Open Access Research

Fredholm alternative for the second-order singular Dirichlet problem

Alexander Lomtatidze12 and Zdeněk Opluštil2*

Author Affiliations

1 Institute of Mathematics, Academy of Sciences of the Czech Republic, branch in Brno, Žižkova 22, Brno, 616 62, Czech Republic

2 Faculty of Mechanical Engineering, Institute of Mathematics, Brno University of Technology, Technická 2, Brno, 616 69, Czech Republic

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Boundary Value Problems 2014, 2014:13  doi:10.1186/1687-2770-2014-13

Published: 13 January 2014

Abstract

Consider the singular Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M2">View MathML</a> are locally Lebesgue integrable functions. It is proved that if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/13/mathml/M3">View MathML</a>

then the Fredholm alternative remains true.

MSC: 34B05.

Keywords:
singular Dirichlet problem; Fredholm alternative