@article{Webb_Lan_2006, title={Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type}, volume={27}, url={https://apcz.umk.pl/TMNA/article/view/TMNA.2006.003}, abstractNote={New criteria are established for the existence of multiple positive
solutions of a Hammerstein integral equation of the form
$$
u(t)= \int_{0}^1 k(t,s)g(s)f(s,u(s))ds \equiv Au(t)
$$
where $k$ can have discontinuities in its second variable and $g \in
L^{1}$. These criteria are determined by the relationship between the
behaviour of $f(t,u)/u$ as $u$ tends to $0^+$ or $\infty$ and the
principal (positive) eigenvalue of the linear Hammerstein integral
operator $$
Lu(t)=\int_{0}^1 k(t,s)g(s)u(s)ds. $$
We obtain new results
on the existence of multiple positive solutions of a second order
differential equation of the form
$$
uāā(t)+g(t)f(t,u(t))=0 \quad\text{a.e. on } [0,1],
$$
subject to general separated boundary conditions and also to nonlocal
$m$-point boundary conditions. Our results are optimal in some cases.
This work contains several new ideas, and gives a {\it unified}
approach applicable to many BVPs.}, number={1}, journal={Topological Methods in Nonlinear Analysis}, author={Webb, Jeffrey R. L. and Lan, Kunquan Q.}, year={2006}, month={Mar.}, pages={91ā115} }