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ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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Periodic Solutions for Third and Fourth Order Delay Differential Equation Impulses with Fredholm Operator of Index Zero

Author(s):

S. Balamuralitharan

Faculty of Engineering and Technology, Department of Mathematics,
SRM University, Kattankulathur - 603 203,Tamil Nadu, INDIA.

balamurali.maths@gmail.com

Abstract:

The purpose of this paper is to study the periodic solution of a certain class of third and fourth order delay differential equation impulses with Fredholm operator of index zero. We obtain the existence of periodic solution and Mawhin’s continuation theorem. The delay conditions for the Schwarz inequality of the periodic solutions are also obtained.An example is also furnished which demonstrates validity of main result. We establish some new sufficient conditions which ensure that every solution of this equation impulses to at least one periodic solution.

Keywords

References:

  1. Zhimin He and Weigao Ge, Oscillations of second-order nonlinear impulsive ordinary differential equations, Journal of Computational and AppliedMathematics, 158 (2), 397-406, 2003
  2. Jiaowan Luo and Lokenath Debnath , Oscillations of Second-Order Nonlinear Ordinary Differential Equations with Impulses, Journal of MathematicalAnalysis and Applications, 240 (1), 105-114, 1999
  3. C. Fabry, J. Mawhin, M. Nkashama; A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations, Bull. London Math. soc., 18, 173-180, 1986
  4. K. Gopalsamy, B. G. Zhang; On delay differential equations with impulses, J. Math. Anal. Appl., 139, 110-122, 1989
  5. I. T. Kiguradze, B. Puza; On periodic solutions of system of differential equations with deviating arguments, Nonlinear Anal., 42, 229-242, 2000
  6. V. Lakshmikantham, D. D. Bainov, P. S. Simeonov; Theory of impulsive differential equations, World Scientific Singapore, 1989
  7. Lijun Pan, Periodic solutions for higher order differential equations with deviating argument, Journal of Mathematical Analysis and Applications, 343 (2), 904-918, 2008
  8. S. Lu, W. Ge; Sufficient conditions for the existence of periodic solutions to some second order differential equation with a deviating argument, J. Math. Anal. Appl., 308, 393-419, 2005
  9. J. H. Shen; The nonoscillatory solutions of delay differential equations with impulses, Appl. Math. comput., 77, 153-165, 1996
  10. Hale, J. K. and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer- Verlag, New York, Applied Mathematical Sciences, 99, 1993
  11. Li, L. M., Stability of linear neutral delay-differential systems, Bull. Aust. Math. Soc., 38, 339-344, 1998
  12. Mahmoud, MS. and Al-Muthairi NF, Quadratic stabilization of continuous time systems with state-delay and norm- bounded time-varying uncertainties, Automatica, 32 , 2135-2139, 1994
  13. Park, Ju-H. and Won, S, A note on stability of neutral delay-differential system, Journal of the Franklin Institute, 336 , 543-548, 1999
  14. R. E Gaines, J. L Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977
  15. S Lu, W Ge, On the existence of periodic solutions of second order differential equations with deviating arguments, Acta. Math. Sinica, 45, 811-818, 2002
  16. S Lu, W Ge, Periodic solutions for a kind of second order differential equations with multiple deviating arguments, Applied Mathematics and Computation, 146 , 195-209, 2003
  17. S. Lu, W. Ge, Some new results on the existence of periodic solutions to a kind of Rayleigh equation with a deviating argument, Nonlinear Anal., 56 , 501-514, 2004
  18. S. W Ma, Z. C Wang, J. S Yu, Coincidence degree and periodic solutions of Duffing equations, Nonlinear Analysis, 34 , 443-460, 1998
  19. Sadek AI, Stability and bovndedness of a kind of Third-order Delay Differential System, Appl. Math. Letters, 91 , 657-662, 2003
  20. Shiping Lu, Weigao Ge, Sufficient conditions for the existence of periodic solutions to some second order differential equations with a deviating argument , Journal of Mathematical Analysis and Applications, 308, 393-419, 2005
  21. Shiping Lu, Weigao Ge, Zuxiou Zheng, Periodic solutions for a kind of Rayleigh equation with a deviating argument, , Applied Mathematics Letters, 17, 443-449, 2004
  22. Wang, GQ. A priori bounds for periodic solutions of a delay Rayleigh equation, Applied Mathematics Letters, 12 , 41-44, 1999
  23. Yoshizawa, T, Stability Theorem by Liapvnov’s Second Method, The Mathematical Society of Japan, 1966.

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