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Abstract and Applied Analysis
Volume 2016, Article ID 5784273, 12 pages
http://dx.doi.org/10.1155/2016/5784273
Research Article

Existence of Infinitely Many Periodic Solutions for Perturbed Semilinear Fourth-Order Impulsive Differential Inclusions

1Department of Law and Economics, Mediterranea University of Reggio Calabria, Via dei Bianchi 2, 89131 Reggio Calabria, Italy
2Department of Economics, University of Messina, Via dei Verdi 75, 98122 Messina, Italy
3Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149, Iran

Received 28 October 2015; Accepted 11 February 2016

Academic Editor: Agacik Zafer

Copyright © 2016 Massimiliano Ferrara et al. 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. G. A. Afrouzi, S. Heidarkhani, and D. O'Regan, “Existence of three solutions for a doubly eigen-value fourth-order boundary value problem,” Taiwanese Journal of Mathematics, vol. 15, no. 1, pp. 201–210, 2011. View at Google Scholar
  2. G. Bonanno and B. Di Bella, “A boundary value problem for fourth-order elastic beam equations,” Journal of Mathematical Analysis and Applications, vol. 343, no. 2, pp. 1166–1176, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. J. R. Graef and B. Yang, “Existnence and nonexistence of positive solutions of fourth order nonlinear boundary value problems,” Applicable Analysis, vol. 74, no. 1-2, pp. 201–214, 2000. View at Publisher · View at Google Scholar
  4. R. Ma, “Multiple positive solutions for a semipositone fourth-order boundary value problem,” Hiroshima Mathematical Journal, vol. 33, no. 2, pp. 217–227, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. D. Bainov and P. Simeonov, Systems with Impulse Effect, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood, Chichester, UK, 1989.
  6. M. Benchohra, J. Henderson, and S. Ntouyas, Theory of Impulsive Differential Equations, vol. 2 of Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York, NY, USA, 2006.
  7. T. E. Carter, “Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion,” Dynamics and Control, vol. 10, no. 3, pp. 219–227, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. S. Heidarkhani, M. Ferrara, and A. Salari, “Infinitely many periodic solutions for a class of perturbed second-order differential equations with impulses,” Acta Applicandae Mathematicae, vol. 139, pp. 81–94, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  9. X. Liu and A. R. Willms, “Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft,” Mathematical Problems in Engineering, vol. 2, no. 4, pp. 277–299, 1996. View at Publisher · View at Google Scholar · View at Scopus
  10. A. Cabada and S. Tersian, “Existence and multiplicity of solutions to boundary value problems for fourth-order impulsive differential equations,” Boundary Value Problems, vol. 2014, article 105, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. J. Sun, H. Chen, and L. Yang, “Variational methods to fourth-order impulsive differential equations,” Journal of Applied Mathematics and Computing, vol. 35, no. 1-2, pp. 323–340, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. J. Xie and Z. Luo, “Solutions to a boundary value problem of a fourth-order impulsive differential equation,” Boundary Value Problems, vol. 2013, article 154, 14 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. S. Heidarkhani, G. A. Afrouzi, A. Hadjian, and J. Henderson, “Existence of infinitely many anti-periodic solutions for second-order impulsive differential inclusions,” Electronic Journal of Differential Equations, vol. 2013, article 97, 13 pages, 2013. View at Google Scholar · View at MathSciNet
  14. A. Iannizzotto, “Three critical points for perturbed nonsmooth functionals and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1319–1338, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. A. Iannizzotto, “Three periodic solutions for an ordinary differential inclusion with two parameters,” Annales Polonici Mathematici, vol. 103, no. 1, pp. 89–100, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. A. Kristály, “Infinitely many solutions for a differential inclusion problem in RN,” Journal of Differential Equations, vol. 220, no. 2, pp. 511–530, 2006. View at Publisher · View at Google Scholar
  17. A. Kristály, W. Marzantowicz, and C. Varga, “A non-smooth three critical points theorem with applications in differential inclusions,” Journal of Global Optimization, vol. 46, no. 1, pp. 49–62, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. Y. Tian and J. Henderson, “Three anti-periodic solutions for second-order impulsive differential inclusions via nonsmooth critical point theory,” Nonlinear Analysis, vol. 75, no. 18, pp. 6496–6505, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. B.-X. Yang and H.-R. Sun, “Periodic solutions for semilinear fourth-order differential inclusions via nonsmooth critical point theory,” Journal of Function Spaces, vol. 2014, Article ID 816490, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  20. B. Ricceri, “A general variational principle and some of its applications,” Journal of Computational and Applied Mathematics, vol. 113, no. 1-2, pp. 401–410, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. S. A. Marano and D. Motreanu, “Infinitely many critical points of non-differentiable functions and applications to a Neumann-type problem involving the p-Laplacian,” Journal of Differential Equations, vol. 182, no. 1, pp. 108–120, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. G. Bonanno and G. M. Bisci, “Infinitely many solutions for a boundary value problem with discontinuous nonlinearities,” Boundary Value Problems, vol. 2009, Article ID 670675, 2009. View at Google Scholar · View at MathSciNet
  23. L. H. Erbe and W. Krawcewicz, “Existence of solutions to boundary value problems for impulsive second order differential inclusions,” The Rocky Mountain Journal of Mathematics, vol. 22, no. 2, pp. 519–539, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, vol. 29 of Nonconvex Optimization and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  25. F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, NY, USA, 1983. View at MathSciNet
  26. S. Tersian and J. Chaparova, “Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations,” Journal of Mathematical Analysis and Applications, vol. 260, no. 2, pp. 490–506, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  27. K. C. Chang, “Variational methods for nondifferentiable functionals and their applications to partial differential equations,” Journal of Mathematical Analysis and Applications, vol. 80, no. 1, pp. 102–129, 1981. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus