Semipositone m-point boundary-value problems

We study the $m$-point nonlinear boundary-value problem $$ displaylines{ -[p(t)u'(t)]' = lambda f(t,u(t)), quad 0 less than t less than 1, cr u'(0) = 0, quad sum_{i=1}^{m-2}alpha_i u(eta_i) = u(1), }$$ where $0$ less than $eta_1$ less than $eta_2$ lessthan $dots$ less than $eta_{m-2}$ less than $1$, $alpha_i$ greater than 0 for $1 leq i leq m-2$ and $sum_{i=1}^{m-2}alpha_i < 1$, $m geq 3$. We assume that $p(t)$ is non-increasing continuously differentiable on $(0,1)$ and $p(t)$ greater than 0 on $[0,1]$. Using a cone-theoretic approach we provide sufficient conditions on continuous $f(t,u)$ under which the problem admits a positive solution.