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Eigenvalues of semi-positone Hammerstein integral equations and applications to boundary value problems. (English) Zbl 1186.45002

Nonlinear eigenvalue problems with positive eigenfunctions for the Hammerstein equation

$\lambda y\left(t\right)={\int }_{0}^{1}k\left(t,s\right)f\left(s,y\left(s\right)\right)ds$

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

$C\left(t\right){\Phi }\left(s\right)\le k\left(t,s\right)\le {\Phi }\left(s\right)$

with $0\le C\left(t\right)\le 1$. (For some reason, the kernel is actually written in the split form $k\left(t,s\right)g\left(s\right)$ in the paper.) Using estimates for the spectral radius (and thus the largest eigenvalue) of the positive linear integral operators $Lu\left(s\right)={\int }_{\alpha }^{\beta }k\left(t,s\right)u\left(s\right)ds$, 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

$\lambda {y}^{\text{'}\text{'}\text{'}}\left(t\right)-g\left(t\right)f\left(t,y\left(t\right)\right)=0,$
$y\left(0\right)={y}^{\text{'}}\left(\beta \right)={y}^{\text{'}\text{'}}\left(1\right)=0·$

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