ORIGINAL ARTICLES

Acta Mathematica Sinica

, Volume 20, Issue 2, pp 273-282

First online:

Positive Solutions for Semipositone m-point Boundary-value Problems

  • Ru Yun Ma*Affiliated withDepartment of Mathematics, Northwest Normal University Email author 
  • , Qiao Zhen MaAffiliated withDepartment of Mathematics, Northwest Normal University

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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.

Keywords

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

MR (2000) Subject Classification

34B10