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

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