DOI QR코드

DOI QR Code

ANTI-PERIODIC SOLUTIONS FOR HIGHER-ORDER LIÉENARD TYPE DIFFERENTIAL EQUATION WITH p-LAPLACIAN OPERATOR

  • Chen, Taiyong (Department of Mathematics China University of Mining and Technology) ;
  • Liu, Wenbin (Department of Mathematics China University of Mining and Technology)
  • Received : 2010.07.28
  • Published : 2012.05.31

Abstract

In this paper, by using degree theory, we consider a kind of higher-order Li$\acute{e}$enard type $p$-Laplacian differential equation as follows $$({\phi}_p(x^{(m)}))^{(m)}+f(x)x^{\prime}+g(t,x)=e(t)$$. Some new results on the existence of anti-periodic solutions for above equation are obtained.

Keywords

References

  1. A. R. Aftabizadeh, N. H. Pavel, and Y. Huang, Anti-periodic oscillations of some second- order differential equations and optimal control problems, J. Comput. Appl. Math. 52 (1994), no. 1-3, 3-21. https://doi.org/10.1016/0377-0427(94)90345-X
  2. C. Ahn and C. Rim, Boundary ows in general coset theories, J. Phys. A 32 (1999), no. 13, 2509-2525. https://doi.org/10.1088/0305-4470/32/13/004
  3. S. Aizicovici, M. McKibben, and S. Reich, Anti-periodic solutions to nonmonotone evo- lution equations with discontinuous nonlinearities, Nonlinear Anal. 43 (2001), no. 2, 233-251. https://doi.org/10.1016/S0362-546X(99)00192-3
  4. H. Chen, Antiperiodic wavelets, J. Comput. Math. 14 (1996), no. 1, 32-39.
  5. Y. Chen, On Massera's theorem for anti-periodic solution, Adv. Math. Sci. Appl. 9 (1999), no. 1, 125-128.
  6. T. Chen, W. Liu, J. Zhang, and H. Zhang, Anti-periodic solutions for higher-order nonlinear ordinary differential equations, J. Korean Math. Soc. 47 (2010), no. 3, 573- 583. https://doi.org/10.4134/JKMS.2010.47.3.573
  7. Y. Chen, J. J. Nieto, and D. ORegan, Anti-periodic solutions for fully nonlinear first- order differential equations, Math. Comput. Modelling 46 (2007), no. 9-10, 1183-1190. https://doi.org/10.1016/j.mcm.2006.12.006
  8. G. Croce and B. Dacorogna, On a generalized Wirtinger inequality, Discrete Contin. Dyn. Syst. 9 (2003), no. 5, 1329-1341. https://doi.org/10.3934/dcds.2003.9.1329
  9. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
  10. R. E. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, in: Lecture Notes in Mathematics, vol. 568, Springer-Verlag, Berlin, New York, 1977.
  11. H. Kleinert and A. Chervyakov, Functional determinants from Wronski Green function, J. Math. Phys. 40 (1999), no. 11, 6044-6051. https://doi.org/10.1063/1.533069
  12. X. Li, Existence and uniqueness of periodic solutions for a kind of high-order p- Laplacian Duffing differential equation with sign-changing coefficient ahead of linear term, Nonlinear Anal. 71 (2009), no. 7-8, 2764-2770. https://doi.org/10.1016/j.na.2009.01.153
  13. B. Liu, Anti-periodic solutions for forced Rayleigh-type equations, Nonlinear Anal. Real World Appl. 10 (2009), no. 5, 2850-2856. https://doi.org/10.1016/j.nonrwa.2008.08.011
  14. W. Liu, J. Zhang, and T. Chen, Anti-symmetric periodic solutions for the third order differential systems, Appl. Math. Lett. 22 (2009), no. 5, 668-673. https://doi.org/10.1016/j.aml.2008.08.004
  15. Z. Lu, Travelling Tube, Shanghai Science and Technology Press, Shanghai, 1962.
  16. Z. Luo, J. Shen, and J. J. Nieto, Antiperiodic boundary value problem for first-order impulsive ordinary differential equation, Comput. Math. Appl. 49 (2005), no. 2-3, 253- 261. https://doi.org/10.1016/j.camwa.2004.08.010
  17. M. Nakao, Existence of an anti-periodic solution for the quasilinear wave equation with viscosity, J. Math. Anal. Appl. 204 (1996), no. 3, 754-764. https://doi.org/10.1006/jmaa.1996.0465
  18. H. Pang, W. Ge, and M. Tian, Solvability of nonlocal boundary value problems for ordinary differential equation of higher order with a p-Laplacian, Comput. Math. Appl. 56 (2008), no. 1, 127-142. https://doi.org/10.1016/j.camwa.2007.11.039
  19. P. Souplet, Optimal uniqueness condition for the antiperiodic solutions of some nonlin- ear parabolic equations, Nonlinear Anal. 32 (1998), no. 2, 279-286. https://doi.org/10.1016/S0362-546X(97)00477-X
  20. H. Su, B. Wang, Z. Wei, and X. Zhang, Positive solutions of four-point boundary value problems for higher-order p-Laplacian operator, J. Math. Anal. Appl. 330 (2007), no. 2, 836-851. https://doi.org/10.1016/j.jmaa.2006.07.017
  21. F. Xu, L. Liu, and Y. Wu, Multiple positive solutions of four-point nonlinear boundary value problems for a higher-order p-Laplacian operator with all derivatives, Nonlinear Anal. 71 (2009), no. 9, 4309-4319. https://doi.org/10.1016/j.na.2009.02.118
  22. Y. Yin, Monotone iterative technique and quasilinearization for some anti-periodic problems, Nonlinear World 3 (1996), no. 2, 253-266.

Cited by

  1. Anti-periodic solutions of Liénard equations with state dependent impulses vol.261, pp.7, 2016, https://doi.org/10.1016/j.jde.2016.06.020
  2. Existence of solutions for 2 n th -order nonlinear p -Laplacian differential equations vol.34, 2017, https://doi.org/10.1016/j.nonrwa.2016.09.018
  3. Existence of anti-periodic solutions with symmetry for some high-order ordinary differential equations vol.2012, pp.1, 2012, https://doi.org/10.1186/1687-2770-2012-108
  4. Antiperiodic oscillations in Chua’s circuits using conjugate coupling vol.75, 2015, https://doi.org/10.1016/j.chaos.2015.02.028
  5. Antiperiodic oscillations in a forced Duffing oscillator vol.78, 2015, https://doi.org/10.1016/j.chaos.2015.08.005
  6. Antiperiodic oscillations vol.3, pp.1, 2013, https://doi.org/10.1038/srep01958