ArticleOriginal scientific text
Title
Nonlinear eigenvalue problems for fourth order ordinary differential equations
Authors 1
Affiliations
- Institute of Mathematics, Academy of Mining and Metallurgy, al. Mickiewicza 30, 30-059 Kraków, Poland
Abstract
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.
Keywords
bifurcation point, bifurcation interval, Leray-Schauder degree, characteristic value
Bibliography
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- J. Bochenek, Nodes of eigenfunctions of certain class of ordinary differential equations of the fourth order, Ann. Polon. Math. 29 (1975), 349-356.
- R. Chiappinelli, On eigenvalues and bifurcation for nonlinear Sturm-Liouville operators, Boll. Un. Mat. Ital. (6) 4-A (1985), 77-83.
- T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966.
- J. Przybycin, Some applications of bifurcation theory to ordinary differential equations of the fourth order, Ann. Polon. Math. 53 (1991), 153-160.
- P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161-202.
- K. Schmitt and H. L. Smith, On eigenvalue problems for nondifferentiable mappings, J. Differential Equations 33 (1979), 294-319.
Pages:
249-253
Main language of publication
English
Received
1993-10-18
Accepted
1994-03-24
Published
1995
Exact and natural sciences