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Periodicity Using Shift Periodic Operators

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Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales

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Summary

In Chap. 7, we had to require the time scale to be additive in order to show the existence of periodic solutions. We devote this chapter to the study of periodic solutions of functional delay dynamical systems. The periodic delays are expressed in terms of shift operators (see Definition 1.4.4) that were developed by Adıvar (Electron J Qual Theory Differ Equ 2010(7):1–22, 2010). We begin the chapter with the discussion of periodicity in shift operators over different time scales and provide examples. We proceed to unifying Floquet theory for homogeneous and nonhomogeneous dynamical systems. Then we consider neutral dynamical systems using Floquet theory and Krasnosel’skiı̆ fixed point theorem. We move to the existence of almost automorphic solutions of delayed neutral dynamic system using exponential dichotomy and Krasnosel’skiı̆ fixed point theorem. This chapter should serve as the foundation and guidance for future research on periodicity using the well-defined shift operators, by the authors. Part of this chapter is new and the rest can be found in Adıvar (Math Slovaca 63(4):817–828, 2013), Adıvar and Koyuncuoğlu (Appl Math Comput 273:1208–1233, 2016; Appl Math Comput 242:328–339, 2014), and Koyuncuoğlu (q-Floquet Theory and Its Extensions to Time Scales Periodic in Shifts. Thesis (Ph.D.), 2016).

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References

  1. L. Adamec, Simple Floquet theory on time scales, in Proceedings of the Eighth International Conference on Difference Equations and Applications (Chapman & Hall/CRC, London/Boca Raton, 2005), pp. 1–6

    Google Scholar 

  2. M. Adıvar, Function bounds for solutions of Volterra integro dynamic equations on the time scales. Electron. J. Qual. Theory Differ. Equ. 2010(7), 1–22 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Adıvar, Principal matrix solutions and variation of parameters for Volterra integro-dynamic equations on time scales. Glasg. Math. J. 53(3), 463–480 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Adıvar, A new periodicity concept for time scales. Math. Slovaca 63(4), 817–828 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Adıvar, H.C. Koyuncuoğlu, Floquet theory based on new periodicity concept for hybrid systems involving q-difference equations. Appl. Math. Comput. 273, 1208–1233 (2016)

    MathSciNet  MATH  Google Scholar 

  6. M. Adıvar, H.C. Koyuncuoğlu, Y.N. Raffoul, Existence of periodic solutions in shifts δ ± for neutral nonlinear dynamic systems. Appl. Math. Comput. 242, 328–339 (2014)

    MathSciNet  MATH  Google Scholar 

  7. R. Agarwal, M. Bohner, A. Domoshnitsky, Y. Goltser, Floquet theory and stability of nonlinear integro-differential equations. Acta Math. Hungar. 109(4), 305–330 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. R.P. Agarwal, V. Lupulescu, D. O’Regan, A. Younus, Floquet theory for a Volterra integro-dynamic system. Appl. Anal. 93(9), 2002–2013 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. C.D. Ahlbrandt, J. Ridenhour, Floquet theory for time scales and Putzer representations of matrix logarithms. J. Differ. Equ. Appl. 9(1), 77–92 (2003). In honour of Professor Allan Peterson on the occasion of his 60th birthday, Part II

    Google Scholar 

  10. P.K. Anh, H.T.N. Yen, Floquet theorem for linear implicit nonautonomous difference systems. J. Math. Anal. Appl. 321(2), 921–929 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. S.A. Belbas, Floquet theory for integral and integro-differential equations. Appl. Math. Comput. 223, 327–345 (2013)

    MathSciNet  MATH  Google Scholar 

  12. M. Bohner, R. Chieochan, Floquet theory for q-difference equations. Sarajevo J. Math. 8(21), 355–366 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications (Birkhäuser Boston, Inc., Boston, 2001)

    Book  MATH  Google Scholar 

  14. M. Bohner, G.Sh. Guseinov, B. Karpuz, Properties of the Laplace transform on time scales with arbitrary graininess. Integral Transform. Spec. Funct. 22(11), 785–800 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. J.J. Dacunha, Lyapunov Stability and Floquet Theory for Nonautonomous Linear Dynamic Systems on Time Scales. Thesis (Ph.D.)–Baylor University (ProQuest LLC, Ann Arbor, 2004)

    Google Scholar 

  16. J.J. Dacunha, Transition matrix and generalized matrix exponential via the Peano-Baker series. J. Differ. Equ. Appl. 11(15), 1245–1264 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. J.J. Dacunha, J.M. Davis, A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems. J. Differ. Equ. 251(11), 2987–3027 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques. Ann. Sci. École Norm. Sup. 12, 47–88 (1883)

    Article  MathSciNet  MATH  Google Scholar 

  19. X. Fu, X. Liu, Existence of periodic solutions for abstract neutral non-autonomous equations with infinite delay. J. Math. Anal. Appl. 325(1), 249–267 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. H.R. Henríquez, M. Pierri, A. Prokopczyk, Periodic solutions of abstract neutral functional differential equations. J. Math. Anal. Appl. 385(2), 608–621 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. S. Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus. Results Math. 18(1–2), 18–56 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  22. M.N. Islam, Y.N. Raffoul, Periodic solutions of neutral nonlinear system of differential equations with functional delay. J. Math. Anal. Appl. 331(2), 1175–1186 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. E.R. Kaufmann, Y.N. Raffoul, Periodic solutions for a neutral nonlinear dynamical equation on a time scale. J. Math. Anal. Appl. 319(1), 315–325 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. W.G. Kelley, A.C. Peterson, Difference Equations: An introduction with applications, 2nd edn. (Harcourt/Academic, Boston/San Diego, 2001) .

    Google Scholar 

  25. H.C. Koyuncuoğlu, q-Floquet Theory and Its Extensions to Time Scales Periodic in Shifts. Thesis (Ph.D.)–Izmir University of Economics, Izmir University of Economics, 2016

    Google Scholar 

  26. P. Kuchment, Floquet Theory for Partial Differential Equations. Operator Theory: Advances and Applications, vol. 60 (Birkhäuser Verlag, Basel, 1993)

    Google Scholar 

  27. C. Pötzsche, S. Siegmund, F. Wirth, A spectral characterization of exponential stability for linear time-invariant systems on time scales. Discrete Contin. Dyn. Syst. 9(5), 1223–1241 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Y.N. Raffoul, Existence of periodic solutions in neutral nonlinear difference systems with delay. J. Differ. Equ. Appl. 11(13), 1109–1118 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Y.N. Raffoul, Existence of positive periodic solutions in neutral nonlinear equations with functional delay. Rocky Mountain J. Math. 42(6), 1983–1993 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Broadwell College of Business and Economics, Fayetteville State University, Fayetteville, NC, USA

    Murat Adıvar

  2. Department of Mathematics, University of Dayton, Dayton, OH, USA

    Youssef N. Raffoul

Authors
  1. Murat Adıvar
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  2. Youssef N. Raffoul
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Adıvar, M., Raffoul, Y.N. (2020). Periodicity Using Shift Periodic Operators. In: Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales. Springer, Cham. https://doi.org/10.1007/978-3-030-42117-5_8

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