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

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.

Linked References

  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  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. View at Publisher · View at Google Scholar
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  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. View at Publisher · View at Google Scholar
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  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. View at Publisher · View at Google Scholar
  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. View at Google Scholar · View at Zentralblatt MATH
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  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. View at Google Scholar
  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. View at Publisher · View at Google Scholar
  16. P. J. Torres, “Weak singularities may help periodic solutions to exist,” Journal of Differential Equations, vol. 232, no. 1, pp. 277–284, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH