Nonlinear Spectra for Parameter Dependent Ordinary Differential Equations
Articles
F. Sadyrbaev
Daugavpils University, Latvia
A. Gritsans
Daugavpils University, Latvia
Published 2007-04-25
https://doi.org/10.15388/NA.2007.12.2.14715
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Keywords

nonlinear spectra
jumping nonlinearity
asymptotically asymmetric nonlinearities
Fuchik spectrum

How to Cite

Sadyrbaev, F. and Gritsans, A. (2007) “Nonlinear Spectra for Parameter Dependent Ordinary Differential Equations”, Nonlinear Analysis: Modelling and Control, 12(2), pp. 253–267. doi:10.15388/NA.2007.12.2.14715.

Abstract

Eigenvalue problems of the form x'' = −λf(x) + µg(x)  (i),  x(0) = 0, x(1) = 0  (ii) are considered. We are looking for (λ, µ) such that the problem (i), (ii) has a nontrivial solution. This problem generalizes the famous Fuchik problem for piece-wise linear equations. In our considerations functions f and g may be super-, sub- and quasi-linear in various combinations. The spectra obtained under the normalization condition (otherwise problems may have continuous spectra) structurally are similar to usual Fuchik spectrum for the Dirichlet problem. We provide explicit formulas for Fuchik spectra for super and super, super and sub, sub and super, sub and sub cases, where superlinear and sublinear parts of equations are of the form |x|2αx and |x|1/(2β+1) respectively (α > 0, β > 0.)

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