SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Global structure of positive solutions for three-point boundary value problems

Jia-Ping Gu1, Liang-Gen Hu1 and Huai-Nian Zhang2*

Author Affiliations

1 Department of Mathematics, Ningbo University, Ningbo, 315211, P.R. China

2 Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing, 102617, P.R. China

For all author emails, please log on.

Boundary Value Problems 2013, 2013:174  doi:10.1186/1687-2770-2013-174

Published: 25 July 2013

Abstract

In this paper, we are concerned with the three-point boundary value problem for second-order differential equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M5">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M8">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M9">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M10">View MathML</a>. The existence of the continuum of a positive solution is established by utilizing the Leray-Schauder global continuation principle. Furthermore, the interval of α about the nonexistence of a positive solution is also given.

MSC: 34B10, 34B18, 34G20.

Keywords:
positive solution; global continuous theorem; continuum; differential equation