Hostname: page-component-5c6d5d7d68-pkt8n Total loading time: 0 Render date: 2024-08-14T20:27:41.315Z Has data issue: false hasContentIssue false
  • English
  • Français

Positive solutions for a class of semilinear two-point boundary value problems

Published online by Cambridge University Press:  17 April 2009

Luis Sanchez
Luis Sanchez
Affiliation:
Instituto Nacional de Investigacao CientificaCentro de Matematica e Aplicacoes Fundamentais 1699 Lisboa CodexPortugal
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the existence of positive solutions of the periodic, Neumann or Dirichlet problem for the semilinear equation u″ + f(t, u) = 0, 0 ≤ t ≤ T, where f is a Carathéodory function. Our assumptions in each case are such that the problem possesses a lower solution or an upper solution.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 45 , Issue 3 , June 1992 , pp. 439 - 451
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Amann, H., ‘Fixed point equations and nonlinear eignevalue problems in ordered Banach spaces’, SIAM Review 18 (1976), 620–709.Google Scholar
[2]Ambrosetti, A. and Rabinowitz, P.H., ‘Dual variational methods in critical point theory and applications’, J. Funct. Anal. 14 (1973), 349–381.CrossRefGoogle Scholar
[3]Brézis, H. and Oswald, L., ‘Remarks on sublinear elliptic equations’, Nonlinear Anal. T.M.A. 10 (1986), 55–64.Google Scholar
[4]Castro, A. and Shivaji, R., ‘Non-negative solutions for a class of non-positone problems’, Proc. Roy. Soc. Edinburgh 108A (1988), 291–302.Google Scholar
[5]Costa, D. and Gonçalves, J.V., ‘On the existence of positive solutions for a class of non-selfadjoint elliptic boundary value problems’, Appl. Anal. 31 (1989), 309–320.Google Scholar
[6]Figueiredo, D.G., ‘Positive solutions for some classes of semilinear elliptic problems’, in Proc. Symp. Pure Mathematics 45, Part 1, Editor F. Browder, pp. 371–379, 1986.CrossRefGoogle Scholar
[7]Habets, P., Ramos, M. and Sanchez, L., ‘Jumping non-linearity for 2nd order ODE with positive forcing’, in Delay Differential Equations and Dynamical Systems: Lecture Notes in Mathematics 1475, Proceedings Claremont 1990, Editors S. Busenbert and M. Martelli, pp. 191–203, 1991.CrossRefGoogle Scholar
[8]Lazer, A.C. and Solimini, S., ‘On periodic solutions of nonlinear differential equations with singularities’, Proc. Amer. Math. Soc. 99 (1987), 109–114.Google Scholar
[9]Mawhin, J., ‘Problèmes de Dirichlet variationels non linéaires’, in Séminaire de Mathématiques Supérieures 104 (Les Presses de l'Université de Montréal, 1987).Google Scholar
[10]Manes, A. and Micheletti, A.M., ‘Un'estensione della teoria variazionale classica degli autovalori per opperatori ellittici del secondo ordine’, Boll. Un. Mat. Ital. (4) 7 (1973), 285–301.Google Scholar
[11]Nkashama, M.N. and Santanilla, J., ‘Existence of multiple solutions for some nonlinear boundary value problems’, J. Differential Equations 84 (1990), 148–164.Google Scholar
[12]Njoku, and Zanolin, F., ‘Positive solutions for two-point BVP's: existence and multiplicity results’, Nonlinear Anal. T.M.A. 13 (1989), 1329–1338.CrossRefGoogle Scholar
[13]Rabinowitz, P.H., ‘Minimax methods in critical point theory with applications to differential equations’, in CBMS Reg. Conf. Ser. in Math. 65 (Amer. Math. Soc., Providence RI, 1986).Google Scholar
[14]Ramos, M. and Sanchez, L., ‘Variational elliptic problems involving noncoercive functionals’, Proc. Roy. Soc. Edinburgh 112A (1989), 177–185.CrossRefGoogle Scholar
[15]Sanchez, L., ‘Boundary value problems for some fourth order ordinary differential equations’, Appl. Anal. 38 (1990), 161–177.CrossRefGoogle Scholar
[16]Schaaf, R. and Schmitt, K., ‘A class of nonlinear Sturm-Liouville problems with infinitely many solutions’, Trans. Amer. Math. Soc. 306 (1988), 853–859.Google Scholar
[17]Smoller, J. and Wasserman, A., ‘Existence of positive solutions for semilinear elliptic equations in general domains’, Arch. Rational Mech. Anal. 95 (1986), 217–225.CrossRefGoogle Scholar
[18]Zanolin, F., ‘Remarks on multiple periodic solutions for non-linear ordinary differential systems of Liénard type’, Boll. Un. Mat. Ital (6) 1-B (1982), 683–698.Google Scholar