Positive solutions of nonlocal boundary value problems: a unified approach, Journal of the London Mathematical Society

We give a unified approach to studying the existence of multiple positive solutions of nonlinear differential equations of the form \begin{equation*} -u''(t)=g(t)f(t,u(t)),\; \text{a.e.} \; t \in (0,1), \end{equation*} where $g, f$ are non-negative functions, subject to various nonlocal boundary conditions. We study these problems via new results for a perturbed integral equation, in the space $C[0,1]$, of the form \begin{equation*} u(t)=\gamma(t){\alpha}[u]+\delta(t){\beta}[u] +\int_{0}^{1}k(t,s)g(s)f(s,u(s))\,ds \end{equation*} where $\alpha[u]$, $\beta[u]$ are linear functionals given by Stieltjes integrals but are \emph{not} assumed to be positive for all positive $u$. This means we actually cover many more differential equations than the simple one written above. Previous results have studied positive functionals only, but even for positive functionals our methods give improvements on previous work. The well known $m$-point BVPs are special cases and we obtain sharp conditions on the coefficients, which allows some to have opposite signs. We also use some optimal assumptions on the nonlinear term.