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.
@article{bwmeta1.element.bwnjournal-article-apmv71z1p39bwm,
author = {Jolanta Przybycin},
title = {On bifurcation intervals for nonlinear eigenvalue problems},
journal = {Annales Polonici Mathematici},
volume = {72},
year = {1999},
pages = {39-46},
zbl = {0930.35118},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv71z1p39bwm}
}
Jolanta Przybycin. On bifurcation intervals for nonlinear eigenvalue problems. Annales Polonici Mathematici, Tome 72 (1999) pp. 39-46. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv71z1p39bwm/
[000] [1] H. Berestycki, On some Sturm-Liouville problems, J. Differential Equations 26 (1977), 375-390. | Zbl 0331.34020
[001] [2] R. Chiappinelli, On eigenvalues and bifurcation for nonlinear Sturm-Liouville operators, Boll. Un. Mat. Ital. A (6) 4 (1985), 77-83. | Zbl 0565.34016
[002] [3] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966.
[003] [4] L. Nirenberg, Topics in Nonlinear Functional Analysis, New York Univ. Lecture Notes, 1973-74.
[004] [5] J. Przybycin, Nonlinear eigenvalue problems for fourth order ordinary differential equations, Ann. Polon. Math. 60 (1995), 249-253. | Zbl 0823.34029
[005] [6] P. H. Rabinowitz, On bifurcation from infinity, J. Differential Equations 14 (1973), 462-475. | Zbl 0272.35017
[006] [7] K. Schmitt and H. L. Smith, On eigenvalue problems for nondifferentiable mappings, J. Differential Equations 33 (1979), 294-319. | Zbl 0389.34019