*(English)*Zbl 1186.45002

Nonlinear eigenvalue problems with positive eigenfunctions for the Hammerstein equation

are studied where $f$ can be negative but is bounded from below (“semi-positone”), $k$ is nonnegative and

with $0\le C\left(t\right)\le 1$. (For some reason, the kernel is actually written in the split form $k(t,s)g\left(s\right)$ in the paper.) Using estimates for the spectral radius (and thus the largest eigenvalue) of the positive linear integral operators $Lu\left(s\right)={\int}_{\alpha}^{\beta}k(t,s)u\left(s\right)ds$, conditions for intervals of eigenvalues (of the nonlinear problem) and in case of appropriately oscillating $f$, also lower bounds on their multiplicities are given in terms of the spectral radius of $L$ and in terms of its estimates. The proof consists in considering an auxiliary positive nonlinear operator on a special cone and proving, using the mentioned quantities, that its fixed point index is 0 or 1 on the intersection of certain balls with the cone.

The result can be applied to a third-order three-point boundary value problem

For the corresponding Green’s function, estimates for the required quantities are calculated.

##### MSC:

45C05 | Eigenvalue problems (integral equations) |

47H30 | Particular nonlinear operators |

34B18 | Positive solutions of nonlinear boundary value problems for ODE |

34B16 | Singular nonlinear boundary value problems for ODE |

45G10 | Nonsingular nonlinear integral equations |

34B15 | Nonlinear boundary value problems for ODE |