`Abstract and Applied AnalysisVolume 2011, Article ID 545264, 19 pageshttp://dx.doi.org/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

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.

1. J. R. Graef, L. Kong, and H. Wang, “Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem,” Journal of Differential Equations, vol. 245, no. 5, pp. 1185–1197, 2008.
2. A. Cabada and N. D. Dimitrov, “Multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities,” Journal of Mathematical Analysis and Applications, vol. 371, no. 2, pp. 518–533, 2010.
3. J. R. Graef, L. Kong, and H. Wang, “A periodic boundary value problem with vanishing Green's function,” Applied Mathematics Letters, vol. 21, no. 2, pp. 176–180, 2008.
4. D. Jiang, J. Chu, and M. Zhang, “Multiplicity of positive periodic solutions to superlinear repulsive singular equations,” Journal of Differential Equations, vol. 211, no. 2, pp. 282–302, 2005.
5. P. J. Torres, “Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem,” Journal of Differential Equations, vol. 190, no. 2, pp. 643–662, 2003.
6. R. Ma, “Nonlinear periodic boundary value problems with sign-changing Green's function,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 5, pp. 1714–1720, 2011.
7. P. J. Torres and M. Zhang, “A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle,” Mathematische Nachrichten, vol. 251, pp. 101–107, 2003.
8. A. Cabada and J. Á. Cid, “On the sign of the Green's function associated to Hill's equation with an indefinite potential,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 303–308, 2008.
9. A. Cabada, J. Á. Cid, and M. Tvrdý, “A generalized anti-maximum principle for the periodic one-dimensional $p$-Laplacian with sign-changing potential,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 7-8, pp. 3436–3446, 2010.
10. M. Zhang, “A relationship between the periodic and the Dirichlet BVPs of singular differential equations,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 128, no. 5, pp. 1099–1114, 1998.
11. M. Zhang, “Optimal conditions for maximum and antimaximum principles of the periodic solution problem,” Boundary Value Problems, vol. 2010, Article ID 410986, 26 pages, 2010.
12. M. A. Krasnoselskii, Positive Solutions of Operator Equations, P. Noordhoff Ltd., Groningen, The Netherlands, 1964.
13. E. Esmailzadeh and G. Nakhaie-Jazar, “Periodic solution of a Mathieu-Duffing type equation,” International Journal of Non-Linear Mechanics, vol. 32, no. 5, pp. 905–912, 1997.
14. V. Bevc, J. L. Palmer, and C. Süsskind, “On the design of the transition region of axi-symmetric magnetically focusing beam valves,” British Institution of Radio Engineers, vol. 18, pp. 696–708, 1958.
15. J. Chu and Z. Zhang, “Periodic solutions of singular differential equations with sign-changing potential,” Bulletin of the Australian Mathematical Society, vol. 82, no. 3, pp. 437–445, 2010.
16. P. J. Torres, “Weak singularities may help periodic solutions to exist,” Journal of Differential Equations, vol. 232, no. 1, pp. 277–284, 2007.