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

#### 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.

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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