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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 631412, 19 pages
http://dx.doi.org/10.1155/2011/631412
Research Article

Boundary-Value Problems for Weakly Nonlinear Delay Differential Systems

1Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkovskaya Street 3, 01601 Kyiv, Ukraine
2Department of Mathematics, University of Žilina, Univerzitná 8215/1, 01026 Žilina, Slovakia
3Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, 602 00 Brno, Czech Republic
4Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 10, 616 00 Brno, Czech Republic
5Department of Complex System Modeling, Faculty of Cybernetics, Taras, Shevchenko National University of Kyiv, Vladimirskaya Street 64, 01033 Kyiv, Ukraine

Received 30 January 2011; Accepted 31 March 2011

Academic Editor: Elena Braverman

Copyright © 2011 A. Boichuk et al. 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. N. V. Azbelev and V. P. Maksimov, “Equations with delayed arguments,” Differential Equations, vol. 18, pp. 1419–1441, 1983, translated in Differentsial'nye Uravneniya, Vol. 18, no. 12, pp. 2027-2050.
  2. A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary Value Problems, VSP, Boston, Mass, USA, 2004.
  3. J. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, Vol. 3, Springer, New York, NY, USA, 2nd edition, 1977.
  4. J. Mallet-Paret, “The fredholm alternative for functional-differential equations of mixed type,” Journal of Dynamics and Differential Equations, vol. 11, no. 1, pp. 1–47, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. D. Ya. Khusainov and G. V. Shuklin, “Relative controllability in systems with pure delay,” International Applied Mechanics, vol. 41, no. 2, pp. 210–221, 2005, translated in Prikladnaya Mekhanika, Vol. 41, no. 2, pp. 118-130. View at Publisher · View at Google Scholar · View at MathSciNet
  6. A. A. Boichuk, J. Diblík, D. Khusainov, and M. Růžičková, “Fredholm's boundary-value problems for differential systems with a single delay,” Nonlinear Analysis, Theory, Methods and Applications, vol. 72, no. 5, pp. 2251–2258, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. A. A. Boichuk, J. Diblík, D. Y. A. Khusainov, and M. Růžičková, “Boundary value problems for delay differential systems,” Advances in Difference Equations, vol. 2010, Article ID 593834, 10 pages, 2010. View at Zentralblatt MATH
  8. M. Medvĕd, M. Pospíšil, and L. Škripková, “Stability and the nonexistence of blowing-up solutions of nonlinear delay systems with linear parts defined by permutable matrices,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 12, pp. 3903–3911, 2011. View at Publisher · View at Google Scholar
  9. A. A. Boichuk and M. K. Grammatikopoulos, “Perturbed fredholm boundary value problems for delay differential systems,” Abstract and Applied Analysis, no. 15, pp. 843–864, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. A. A. Boichuk, V. F. Zhuravlev, and A. M. Samoilenko, “Generalized inverse operators and noether boundary-value problems,” in Proceedings of the Institute of Mathematics of the National Academy of Sciences of the Ukraine, vol. 13, p. 320, Kyiv, Ukrania, 1995.
  11. I. G. Malkin, Some Problems in the Theory of Nonlinear Oscillations, Gostekhizdat, Moscow, Russia, 1956.
  12. M. A. Krasnosel'skii, G. M. Vainikko, P. P. Zabreiko, Y. A. B. Rutitskii, and V. Ya. Stetsenko, Approximate Solution of Operator Equations, Nauka, Moscow, Russia, 1968, Translation: Noordhoff, Groningen, 1972.