Two-parametric nonlinear eigenvalue problems

Eigenvalue problems of the form $x'' = -\lambda f(x^+) + \mu g(x^-),$ $\quad (i),$ $x(0) = 0, \; x(1) = 0,$ $\quad (ii)$ are considered, where $x^+$ and $x^-$ are the positive and negative parts of $x$ respectively. We are looking for $(\lambda, \mu)$ such that the problem $(i), (ii)$ has a nontrivial solution. This problem generalizes the famous Fu\v{c}\'{i}k problem for piece-wise linear equations. In our considerations functions $f$ and $g$ may be nonlinear functions of super-, sub- and quasi-linear growth in various combinations. The spectra obtained under the normalization condition $|x'(0)|=1$ are sometimes similar to usual Fu\v{c}\'{i}k spectrum for the Dirichlet problem and sometimes they are quite different. This depends on monotonicity properties of the functions $\xi t_1 (\xi)$ and $\eta \tau_1 (\eta),$ where $t_1 (\xi)$ and $\tau_1 (\eta)$ are the first zero functions of the Cauchy problems $x''= -f(x),$ $\: x(0)=0,$ $\: x'(0)=\xi> 0,$ $y''= g(y),$ $\: y(0)=0,$ $\: y'(0)=-\eta,$ $(\eta > 0)$ respectively.