Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 119, pp. 1-7.
Title: Semipositone $m$-point boundary-value problems
Author: Nickolai Kosmatov (Univ. of Arkansas at Little Rock, AR, USA)
Abstract:
We study the $m$-point nonlinear boundary-value problem
$$
\displaylines{
-[p(t)u'(t)]' = \lambda f(t,u(t)), \quad 0 < t < 1, \cr
u'(0) = 0, \quad \sum_{i=1}^{m-2}\alpha_i u(\eta_i) = u(1),
}$$
where $0 < \eta_1 < \eta_2 < \dots < \eta_{m-2} < 1$, $\alpha_i > 0$
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 $p(t) > 0$ 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.
Submitted April 23, 2004. Published October 10, 2004.
Math Subject Classifications: 34B10, 34B18.
Key Words: Green's function; fixed point theorem;
positive solutions; multi-point boundary-value problem.