POSITIVE SOLUTIONS OF NONLOCAL BOUNDARY VALUE PROBLEMS: A UNIFIED APPROACH

Abstract

We give a unified approach for studying the existence of multiple positive solutions of nonlinear differential equations of the form

$$-u''(t)=g(t)f(t,u(t)),\quad \text{for almost every } t \in (0,1),$$

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

$$u(t)=\gamma(t){\alpha}[u]+\delta(t){\beta}[u]+\int_{0}^{1}k(t,s)g(s)f(s,u(s))\,ds$$

where $\alpha[u]$, $\beta[u]$ are linear functionals given by Stieltjes integrals but are not assumed to be positive for all positive $u$. This means we actually cover many more differential equations than the simple equation 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 boundary value problems are special cases and we obtain sharp conditions on the coefficients, which allows some of them to have opposite signs. We also use some optimal assumptions on the nonlinear term.