method; a priori bounds technique
UR - http://eudml.org/doc/262452
ER -

References

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  1. [1] J.-P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin 1984. 
  2. [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. Zbl0253.34020
  3. [3] J.-M. Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris 265 (1967), 333-336. Zbl0164.16803
  4. [4] H. Brezis, Analyse fonctionnelle, Théorie et applications, Masson, Paris 1982. 
  5. [5] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer, 1977. 
  6. [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. Zbl0511.34039
  7. [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. Zbl0442.34016
  8. [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. [9] K. C. Chang, The obstacle problems and partial differential equations with discontinuous nonlinearities, Comm. Pure Appl. Math. 33 (1980), 117-146. Zbl0405.35074
  10. [10] J. Dugundji and A. Granas, Fixed Point Theory, Vol. 1, PWN, Warszawa 1982. 
  11. [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. Zbl0582.34024
  12. [12] L. H. Erbe, W. Krawcewicz and S. Chen, Some existence results for solutions of differential inclusions with retardations, this issue, 227-239. Zbl0731.34079
  13. [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. Zbl0635.34017
  14. [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. Zbl0635.34016
  15. [15] C. Fabry, Nagumo conditions for systems of second order differential equations, J. Math. Anal. Appl. 107 (1985), 132-143. Zbl0604.34002
  16. [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. Zbl0439.34018
  17. [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). Zbl0728.34017
  18. [18] R. E. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math. 568, Springer, 1977. Zbl0339.47031
  19. [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. Zbl0585.47049
  20. [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. Zbl0092.32302
  21. [21] A. Granas, Sur la méthode de continuité de Poincaré, C. R. Acad. Sci. Paris 287 (1976), 983-985. 
  22. [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. [23] A. Granas, R. B. Guenther and J. W. Lee, On a theorem of S. Bernstein, Pacific. J. Math. 74 (1978), 78-82. 
  24. [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. Zbl0476.34017
  25. [25] A. Granas, R. B. Guenther and J. W. Lee, Nonlinear boundary value problems for ordinary differential equations, Dissertationes Math. 244 (1981). 
  26. [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. Zbl0534.34006
  27. [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. Zbl0594.34019
  28. [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. Zbl0539.34031
  29. [29] P. Hartman, Ordinary Differential Equations, Wiley, New York 1964. Zbl0125.32102
  30. [30] T. Kaczyński, Topological transversality and nonlinear equations in locally convex spaces, 1987, preprint. 
  31. [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. [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. Zbl0368.34009
  33. [33] W. Krawcewicz, Contribution à la théorie des équations non linéaires dans les espaces de Banach, Dissertationes Math. 273 (1988). Zbl0677.47038
  34. [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. Zbl0573.35029
  35. [35] T. Pruszko, Topological degree methods in multivalued boundary value problems, Nonlinear Anal. 5 (9) (1981), 953-973. 
  36. [36] T. Pruszko, Some applications of the topological degree theory to multivalued boundary value problems, Dissertationes Math. 229 (1984). Zbl0543.34008
  37. [37] K. Schmitt, Boundary value problems for quasilinear second order elliptic equations, Nonlinear Anal. 2 (1978), 263-309. Zbl0378.35022
  38. [38] C. A. Stuart, Differential equations with discontinuous nonlinearities, Arch. Rational Mech. Anal. 63 (1976), 59-75. Zbl0393.34010
  39. [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. Zbl0443.34060

Citations in EuDML Documents

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  1. Ralf Bader, Nikolaos S. Papageorgiou, Nonlinear multivalued boundary value problems
  2. Smaïl Djebali, Abdelghani Ouahab, Existence results for ϕ-Laplacian Dirichlet BVP of differential inclusions with application to control theory
  3. S. Carl, S. Heikkilä, On discontinuous implicit differential equations in ordered Banach spaces with discontinuous implicit boundary conditions
  4. Zdzisław Dzedzej, Equivariant degree of convex-valued maps applied to set-valued BVP
  5. Christopher C. Tisdell, Systems of differential inclusions in the absence of maximum principles and growth conditions

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