Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type

Jeffrey R. L. Webb, Kunquan Q. Lan

DOI: http://dx.doi.org/10.12775/TMNA.2006.003

Abstract


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
$$
u''(t)+g(t)f(t,u(t))=0 \quad\text{a.e. on } [0,1],
$$
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.

Keywords


Fixed point index; positive solution; eigenvalue criteria

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