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Positive Solutions for Semipositone <Emphasis Type="Italic">m</Emphasis>-point Boundary-value Problems

Authors:

Email author" target="_blank">Ru?Yun?Ma Email author Qiao?Zhen?Ma

Institution:

(1) Department of Mathematics, Northwest Normal University, Lanzhou, 730070, P. R. China

Abstract:

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. *Supported by the NSFC (10271095). GG-110-10736-1003, NWNU-KJCXGC-212 and the Foundation of Major Project of Science and Technology of Chinese Education Ministry

Keywords:

" target="_blank"> Ordinary differential equation Existence of solutions Multi-point boundary value problems Fixed point theorem in cones

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