Acta Mathematica Sinica

, Volume 20, Issue 2, pp 273–282

# Positive Solutions for Semipositone m-point Boundary-value Problems

ORIGINAL ARTICLES

## Abstract

Let ξi ∈ (0, 1) with 0 < ξ1 < ξ2 < ··· < ξm−2 < 1, ai, bi ∈ [0,∞) with $$0 < {\sum\nolimits_{i = 1}^{m - 2} {a_{i} < 1} }$$ and $${\sum\nolimits_{i = 1}^{m - 2} {b_{i} < 1} }$$. We consider the m-point boundary-value problem
$${u}\ifmmode{''}\else''\fi + \lambda f{\left( {t,u} \right)} = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} t \in {\left( {0,1} \right)},$$
$${x}\ifmmode{'}\else'\fi{\left( 0 \right)} = {\sum\limits_{i = 1}^{m - 2} {b_{i} {x}\ifmmode{'}\else'\fi{\left( {\xi _{i} } \right)},{\kern 1pt} {\kern 1pt} {\kern 1pt} x{\left( 1 \right)} = {\sum\limits_{i = 1}^{m - 2} {a_{i} x{\left( {\xi _{i} } \right)},} }} }$$
where f(x, y) ≥ −M, and M is a positive constant. We show the existence and multiplicity of positive solutions by applying the fixed point theorem in cones.

### Keywords

Ordinary differential equation Existence of solutions Multi-point boundary value problems Fixed point theorem in cones

34B10

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