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

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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