Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Existence of solutions
for first order singular problems

Authors: M. Cherpion and C. De Coster
Journal: Proc. Amer. Math. Soc. 128 (2000), 1779-1791
MSC (1991): Primary 34A12, 34B15
Published electronically: December 8, 1999
MathSciNet review: 1694454
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We develop the lower and upper solutions method for first order initial value problems as well as for first order periodic problems in case the nonlinearity presents singularities. More precisely we prove that if we have a lower solution $\alpha$ and an upper solution $\beta$ of these problems, which are not necessarily continuous nor ordered, we have a solution wedged between $\min\{\alpha,\beta\}$ and $\max\{\alpha,\beta\}$.

References [Enhancements On Off] (What's this?)

  • 1. A. Adje, Existence et multiplicité des solutions d'équations différentielles ordinaires du premier ordre à non-linéarité discontinue, Annales de la Société Scientifique de Bruxelles 101 (1987), 69-87. MR 90h:34017
  • 2. A. Cabada, The monotone method for first order problems with linear and nonlinear boundary conditions, Appl. Math. and Comp. 63 (1994), 163-186. MR 95h:34032
  • 3. F. Cafiero, Su un problema ai limiti relativo all'equazione $y'=f(x,y,\lambda)$, Giornale di Mat. di Battaglini 77 (1947), 145-163. MR 10:194f
  • 4. C. De Coster, La méthode des sur et sous solutions dans l'étude de problèmes aux limites, Thèse de doctorat, Louvain-la-Neuve, 1994.
  • 5. C. De Coster, M. R. Grossinho and P. Habets, On pairs of positive solutions for a singular boundary value problem, Applicable Analysis 59 (1995), 241-256. MR 97d:34016
  • 6. M. Frigon and D. O'Regan, Existence results for some initial and boundary value problems without growth restriction, Proc. A.M.S. 123 (1995), 207-216. MR 95c:34035
  • 7. P. Habets and L. Sanchez, Periodic solutions of some Liénard equations with singularities, Proc. A.M.S. 109 (1990), 1035-1044. MR 90k:34049
  • 8. P. Habets and F. Zanolin, Upper and lower solutions for a generalized Emden-Fowler equation, J. Math. Anal. Appl. 181 (1994), 684-700. MR 95g:34034
  • 9. P. Habets and F. Zanolin, Positive solutions for a class of singular boundary value problem, Boll. U.M.I. 9A (1995), 273-286. MR 96j:34031
  • 10. J.K. Hale, Ordinary differential equations, Wiley-Intersciences, New-York (1969).
  • 11. I.T. Kiguradze and B.L. Shekhter, Singular boundary-value problems for ordinary second-order differential equations, Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya 30 (1987), 105-201, translated in J. Soviet Math. 43 (1988), 2340-2417. MR 89f:34022
  • 12. H.W. Knobloch, An existence theorem for periodic solutions on nonlinear ordinary differential equations, Michigan Math. J. 9 (1962), 303-309. MR 26:2694
  • 13. T.C. Lee and D. Willett, Second order singular boundary value problems, SIAM J. Math. Anal. 8 (1977), 741-755. MR 55:12988
  • 14. A.G. Lomtatidze, Positive solutions of boundary value problems for second order ODE with singular points, Diff. Urav. 23 (1987), 1685-1692. MR 89d:34040
  • 15. C. Marcelli and P. Rubbioni, A new extension of classical Müller's theorem, Nonlinear Anal. T.M.A. 28 (1997), 1759-1767. MR 97k:34003
  • 16. J. Mawhin, Nonlinear functional analysis and periodic solutions of ordinary differential equations, Summer School ``Difford 74", Stara Lesna, Czechoslovaquia (1974), 37-60.
  • 17. J. Mawhin, Nonlinear perturbations of Fredholm mappings in normed spaces and applications to differential equations, Univ. de Brasilia, Trabalho de Matem. 61 (1974).
  • 18. S. Moretto, Sull'esistenza di soluzioni periodiche per l'equazione $y'=f(x,y)$, Annali Univ. Ferrara 8 (1958-1959), 61-67. MR 22:6914
  • 19. M.N. Nkashama, A generalized upper and lower solutions method and multiplicity results for nonlinear first-order ordinary differential equations, J. Math. Anal. Appl. 140 (1989), 381-395. MR 90e:34006
  • 20. G. Peano, Sull'integrabilità delle equazioni differenziali di primo ordine, Atti Acad. Torino 21 (1885), 677-685.
  • 21. O. Perron, Ein neuer existenzbeweis für die integrale der differentialgleichung $y'=f(x,y)$, Math. Ann. 76 (1915), 471-484.
  • 22. G. Prodi, Teoremi di esistenza per equazioni alle derivate parziali non lineari di tipo parabolico I, II, Rend. Ist. Lombardo 86 (1953), 1-47. MR 16:259c
  • 23. G. Scorza Dragoni, Sugli integrali dei sistemi di equazioni differenziali, Rend. Ist. Lombardo de Sc. e Let. 64 (1931), 659-682.
  • 24. F. Tricomi, Equazioni differenziali 2e Ed., Torino (1953). MR 15:793a

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34A12, 34B15

Retrieve articles in all journals with MSC (1991): 34A12, 34B15

Additional Information

M. Cherpion
Affiliation: Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée, Chemin du Cyclotron 2, 1348 Louvain-La-Neuve, Belgique

C. De Coster
Affiliation: Université du Littoral - Côte d’Opale, Centre Universitaire de la Mi-Voix, 50 Rue F. Buisson, B.P. 699, 62228 Calais Cédex, France

Keywords: Singular first order initial value problem, singular first order periodic BVP, lower and upper solutions method
Received by editor(s): July 24, 1998
Published electronically: December 8, 1999
Communicated by: Hal L. Smith
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society