Nonlinear Spectra for Parameter Dependent Ordinary Differential Equations - Sadyrbaev, F.; Gritsans, A. (Nonlinear Analysis: Modelling and Control, 2007)
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.)