# On bifurcation intervals for nonlinear eigenvalue problems

Annales Polonici Mathematici (1999)

- Volume: 71, Issue: 1, page 39-46
- ISSN: 0066-2216

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topJolanta Przybycin. "On bifurcation intervals for nonlinear eigenvalue problems." Annales Polonici Mathematici 71.1 (1999): 39-46. <http://eudml.org/doc/262735>.

@article{JolantaPrzybycin1999,

abstract = {We give a sufficient condition for [μ-M, μ+M] × \{0\} to be a bifurcation interval of the equation u = L(λu + F(u)), where L is a linear symmetric operator in a Hilbert space, μ ∈ r(L) is of odd multiplicity, and F is a nonlinear operator. This abstract result provides an elementary proof of the existence of bifurcation intervals for some eigenvalue problems with nondifferentiable nonlinearities. All the results obtained may be easily transferred to the case of bifurcation from infinity.},

author = {Jolanta Przybycin},

journal = {Annales Polonici Mathematici},

keywords = {bifurcation interval; symmetric operator; Sturm-Liouville problem; Dirichlet problem; Leray-Schauder degree; characteristic values; odd multiplicity; nondifferentiable nonlinearities; bifurcation from infinity},

language = {eng},

number = {1},

pages = {39-46},

title = {On bifurcation intervals for nonlinear eigenvalue problems},

url = {http://eudml.org/doc/262735},

volume = {71},

year = {1999},

}

TY - JOUR

AU - Jolanta Przybycin

TI - On bifurcation intervals for nonlinear eigenvalue problems

JO - Annales Polonici Mathematici

PY - 1999

VL - 71

IS - 1

SP - 39

EP - 46

AB - We give a sufficient condition for [μ-M, μ+M] × {0} to be a bifurcation interval of the equation u = L(λu + F(u)), where L is a linear symmetric operator in a Hilbert space, μ ∈ r(L) is of odd multiplicity, and F is a nonlinear operator. This abstract result provides an elementary proof of the existence of bifurcation intervals for some eigenvalue problems with nondifferentiable nonlinearities. All the results obtained may be easily transferred to the case of bifurcation from infinity.

LA - eng

KW - bifurcation interval; symmetric operator; Sturm-Liouville problem; Dirichlet problem; Leray-Schauder degree; characteristic values; odd multiplicity; nondifferentiable nonlinearities; bifurcation from infinity

UR - http://eudml.org/doc/262735

ER -

## References

top- [1] H. Berestycki, On some Sturm-Liouville problems, J. Differential Equations 26 (1977), 375-390. Zbl0331.34020
- [2] R. Chiappinelli, On eigenvalues and bifurcation for nonlinear Sturm-Liouville operators, Boll. Un. Mat. Ital. A (6) 4 (1985), 77-83. Zbl0565.34016
- [3] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966.
- [4] L. Nirenberg, Topics in Nonlinear Functional Analysis, New York Univ. Lecture Notes, 1973-74.
- [5] J. Przybycin, Nonlinear eigenvalue problems for fourth order ordinary differential equations, Ann. Polon. Math. 60 (1995), 249-253. Zbl0823.34029
- [6] P. H. Rabinowitz, On bifurcation from infinity, J. Differential Equations 14 (1973), 462-475. Zbl0272.35017
- [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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