Positive solutions of nonlocal boundary value problems involving integral conditions

We establish the existence of multiple positive solutions of nonlinear equations of the form \begin{equation*} -u''(t)=g(t)f(t,u(t)),\ t \in (0,1), \end{equation*} where $g, f$ are non-negative functions, subject to various nonlocal boundary conditions. The common feature is that each can be written as an integral equation, in the space $C[0,1]$, of the form $$ u(t)=\gamma(t){\alpha}[u]+\int_{0}^{1}k(t,s)g(s)f(s,u(s))\,ds $$ where $\alpha[u]$ is a linear functional given by a Stieltjes integral but is \emph{not} assumed to be positive for all positive $u$. Our new results cover many nonlocal boundary conditions previously studied on a case by case basis for particular positive functionals only, for example, many $m$-point BVPs are special cases. Even for positive functionals our methods give improvements on previous work. Also we allow weaker assumptions on the nonlinear term than were previously imposed.