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Method of Dynamic Integral Inequalities

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Stability Theory for Dynamic Equations on Time Scales

Part of the book series: Systems & Control: Foundations & Applications ((SCFA))

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Abstract

In this chapter the reader will find the main integral inequalities on time scales (dynamic integral inequalities), used in stability analysis of solutions to the corresponding systems of equalities on time scales.

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References

  1. Anderson D., Bullock J., Erbe L., Peterson A., and Tran H. Nabla dynamic equations on time scales. Pan-Amer. Math. J. 14 (2003) 1–47.

    Google Scholar 

  2. Aulbach B., Hilger S. A unified approach to continuous and discrete dynamics. In: Qualitative Theory of Differential Equations (Szeged, 1988), volume 53 of Colloq. Math. Soc. János Bolyai, pp. 37–56. North-Holland, Amsterdam, 1990.

    Google Scholar 

  3. Babenko, S.V., Martynyuk, A.A. Nonlinear dynamic inequalities and stability of quasilinear systems on time scales. Nonlinear Dynamics and Systems Theory13 (1) (2013) 13–23.

    MathSciNet  MATH  Google Scholar 

  4. Beesack P.R. Gronwall Inequalities. Carleton Mathematical Lecture Notes. No 11, 1975.

    Google Scholar 

  5. Bohner M., Martynyuk A.A. Elements of stability theory of A.M. Lyapunov for dynamic equations on time scales. Int. Appl. Mech.43 (9) (2007) 949–970.

    Google Scholar 

  6. Bohner M., Peterson A. Dynamic Equations on Time Scales: An Introduction with Applications. Boston: Birkhäuser, 2001.

    Book  MATH  Google Scholar 

  7. Choi S.K., Im D.M., Koo N. Stability of linear dynamic systems on time scales. Advances in Difference Equations. Vol. 2008, Article ID 670203. — 12 p.

    Google Scholar 

  8. DaCunha J.J. Lyapunov stability theory and Floquet theory for nonautonomous linear dynamic systems on time scales. Waco, Texas, PhD Thesis, 2004.

    Google Scholar 

  9. Doan T.S., Kalauch A., Siegmund S. Exponential stability of linear time-invariant systems on time scales. Nonlinear Dynamics and Systems Theory 9(1) (2009) 37–50.

    MathSciNet  MATH  Google Scholar 

  10. Dragomir S.S. The Gronwall Type Lemmas and Applications. Timisoara: Tipografia Universitatii din Timisoara, 1987.

    MATH  Google Scholar 

  11. Du Hguyen Hun, Tien Le Huy On the exponential stability of dynamic equations on time scales. J. Math. Anal. Appl. 331 (2007) 1159–1174.

    Google Scholar 

  12. Feng Q., Meng F., Zhang Y., Zhou J. and Zheng B. Some delay integral inequalities on time scales and their applications in the theory of dynamic equations. Abstract and Applied Analysis2013 Article ID 538247.

    Google Scholar 

  13. Gard T., Hoffacker J. Asymptotic behavior of natural growth on time scales. Dynamic Systems and Appl. 12 (1–2) (2003) 131–148.

    Google Scholar 

  14. Hamza A.E. and Oraby K.M. Stability of abstract dynamic equations on time scales. Advances in Difference Equations (2012) 2012:143.

    Google Scholar 

  15. Hilger S. Analysis on measure chains: a unified approach to continuous and discrete calculus. Res. in Mathematics 18 (1990) 18–56.

    Google Scholar 

  16. Huu Du Nguyen, Thanh Dieu Nguyen and Anh Tuan Le Exponential P-stability of stochastic ∇-dynamic equations on disconnected sets. Electronic Journal of Differential Eqns. 2015 (285) (2015) 1–23.

    Google Scholar 

  17. Izobov N.A., Prokhorova R.A. Coppel-Conti Linear Differential Systems. Minsk: Belorusskaya Nauka, 2008.

    MATH  Google Scholar 

  18. Kaymakçalan B., Özgün S.A., Zafer A. Gronwall and Bihari type inequalities on time scales. In: Conference Proceedings of the Second Int. Conf. on Difference Equations (Veszprém, 1995). Amsterdam: Gordon and Breach, 1997, pp. 481–490.

    Google Scholar 

  19. Keller S. Asymptotisches Verhalten invarianter Faserbündel bei Diskretisierung und Mittelwerbildung im Rahmen der Analysis auf Zeitskalen. PhD thesis, Universität Augsburg, 1999.

    Google Scholar 

  20. Lin X.M A note on Gronwall’s inequality on time scales. Abstract and Applied Analysis2014 Article ID 623726.

    Google Scholar 

  21. Luk’yanova T.A. and Martynyuk A.A. Integral inequalities and stability of an equilibrium state on a time scale. Ukrainian Mathematical Journal62 (11) (2011) 1490–1499.

    MathSciNet  MATH  Google Scholar 

  22. Marks II R.J., Gravagne I., Davis J.M., and DaCunha J.J. Nonregressivity in switched linear circuits and mechanical systems. Mathematical and Computer Modeling43 (2006) 1383–1392.

    Article  MathSciNet  MATH  Google Scholar 

  23. Martynyuk A.A., Gutowski R. Integral Inequalities and Stability of Motion. Kiev: Naukova dumka, 1979.

    Google Scholar 

  24. Martynyuk A.A., Lakshmikantham V., Leela S. The Stability of Motion: The Method of Integral Inequalities. Kiev: Naukova dumka, 1989.

    MATH  Google Scholar 

  25. Martynyuk A.A., Martynyuk-Chernienko Yu.A. Uncertain Dynamical Systems. Stability and Motion Control. Boca Raton: CRC Press Taylor and Francis Group, 2012.

    MATH  Google Scholar 

  26. Martynyuk A.A. and Slyn’ko V.I. On a nonlinear inequality on the time scale. Differential Eqns.44 (10) (2008) 1420–1426.

    MathSciNet  MATH  Google Scholar 

  27. Martynyuk-Chernienko Yu.A. On stability of dynamic systems on time scales. Dokl. Akad. Nauk413 (1) (2007) 11–15.

    MathSciNet  MATH  Google Scholar 

  28. Nasser B.B., Boukerrioua K. and Hammami M.A. On stability and stabilization of perturbed time scale systems with Gronwall inequalities J. Math. Physics, Analysis, Geometry11 (3) (2015) 207–235.

    Google Scholar 

  29. Pachpatte B.G. Inequalities for Differential and Integral Equations. San Diego and Boston: Academic Press, 1998.

    MATH  Google Scholar 

  30. Rashford M., Siloti J., Wrolstad J. Exponential stability of dynamic equations on time scales Pan American Math.Journal16 (2) (2006) 61–73.

    Google Scholar 

  31. Tariboon J., Thiramanus Ph. and Ntouyas S.K Dynamic integral inequalities on time scales with “maxima”. J. Inequalities and Appl. (2013) 2013:564. 415–422. www.journalofinequalitiesandapplications.com/content/2013/1/564

  32. Wong Fu Hsiang, Yen Chen-Chin, Hong Chen-Huang Gronwall inequalities on time scales. Mathematical Inequalities Applications9 (1) (2006) 75–86.

    Google Scholar 

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Martynyuk, A.A. (2016). Method of Dynamic Integral Inequalities. In: Stability Theory for Dynamic Equations on Time Scales. Systems & Control: Foundations & Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-42213-8_2

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