Multiple positive solutions for semi-positone m-point boundary value problems

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Abstract

In this article, we establish the existence of at least two positive solutions for the semi-positone m-point boundary value problem with a parameter
$$\begin{array}{*{20}c} {u''\left( t \right) + \lambda f\left( {t,u} \right) = 0} & {t \in \left( {0,1} \right),} \\ {u'\left( 0 \right) = \sum\limits_{i = 1}^{m - 2} {b_i u'} \left( {\xi _i } \right),} & {u\left( 1 \right) = \sum\limits_{i = 1}^{m - 2} {a_i u\left( {\xi _i } \right),} } \\ \end{array}$$
where λ > 0 is a parameter, 0 < ξ 1 < ξ 2 < … < ξ m−2 < 1 with \(0 < \sum\limits_{i = 1}^{m - 2} {a_i < 1, \sum\limits_{i = 1}^{m - 2} {b_i < 1, a_i , b_i \in \left[ {0,\infty } \right)} }\) and f(t, u) ≥ − M with M is a positive constant. The method employed is the Leggett-Williams fixed-point theorem. As an application, an example is given to demonstrate the main result.

Keywords

Multiple positive solutions cone semi-positone m-point boundary value problem concave functional parameter 

2000 MR Subject Classification

34B10 
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Copyright information

© Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.School of Mathematical ScienceShanxi UniversityShanxiChina
  2. 2.Business College of Shanxi UniversityShanxiChina

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