Nonlinear Spectra for Parameter Dependent Ordinary Differential Equations

Eigenvalue problems of the form x 00 = f(x) + µg(x), (i), x(0) = 0, x(1) = 0 (ii) are considered. We are looking for (,µ) such that the problem (i), (ii) has a nontrivial solution. This problem generalizes the fa mous Fuchik problem for piece-wise linear equations. In our considerations function s 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| 2 x and |x| 1 2+1 respectively ( > 0, > 0.)