Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지

EXISTENCE OF THREE POSITIVE PERIODIC SOLUTIONS OF NEUTRAL IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS

  • Liu, Yuji (Department of Mathematics, Guangdong University of Business Studies) ;
  • Xia, Jianye (Department of Mathematics, Guangdong University of Finance)
  • Published : 2010.01.30
Download PDF
⟨ Previous Next ⟩

Abstract

This paper is concerned with the neutral impulsive functional differential equations $$\{{x'(t)\;+\;a(t)x(t)\;=\;f(t,\;x(t\;-\;\tau(t),\;x'(t\;-\;\delta(t))),\;a.e.\;t\;{\in}\;R, \atop {\Delta}x(t_k)\;=\;b_kx(t_k),\;k\;{\in}\;Z.$$ Sufficient conditions for the existence of at least three positive T-periodic solution are established. Our results generalize and improve the known ones. Some examples are presented to illustrate the main results.

Keywords

References

  1. Y. Liu, Positive periodic solutions of nonlinear differential equations, Applied Mathematics A Journal of Chinese Universities, B(4) (2003), 1-5.
  2. J. Yu, B. Liu, Existence of periodic solutions 10r nonlinear neutral delay differential equations, Applied Mathematics A Journal of Chinese Universities, A(3)(2001), 11-15.
  3. Y. Liu, W. Ge, Stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients, Nonl. Anal., 57(2004), 363-399. https://doi.org/10.1016/j.na.2004.02.020
  4. X. Li, X. Lin, D. Jiang, X. Zhang, Existence and multiplicity of positive periodic solutions to 1unctional differential equations with impulse effects, Nonl. Anal., 62 (2005), 683-701. https://doi.org/10.1016/j.na.2005.04.005
  5. G. S. Ladde, V. Lakshmikantham, A. S. Vatsala, Monotone iterative techniques for nonlinear differential equations, Pitman Advanced Publishing Program, 1985.
  6. X. Fan, Y. Li, Positive periodic solutions of neutral functional differential equations with a parameter and impulse, Electronic J. of Differential Equations, 38(2008), 1-8.
  7. J. Yan, Existence of positive periodic solutions of impulsive functional differential equations with two parameters, J. Math. Anal. Appl , 327(2007), 854-868. https://doi.org/10.1016/j.jmaa.2006.04.018
  8. J. O. Alzabut, A necessary and sufficient conditions for the existence of periodic solutions of linear impulsive distributed delay, Discrete and Continuous Dynamical Systems, Website: www.AIMSciences.org, 15 (2007), 35-43.
  9. S. Cheng, G. Zhang, Existence of positive periodic solutions for non-autonomous functional differential equations, Electron J. Diff. Eqns., 59(2001), 1-8.
  10. S. Kang, G. Zhang, Existence of nontrivial periodic solutions for first order functional differential equations, Appl. Math. Letters, 18(2005), 101-107. https://doi.org/10.1016/j.aml.2004.07.018
  11. S. Kang, G. Zhang, B. Shi, Existence of three periodic positive solutions for a class of integral equations with parameters, J. Math. Anal. Appl., 323(2006), 654-665. https://doi.org/10.1016/j.jmaa.2005.10.072
  12. S. Padhi, S. Srivastava, Multiple periodic solutions for a nonlinear first order functional differential equations with applications to population dynamics, Appl. Math. Comput., 203(2008), 1-6. https://doi.org/10.1016/j.amc.2008.03.031
  13. X.H. Tang, Z. Jiang, Periodic solutions of first-order nonlinear functional differential equations, Nonl. Anal., 68(2008), 845-861. https://doi.org/10.1016/j.na.2006.11.041
  14. X. Han, S. Ji, Z. Ma, On the existence and multiplicity of positive periodic solutions for first-order vector differential equation, J. Math. Anal. Appl., 329(2007), 977-986. https://doi.org/10.1016/j.jmaa.2006.07.009
  15. D. Bai, Y. Xu, Periodic solutions of first order functional differential equations with periodic deviations, Comput. Math. Appl., 53(2007), 1361-1366. https://doi.org/10.1016/j.camwa.2006.12.012
  16. B. Liu, L. Huang, Existence and uniqueness of periodic for a kind of first order neutral functional differential equations, J. Math. Anal. Appl., 322(2006), 121-132. https://doi.org/10.1016/j.jmaa.2005.08.069
  17. G. Wang, Existence theorem of periodic solutions for a delay nonlinear differential equation with piecewise constant arguments, J. Math. Anal. Appl., 298(2004), 298-307. https://doi.org/10.1016/j.jmaa.2004.05.016
  18. A. Cabada, J.B. Ferreiro, J.J. Nieto, Green's function and comparinciples for first order periodic differential equations with piecewise constant arguments, J. Math. Anal. Appl., 291(2004), 690-697. https://doi.org/10.1016/j.jmaa.2003.11.022
  19. D. Li, Y. Xu, Existence of periodic solutions for fully nonlinear first-order differential equations, Nonl. Anal., 52(2003), 1095-1109. https://doi.org/10.1016/S0362-546X(02)00153-0
  20. R. Hakl, A. Lomtatidze, B. Puza, On periodic solutions 0f first order linear functional differential equations, Nonl. Anal., 49(2002), 929-945. https://doi.org/10.1016/S0362-546X(01)00147-X
  21. Ch. G. Philos, I. K. Purnaras, Periodic first order linear neutral delay differential equations, Appl. Math. Comput., 117(2001), 203-222. https://doi.org/10.1016/S0096-3003(99)00174-5