About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2011 (2011), Article ID 178568, 19 pages
http://dx.doi.org/10.1155/2011/178568
Research Article

New Stability Conditions for Linear Differential Equations with Several Delays

1Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel
2Department of Mathematics and Statistics, University of Calgary, 2500 University Drive Northwest, Calgary, AB, Canada T2N 1N4

Received 28 January 2011; Accepted 4 April 2011

Academic Editor: Josef Diblík

Copyright © 2011 Leonid Berezansky and Elena Braverman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. L. Berezansky and E. Braverman, “On stability of some linear and nonlinear delay differential equations,” Journal of Mathematical Analysis and Applications, vol. 314, no. 2, pp. 391–411, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. L. Berezansky and E. Braverman, “On exponential stability of linear differential equations with several delays,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 1336–1355, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. L. Berezansky and E. Braverman, “Explicit exponential stability conditions for linear differential equations with several delays,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 246–264, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. T. Yoneyama and J. Sugie, “On the stability region of scalar delay-differential equations,” Journal of Mathematical Analysis and Applications, vol. 134, no. 2, pp. 408–425, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. T. Yoneyama, “The 3/2 stability theorem for one-dimensional delay-differential equations with unbounded delay,” Journal of Mathematical Analysis and Applications, vol. 165, no. 1, pp. 133–143, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. T. Krisztin, “On stability properties for one-dimensional functional-differential equations,” Funkcialaj Ekvacioj, vol. 34, no. 2, pp. 241–256, 1991. View at Zentralblatt MATH
  7. J. W.-H. So, J. S. Yu, and M.-P. Chen, “Asymptotic stability for scalar delay differential equations,” Funkcialaj Ekvacioj, vol. 39, no. 1, pp. 1–17, 1996. View at Zentralblatt MATH
  8. I. Győri, F. Hartung, and J. Turi, “Preservation of stability in delay equations under delay perturbations,” Journal of Mathematical Analysis and Applications, vol. 220, no. 1, pp. 290–312, 1998. View at Publisher · View at Google Scholar
  9. I. Győri and F. Hartung, “Stability in delay perturbed differential and difference equations,” in Topics in Functional Differential and Difference Equations (Lisbon, 1999), vol. 29 of Fields Inst. Commun., pp. 181–194, American Mathematical Society, Providence, RI, USA, 2001.
  10. S. A. Gusarenko and A. I. Domoshnitskiĭ, “Asymptotic and oscillation properties of first-order linear scalar functional-differential equations,” Differential Equations, vol. 25, no. 12, pp. 1480–1491, 1989. View at Zentralblatt MATH
  11. T. Wang, “Inequalities and stability for a linear scalar functional differential equation,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 33–44, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. J. H. Shen and J. S. Yu, “Asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients,” Journal of Mathematical Analysis and Applications, vol. 195, no. 2, pp. 517–526, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. X. Wang and L. Liao, “Asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients,” Journal of Mathematical Analysis and Applications, vol. 279, no. 1, pp. 326–338, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. Z. Zhang and Z. Wang, “Asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients,” Annals of Differential Equations, vol. 17, no. 3, pp. 295–305, 2001. View at Zentralblatt MATH
  15. Z. Zhang and J. Yu, “Asymptotic behavior of solutions of neutral difference equations with positive and negative coefficients,” Mathematical Sciences Research Hot-Line, vol. 2, no. 6, pp. 1–12, 1998. View at Zentralblatt MATH
  16. N. V. Azbelev and P. M. Simonov, Stability of Differential Equations with Aftereffect, vol. 20 of Stability and Control: Theory, Methods and Applications, Taylor & Francis, London, UK, 2003.
  17. A. Ivanov, E. Liz, and S. Trofimchuk, “Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima,” The Tohoku Mathematical Journal, vol. 54, no. 2, pp. 277–295, 2002. View at Zentralblatt MATH
  18. E. Liz, V. Tkachenko, and S. Trofimchuk, “A global stability criterion for scalar functional differential equations,” SIAM Journal on Mathematical Analysis, vol. 35, no. 3, pp. 596–622, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. X. H. Tang, “Asymptotic behavior of delay differential equations with instantaneously terms,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 342–359, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. V. V. Malygina, “Some criteria for stability of equations with retarded argument,” Differential Equations, vol. 28, no. 10, pp. 1398–1405, 1992. View at Zentralblatt MATH
  21. V. V. Malygina, “Stability of solutions of some linear differential equations with aftereffect,” Russian Mathematics, vol. 37, no. 5, pp. 63–75, 1993.
  22. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1993.
  23. N. V. Azbelev, L. Berezansky, and L. F. Rahmatullina, “A linear functional-differential equation of evolution type,” Differential Equations, vol. 13, no. 11, pp. 1331–1339, 1977.
  24. I. Győri and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, NY, USA, 1991.
  25. L. Berezansky and E. Braverman, “On non-oscillation of a scalar delay differential equation,” Dynamic Systems and Applications, vol. 6, no. 4, pp. 567–580, 1997. View at Zentralblatt MATH
  26. L. Berezansky and E. Braverman, “Preservation of exponential stability for linear non-autonomous functional differential systems,” Automatica, vol. 46, no. 12, pp. 2077–2081, 2010. View at Publisher · View at Google Scholar
  27. L. Berezansky and E. Braverman, “Preservation of the exponential stability under perturbations of linear delay impulsive differential equations,” Zeitschrift für Analysis und ihre Anwendungen, vol. 14, no. 1, pp. 157–174, 1995. View at Zentralblatt MATH
  28. G. Ladas, Y. G. Sficas, and I. P. Stavroulakis, “Asymptotic behavior of solutions of retarded differential equations,” Proceedings of the American Mathematical Society, vol. 88, no. 2, pp. 247–253, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. A. D. Myshkis, Differential Equations with Retarded Argument, Nauka, Moscow, Russia, 1951.
  30. N. Krasovskii, Stability of Motion, Nauka, Moscow, Russia, 1959, translation, Stanford University Press, 1963.
  31. E. Liz and M. Pituk, “Exponential stability in a scalar functional differential equation,” Journal of Inequalities and Applications, Article ID 37195, 10 pages, 2006. View at Zentralblatt MATH
  32. I. Győri, “Interaction between oscillations and global asymptotic stability in delay differential equations,” Differential and Integral Equations, vol. 3, no. 1, pp. 181–200, 1990.
  33. T. Faria and W. Huang, “Special solutions for linear functional differential equations and asymptotic behaviour,” Differential and Integral Equations, vol. 18, no. 3, pp. 337–360, 2005.
  34. I. Győri and M. Pituk, “Stability criteria for linear delay differential equations,” Differential and Integral Equations, vol. 10, no. 5, pp. 841–852, 1997. View at Zentralblatt MATH
  35. L. Berezansky and E. Braverman, “On exponential stability of a linear delay differential equation with an oscillating coefficient,” Applied Mathematics Letters, vol. 22, no. 12, pp. 1833–1837, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH