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International Journal of Differential Equations
Volume 2017 (2017), Article ID 7958398, 5 pages
https://doi.org/10.1155/2017/7958398
Research Article

Existence and Uniqueness of Solution of Stochastic Dynamic Systems with Markov Switching and Concentration Points

Department of the System Analysis and Insurance and Financial Mathematics, Yuriy Fedkovych Chernivtsi National University, 28 Unversitetska St., Chernivtsi 58012, Ukraine

Correspondence should be addressed to Taras Lukashiv

Received 5 December 2016; Revised 21 March 2017; Accepted 28 March 2017; Published 30 April 2017

Academic Editor: Elena Braverman

Copyright © 2017 Taras Lukashiv and Igor Malyk. 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. K. Gopalsamy and B. G. Zhang, “On delay differential equations with impulses,” Journal of Mathematical Analysis and Applications, vol. 139, no. 1, pp. 110–122, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. D. Xu and Z. Yang, “Impulsive delay differential inequality and stability of neural networks,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 107–120, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
  4. A. M. Samoilenko and Y. V. Teplinskii, CoUntable Systems of Differential Equations, VSP, Boston, 2003.
  5. A. M. Samoilenko and O. Stanzhytskyi, Qualitative and Asymptotic Analysis of Differential Equations with Random Perturbations, World Scientific, Singapore, 2011.
  6. T. O. Lukashiv, I. V. Yurchenko, and V. K. Yasinskii, “Lyapunov function method for investigation of stability of stochastic Ito random-structure systems with impulse Markov switchings. I. General theorems on the stability of stochastic impulse systems,” Cybernetics and Systems Analysis, vol. 45, no. 3, pp. 464–476, 2009. View at Publisher · View at Google Scholar · View at Scopus
  7. T. O. Lukashiv, V. K. Yasinskiy, and E. V. Yasinskiy, “Stabilization of stochastic diffusive dynamical systems with impulse markov switchings and parameters. part I. stability of impulse stochastic systems with markov parameters,” Journal of Automation and Information Sciences, vol. 41, no. 2, pp. 1–24, 2009. View at Google Scholar · View at MathSciNet
  8. T. O. Lukashiv and V. K. Yasynskyy, “Probabilistic stability in the whole of the stochastic dynamical systems of the random structure with constant delay,” Volyn' Mathematical Visnyk. Applied Mathematics, vol. 10, no. 191, pp. 140–151, 2013. View at Google Scholar
  9. R. Iwankiewicz, “Equation for probability density of the response of a dynamic system to Erlang reneval random impulse processes,” in Advances in Reliability and Optimization of Structural Systems, pp. 107–113, Taylor & Francis, Aalborg, Denmark, 2005. View at Google Scholar
  10. V. S. Denyssenko, “Stability of fuzzy impulsive Takagi-Sugeno' systems: method of linear matrix inequalities,” Reports of the National Academy of Sciences of Ukraine, vol. 11, pp. 66–73, 2008. View at Google Scholar
  11. Å. P. Trofymchuk and Å. P. Trofymchuk, “Switching systems with fixed moments shocks the general location: existence, uniqueness of the solution and the correctness of the Cauchy problem,” Ukrainian Mathematical Journal, vol. 42, no. 2, pp. 230–237, 1990. View at Google Scholar
  12. R. F. Nagayev, in Mechanical processes with repeated damped collisions, Nauka, Moskow, Russia, 1985.
  13. J. L. Doob, Stochastic Processes, John Wiley & Sons, New York, NY, USA, 1953. View at MathSciNet
  14. À. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Naukova dumka, Êiev, 1987.
  15. V. K. Yasynskyy and I. V. Malyk, “Analysis of fluctuations of a parametric vacuum tube oscillator with delayed feedback,” Cybernetics and Systems Analysis, vol. 51, no. 3, pp. 400–409, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus