Two-parametric nonlinear eigenvalue problems

Eigenvalue problems of the form x′′ = −λf(x) + μg(x−), (i), x(0) = 0, x(1) = 0, (ii) are considered, where x and x− are the positive and negative parts of x respectively. We are looking for (λ, μ) such that the problem (i), (ii) has a nontrivial solution. This problem generalizes the famous Fuč́ık problem for piece-wise linear equations. In our considerations functions f and g may be nonlinear functions of super-, suband quasi-linear growth in various combinations. The spectra obtained under the normalization condition |x′(0)| = 1 are sometimes similar to usual Fuč́ık spectrum for the Dirichlet problem and sometimes they are quite different. This depends on monotonicity properties of the functions ξt1(ξ) and ητ1(η), where t1(ξ) and τ1(η) are the first zero functions of the Cauchy problems x′′ = −f(x), x(0) = 0, x′(0) = ξ > 0, y′′ = g(y), y(0) = 0, y′(0) = −η, (η > 0) respectively.