Open Access Research

Positive solutions for singular boundary value problems involving integral conditions

Liang-Gen Hu

Author affiliations

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

Citation and License

Boundary Value Problems 2012, 2012:72  doi:10.1186/1687-2770-2012-72

Published: 5 July 2012

Abstract

We are interested in the following singular boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M2">View MathML</a> is a parameter and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M3">View MathML</a> is the Stieltjes integral. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M4">View MathML</a> and w may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M5">View MathML</a> and/or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M8">View MathML</a>. Some a priori estimates and the existence, multiplicity and nonexistence of positive solutions are obtained. Our proofs are based on the method of global continuous theorem, the lower-upper solutions methods and fixed point index theory. Furthermore, we also discuss the interval of parameter μ such that the problem has a positive solution.

Keywords:
singularity; global continuous theorem; solution of boundedness; fixed point index; positive solution