Nonlinear eigenvalue problems for fourth order ordinary differential equations

• Autorzy: Jolanta Przybycin
• Języki publikacji: 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.

Słowa kluczowe

EN

bifurcation point; bifurcation interval; Leray-Schauder degree; characteristic value

Kategorie tematyczne

• 34B15: Nonlinear boundary value problems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1994-1995
• Tom: 60
• Numer: 3
• Strony: 249-253

Daty

wydano

1995

otrzymano

1993-10-18

poprawiono

1994-03-24

Twórcy

autor

Jolanta Przybycin

• 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.