Electron. J. Diff. Eqns., Vol. 2004(2004), No. 119, pp. 1-7.

Semipositone $m$-point boundary-value problems

Nickolai Kosmatov

Abstract:
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.

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.

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Nickolai Kosmatov
Department of Mathematics and Statistics
University of Arkansas at Little Rock
Little Rock, AR 72204-1099, USA
email: nxkosmatov@ualr.edu

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