Positive Solutions for Semipositone m-point Boundary-value Problems

Abstract

Let ξ _i ∈ (0, 1) with 0 < ξ₁ < ξ₂ < ··· < ξ _m−2 < 1, a _i , b _i ∈ [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.