Abstract and Applied Analysis
Volume 2011 (2011), Article ID 545264, 19 pages
doi:10.1155/2011/545264
Research Article

Existence and Multiplicity of Solutions for a Periodic Hill's Equation with Parametric Dependence and Singularities

1Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
2Departamento de Matemáticas, Universidad de Jaén, Campus Las Lagunillas, Ed. B3, 23071 Jaén, Spain

Received 5 July 2010; Revised 27 January 2011; Accepted 24 February 2011

Academic Editor: Pavel Drábek

Copyright © 2011 Alberto Cabada and José Ángel Cid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We deal with the existence and multiplicity of solutions for the periodic boundary value problem 𝑥 ( 𝑡 ) + 𝑎 ( 𝑡 ) 𝑥 ( 𝑡 ) = 𝜆 𝑔 ( 𝑡 ) 𝑓 ( 𝑥 ) + 𝑐 ( 𝑡 ) , 𝑥 ( 0 ) = 𝑥 ( 𝑇 ) , 𝑥 ( 0 ) = 𝑥 ( 𝑇 ) , where 𝜆 is a positive parameter. The function 𝑓 ( 0 , ) ( 0 , ) is allowed to be singular, and the related Green's function is nonnegative and can vanish at some points.