# Nonlinear eigenvalue problems for fourth order ordinary differential equations

• Volume: 60, Issue: 3, page 249-253
• ISSN: 0066-2216

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## Abstract

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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.

## How to cite

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Jolanta Przybycin. "Nonlinear eigenvalue problems for fourth order ordinary differential equations." Annales Polonici Mathematici 60.3 (1995): 249-253. <http://eudml.org/doc/262442>.

@article{JolantaPrzybycin1995,
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.},
author = {Jolanta Przybycin},
journal = {Annales Polonici Mathematici},
keywords = {bifurcation point; bifurcation interval; Leray-Schauder degree; characteristic value; nonlinear eigenvalue problem; Leray-Schauder degree theory; location of the eigenvalues},
language = {eng},
number = {3},
pages = {249-253},
title = {Nonlinear eigenvalue problems for fourth order ordinary differential equations},
url = {http://eudml.org/doc/262442},
volume = {60},
year = {1995},
}

TY - JOUR
AU - Jolanta Przybycin
TI - Nonlinear eigenvalue problems for fourth order ordinary differential equations
JO - Annales Polonici Mathematici
PY - 1995
VL - 60
IS - 3
SP - 249
EP - 253
AB - 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.
LA - eng
KW - bifurcation point; bifurcation interval; Leray-Schauder degree; characteristic value; nonlinear eigenvalue problem; Leray-Schauder degree theory; location of the eigenvalues
UR - http://eudml.org/doc/262442
ER -

## References

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1. [1] H. Berestycki, On some Sturm-Liouville problems, J. Differential Equations 26 (1977), 375-390. Zbl0331.34020
2. [2] J. Bochenek, Nodes of eigenfunctions of certain class of ordinary differential equations of the fourth order, Ann. Polon. Math. 29 (1975), 349-356. Zbl0316.34020
3. [3] R. Chiappinelli, On eigenvalues and bifurcation for nonlinear Sturm-Liouville operators, Boll. Un. Mat. Ital. (6) 4-A (1985), 77-83. Zbl0565.34016
4. [4] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966.
5. [5] J. Przybycin, Some applications of bifurcation theory to ordinary differential equations of the fourth order, Ann. Polon. Math. 53 (1991), 153-160. Zbl0729.34022
6. [6] P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161-202. Zbl0255.47069
7. [7] K. Schmitt and H. L. Smith, On eigenvalue problems for nondifferentiable mappings, J. Differential Equations 33 (1979), 294-319. Zbl0389.34019

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