309
Views
22
CrossRef citations to date
0
Altmetric
Original Articles

Advanced impulsive differential equations with piecewise constant arguments

, &
Pages 175-187 | Received 13 Jul 2009, Published online: 09 Jun 2011

References

  • Aftabizadeh , A.R. and Wiener , J. 1985 . Oscillatory properties of first order linear functional differential equations . Applicable Anal., , 20 : 165 – 187 . Doi:10.1080/00036818508839568
  • Aftabizadeh , A.R. and Wiener , J. 1988 . Oscillatory and periodic solutions for systems of two first order linear differential equations with piecewise constant argument . Appl. Anal., , 26 : 327 – 333 . Doi:10.1080/00036818808839717
  • Aftabizadeh , A.R. , Wiener , J. and Xu , J.Ming . 1987 . Oscillatory and periodic solutions of delay differential equations with piecewise constant argument . Proc. of American Math. Soc., , 99 : 673 – 679 . Doi:10.2307/2046474
  • Alwan , M.S. and Liu , X. 2009 . Stability of singularly perturbed switched systems with time delay and impulsive effects . Nonlinear Analysis, , 71 : 4297 – 4308 . Doi:10.1016/j.na.2009.02.131.
  • Bereketoglu , H. and Karakoc , F. 2008 . Asymptotic constancy for impulsive delay differential equations . Dynamic Systems and Applications, , 17 : 71 – 84 .
  • Bereketoglu , H. and Pituk , M. 2003 . Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays . Discrete and Continuous Dynamical Systems, Supp , : 100 – 107 .
  • Busenberg , S. and Cooke , K.L. 1982 . “ Models of vertically transmitted diseases with sequential‐continuous dynamics ” . In Nonlinear Phenomena in Mathematical Sciences , Edited by: Lakshmikantham , V. 179 – 187 . Academic Press .
  • Cooke , K.L. and Wiener , J. 1984 . Retarded differential equations with piecewise constant delays . J. Math. Anal. Appl., , 99 : 265 – 297 . Doi:10.1016/0022–247X(84)90248–8.
  • De la Sen , M. 2006 . Stability of impulsive time‐varying systems and compactness of the operators mapping the input space into the state and output spaces . J. Math. Anal. Appl., , 321 : 621 – 650 . Doi:10.1016/j.jmaa.2005.08.038.
  • Gopalsamy , K. and Zhang , B.G. 1989 . On delay differential equations with impulses . J. Math. Anal. Appl., , 139 : 110 – 122 . Doi:10.1016/0022–247X(89)90232–1.
  • Gyori , I. 1991 . On the approximation of the solutions of delay differential equations by using piecewise constant arguments . Internat. J. Math. Math. Sci., , 14 : 111 – 126 . Doi:10.1155/S016117129100011X.
  • Gyori , I. and Ladas , G. 1989 . Linearized oscillations for equations with piecewise constant argument . Differential and Integral Equations, , 2 : 123 – 131 .
  • Huang , Y.K. 1990 . Oscillations and asymptotic stability of solutions of first order neutral differential equations with piecewise constant argument . J. Math. Anal. Appl., , 149 : 70 – 85 . Doi:10.1016/0022–247X(90)90286‐O.
  • Karakoc , F. and Bereketoglu , H. 2009 . Some results for linear impulsive delay differential equations . Dynamics of Cont. Discrete and Impulsive Systems Series A: Mathematical Analysis, , 16 : 313 – 326 .
  • Li , J. and Shen , J. 2002 . Periodic boundary value problems of impulsive differential equations with piecewise constant argument . J. Nat. Sci. Hunan Norm. Univ., , 25 : 5 – 9 .
  • Li , X. and Weng , P. 2004 . Impulsive stabilization of two kinds of second‐order linear delay differential equations . J. Math. Anal. Appl., , 291 : 270 – 281 . Doi:10.1016/j.jmaa.2003.11.002.
  • Liang , H. and Wang , G. 2009 . Existence and uniqueness of periodic solutions for a delay differential equation with piecewise constant arguments . Port. Math., , 66 (1) : 1 – 12 . Doi:10.4171/PM/1827
  • Luo , Z. and Luo , Y. 2009 . Asymptotic stability for impulsive functional differential equations . Appl. Math. Mech., , 30 (10) : 1317 – 1324 . Doi:10.1007/s10483–009–1011‐z.
  • Muroya , Y. 2008 . New contractivity condition in a population model with piecewise constant arguments . J. Math. Anal. Appl., , 346 (1) : 65 – 81 . Doi:10.1016/j.jmaa.2008.05.025.
  • Pinto , M. 2009 . Asymptotic equivalence of nonlinear and quasilinear differential equations with piecewise constant arguments . Math. Comput. Modelling, , 49 (9–10) : 1750 – 1758 . Doi:10.1016/j.mcm.2008.10.001.
  • Shah , S.M. and Wiener , J. 1983 . Advanced differential equations with piecewise constant argument deviations . Internat. J. Math. Math. Sci., , 6 : 671 – 703 . Doi:10.1155/S0161171283000599
  • Wang , Y. and Yan , J. 2006 . Oscillation of a differential equation with fractional delay and piecewise constant arguments . Computers and Mathematics with Applications, , 52 : 1099 – 1106 . Doi:10.1016/j.camwa.2006.03.032.
  • Weng , A. and Sun , J. 2009 . Impulsive stabilization of second‐order nonlinear delay differential systems . Applied Mathematics and Computation, , 214 : 95 – 101 . Doi:10.1016/j.amc.2009.03.071.
  • Wiener , J. 1983 . “ Differential equations with piecewise constant delays ” . In Trends in the Theory and Practice of Nonlinear Differential Equations , Edited by: Lakshmikantham , V. 547 – 552 . Marcel Dekker .
  • Wiener , J. 1994 . Generalized Solutions of Functional Differential Equations. , Singapore : World Scientific .
  • Wiener , J. and Lakshmikantham , V. 2000 . Differential equations with piecewise constant argument and impulsive equations . Nonlinear Stud., , 7 : 60 – 69 .
  • Yuan , R. 2002 . The existence of almost periodic solutions of retarded differential equations with piecewise constant argument . Nonlinear Analysis, , 48 : 1013 – 1032 . Doi:10.1016/S0362–546X(00)00231–5.
  • Zhang , Y. , Zhao , A. and Yan , J. 1997 . Oscillation criteria for impulsive delay differential equations . J. Math. Anal. Appl., , 205 : 461 – 470 . Doi:10.1006/jmaa.1997.5211

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.