Open Access Research

Application of the shooting method to second-order multi-point integral boundary-value problems

Huilan Wang*, Zigen Ouyang and Liguang Wang

Author Affiliations

School of Mathematics and Physics, University of South China, Hengyang, 421001, P.R. China

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Boundary Value Problems 2013, 2013:205  doi:10.1186/1687-2770-2013-205

Published: 9 September 2013

Abstract

In this paper, we focus on the following second-order multi-point integral boundary-value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M3">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M5">View MathML</a> are given constants. The proof is based on the shooting method. By constructing a quadratic function and a sine function as the shooting objects and combining the integral mean value theorem with the comparison principle, we consider the existence of positive solutions to the BVP respectively under the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M6">View MathML</a> and the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M7">View MathML</a>. The method is concise and some new criteria are established.

MSC: 34B10, 34B15, 34B18.

Keywords:
shooting method; integral boundary-value problem; positive solution