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1994-1995 | 60 | 3 | 249-253
Tytuł artykułu

Nonlinear eigenvalue problems for fourth order ordinary differential equations

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EN
Abstrakty
EN
This paper was inspired by the works of Chiappinelli ([3]) and Schmitt and Smith ([7]). We study the problem ℒu = λau + f(·,u,u',u'',u''') with separated boundary conditions on [0,π], where ℒ is a composition of two operators of Sturm-Liouville type. We assume that the nonlinear perturbation f satisfies the inequality |f(x,u,u',u'',u''')| ≤ M|u|. Because of the presence of f the considered equation does not in general have a linearization about 0. For this reason the global bifurcation theorem of Rabinowitz ([5], [6]) is not applicable here. We use the properties of Leray-Schauder degree to establish the existence of nontrivial solutions and describe their location. The results obtained are similar to those proved by Chiappinelli for Sturm-Liouville operators.
Rocznik
Tom
60
Numer
3
Strony
249-253
Opis fizyczny
Daty
wydano
1995
otrzymano
1993-10-18
poprawiono
1994-03-24
Twórcy
  • Institute of Mathematics, Academy of Mining and Metallurgy, al. Mickiewicza 30, 30-059 Kraków, Poland
Bibliografia
  • [1] H. Berestycki, On some Sturm-Liouville problems, J. Differential Equations 26 (1977), 375-390.
  • [2] J. Bochenek, Nodes of eigenfunctions of certain class of ordinary differential equations of the fourth order, Ann. Polon. Math. 29 (1975), 349-356.
  • [3] R. Chiappinelli, On eigenvalues and bifurcation for nonlinear Sturm-Liouville operators, Boll. Un. Mat. Ital. (6) 4-A (1985), 77-83.
  • [4] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966.
  • [5] J. Przybycin, Some applications of bifurcation theory to ordinary differential equations of the fourth order, Ann. Polon. Math. 53 (1991), 153-160.
  • [6] P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161-202.
  • [7] K. Schmitt and H. L. Smith, On eigenvalue problems for nondifferentiable mappings, J. Differential Equations 33 (1979), 294-319.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-apmv60z3p249bwm
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