Abstract and Applied Analysis

Subharmonic Solutions of Nonautonomous Second Order Differential Equations with Singular Nonlinearities

N. Daoudi-Merzagui, F. Derrab, and A. Boucherif

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We discuss the existence of subharmonic solutions for nonautonomous second order differential equations with singular nonlinearities. Simple sufficient conditions are provided enable us to obtain infinitely many distinct subharmonic solutions. Our approach is based on a variational method, in particular the saddle point theorem.

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Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 903281, 20 pages.

First available in Project Euclid: 5 April 2013

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Daoudi-Merzagui, N.; Derrab, F.; Boucherif, A. Subharmonic Solutions of Nonautonomous Second Order Differential Equations with Singular Nonlinearities. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 903281, 20 pages. doi:10.1155/2012/903281. https://projecteuclid.org/euclid.aaa/1365125157

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  • J. Chu and J. J. Nieto, “Recent existence results for second-order singular periodic differential equa- tions,” Boundary Value Problems, vol. 2009, Article ID 540863, 20 pages, 2009.
  • A. Boucherif and N. Daoudi-Merzagui, “Periodic solutions of singular nonautonomous second order differential equations,” Nonlinear Differential Equations and Applications, vol. 15, no. 1-2, pp. 147–158, 2008.
  • N. Daoudi-Merzagui, “Periodic solutions of nonautonomous second order differential equations with a singularity,” Applicable Analysis, vol. 73, no. 3-4, pp. 449–462, 1999.
  • X. Li and Z. Zhang, “Periodic solutions for second-order differential equations with a singular non- linearity,” Nonlinear Analysis, vol. 69, no. 11, pp. 3866–3876, 2008.
  • J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.
  • P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986.
  • A. Fonda, Periodic Solutions of Scalar Second Order Differential Equations with a Singularity, Académie Royale de Belgique, Brussels, Belgium, 1993.
  • A. Fonda, R. Manásevich, and F. Zanolin, “Subharmonic solutions for some second-order differential equations with singularities,” SIAM Journal on Mathematical Analysis, vol. 24, no. 5, pp. 1294–1311, 1993.
  • A. Fonda and M. Ramos, “Large-amplitude subharmonic oscillations for scalar second-order dif- ferential equations with asymmetric nonlinearities,” Journal of Differential Equations, vol. 109, no. 2, pp. 354–372, 1994.
  • E. Serra, M. Tarallo, and S. Terracini, “Subharmonic solutions to second-order differential equations with periodic nonlinearities,” Nonlinear Analysis, vol. 41, pp. 649–667, 2000.
  • C.-L. Tang, “Periodic solutions for nonautonomous second order systems with sublinear nonlinearity,” Proceedings of the American Mathematical Society, vol. 126, no. 11, pp. 3263–3270, 1998.
  • J. Yu, “Subharmonic solutions with prescribed minimal period of a class of nonautonomous Hamil- tonian systems,” Journal of Dynamics and Differential Equations, vol. 20, no. 4, pp. 787–796, 2008.
  • X. Zhang and X. Tang, “Subharmonic solutions for a class of non-quadratic second order Hamiltonian systems,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 113–130, 2012.
  • L. D. Humphreys, P. J. McKenna, and K. M. O'Neill, “High frequency shaking induced by low freq- uency forcing: periodic oscillations in a spring-cable system,” Nonlinear Analysis. Real World Ap- plications, vol. 11, no. 5, pp. 4312–4325, 2010.
  • P. Omari, G. Villari, and F. Zanolin, “Periodic solutions of the Liénard equation with one-sided growth restrictions,” Journal of Differential Equations, vol. 67, no. 2, pp. 278–293, 1987.