| A. Gritsans and F. Sadyrbaev. On nonlinear eigenvalue problems |
We consider the second order nonlinear eigenvalue problems depending on one or two parameters. First we are looking for positive solutions of equations $x''= - f(x)$ and $x''= - \lambda f(x),$ which are considered together with the Dirichlet boundary conditions $x(0)=0,$ $x(1) = 0, \quad (i).$ Function $f(x)$ is supposed to be convex. The relation between the parameter $\lambda$ and the Nehari number $\lambda_0 (0,1)$ is established (\cite{Nehari1}, \cite{Trakai}). Fu\v{c}ik like nonlinear problem is treated for equation $x''= - \lambda f (x) + \mu g(x).$ We construct the set of points $(\lambda, \mu)$ such that this equation has a nontrivial normalized (by a condition $x'(0)=1$) solution which satisfies the boundary conditions $(i).$ |