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Piecewise Almost Periodic Functions and Generalizations on Translation Time Scales

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Theory of Translation Closedness for Time Scales

Part of the book series: Developments in Mathematics ((DEVM,volume 62))

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Abstract

This chapter is mainly concerned with piecewise almost periodic functions and their generalizations on various types of translation time scales and is divided into four sections. Section 4.1 establishes some basic results of piecewise almost periodic functions on complete-closed time scales (CCTS). In Sect. 4.2, the properties of weighted piecewise almost periodic functions are provided on CCTS. In Sect. 4.3, based on almost complete-closed time scales (ACCTS), the concept of double-almost periodic functions are introduced and some fundamental results are established. Moreover, the generalization of piecewise double-almost periodic functions is introduced and investigated.

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References

  1. Akhmet, M.U., Turan, M.: The differential equations on time scales through impulsive differential equations. Nonlinear Anal. Theory Methods Appl. 65, 2043–2060 (2006)

    Article  MathSciNet  Google Scholar 

  2. Akhmet, M.U., Turan, M.: Differential equations on variable time scales. Nonlinear Anal. Theory Methods Appl. 70, 1175–1192 (2009)

    Article  MathSciNet  Google Scholar 

  3. Akhmet, M.U., Yılmaz, E.: Impulsive Hopfield-type neural network system with piecewise constant argument. Nonlinear Anal. Real World Appl. 11, 2584–2593 (2010)

    Article  MathSciNet  Google Scholar 

  4. Akhmet, M.U., Aruğaslan, D., Yılmaz, E..: Method of Lyapunov functions for differential equations with piecewise constant delay. J. Comput. Appl. Math. 235, 4554–4560 (2011)

    Article  MathSciNet  Google Scholar 

  5. Akhmet, M.U., Turan, M.: Bifurcation of three-dimensional discontinuous cycles. Nonlinear Anal. Theory Methods Appl. 71, 2090–2102 (2009)

    Article  MathSciNet  Google Scholar 

  6. Afonso, S.M.S., Bonotto, E.M., Federson, M., Schwabik, S̆.: Discontinuous local semiflows for Kurzweil equations leading to LaSalle’s Invariance Principle for differential systems with impulses at variable times. J. Differ. Equ. 250, 2969–3001 (2011)

    Google Scholar 

  7. Afonso, S.M.S., Bonotto, E.M., Federson, M., Gimenes, L.P.: Boundedness of solutions of functional differential equations with variable impulses via generalized ordinary differential equations. Math. Nachrichten 285, 545–561 (2012)

    Article  MathSciNet  Google Scholar 

  8. Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: On first order impulsive dynamic equations on time scales. J. Differ. Equ. Appl. 10, 541–548 (2004)

    Article  MathSciNet  Google Scholar 

  9. Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Impulsive functional dynamic equations on time scales. Dyn. Syst. Appl. 15, 43–52 (2006)

    MathSciNet  MATH  Google Scholar 

  10. Bainov, D., Milusheva, S.D.: Justification of the averaging method for a class of functional differential equations with impulses. J. Lond. Math. Soc. 25, 309–331 (1982)

    MathSciNet  MATH  Google Scholar 

  11. Chang, Y.K., Li, W.: Existence results for impulsive dynamic equations on time scales with nonlocal initial conditions. Math. Comput. Model. 43, 377–384 (2006)

    Article  MathSciNet  Google Scholar 

  12. Graef, J.R., Ouahab, A.: Nonresonance impulsive functional dynamic equations on times scales. Int. J. Appl. Math. Sci. 2, 65–80 (2005)

    MathSciNet  MATH  Google Scholar 

  13. Hatipoǧlu, V.F., Uçar, D., Koçak, Z.F.: ψ-exponential stability of nonlinear impulsive dynamic equations on time scales. Abstr. Appl. Anal. 2013, Article ID 103894, 1–5 (2013)

    Google Scholar 

  14. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S., Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)

    Book  Google Scholar 

  15. Mesquita, J.G.: Measure functional differential equations and impulsive functional dynamic equations on time scales. Doctoral dissertation, Instituto de Ciencias Matematicas e de Computacao (2012)

    Google Scholar 

  16. Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995)

    Book  Google Scholar 

  17. Tang, S., Cheke, R.A., Xiao, Y.: Optimal impulsive harvesting on non-autonomous Beverton-Holt difference equations. Nonlinear Anal. Theory Methods Appl. 65, 2311–2341 (2006)

    Article  Google Scholar 

  18. Wang, C., Agarwal, R.P., O’Regan, D.: Weighted piecewise pseudo double-almost periodic solution for impulsive evolution equations. J. Nonlinear Sci. Appl. 10, 3863–3886 (2017)

    Article  MathSciNet  Google Scholar 

  19. Wang, C., Agarwal, R.P., O’Regan, D.: Compactness criteria and new impulsive functional dynamic equations on time scales. Adv. Differ. Equ. 2016, 197, 1–41 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Wang, C., Agarwal, R.P., Sakthivel, R.: Almost periodic oscillations for delay impulsive stochastic Nicholson’s blowflies timescale model. Comput. Appl. Math. 37, 3005–3026 (2018)

    Article  MathSciNet  Google Scholar 

  21. Wang, C., Agarwal, R.P.: Almost periodic solution for a new type of neutral impulsive stochastic Lasota-Wazewska timescale model. Appl. Math. Lett. 70, 58–65 (2017)

    Article  MathSciNet  Google Scholar 

  22. Wang, C., Agarwal, R.P., O’Regan, D.: Matrix measure on time scales and almost periodic analysis of the impulsive Lasota-Wazewska model with patch structure and forced perturbations. Math. Methods Appl. Sci. 39, 5651–5669 (2016)

    Article  MathSciNet  Google Scholar 

  23. Wang, C., Agarwal, R.P.: Almost periodic dynamics for impulsive delay neural networks of a general type on almost periodic time scales. Commun. Nonlinear Sci. Numer. Simul. 36, 238–251 (2016)

    Article  MathSciNet  Google Scholar 

  24. Wang, C.: Piecewise pseudo almost periodic solution for impulsive non-autonomous high-order Hopfield neural networks with variable delays. Neurocomputing 171, 1291–1301 (2016)

    Article  Google Scholar 

  25. Wang, C., Agarwal, R.P.: Uniformly rd-piecewise almost periodic functions with applications to the analysis of impulsive Δ-dynamic system on time scales. Appl. Math. Comput. 259, 271–292 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Wang, C.: Existence and exponential stability of piecewise mean-square almost periodic solutions for impulsive stochastic Nicholson’s blowflies model on time scales. Appl. Math. Comput. 248, 101–112 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Wang, C., Agarwal, R.P., O’Regan, D., Sakthivel, R.:Discontinuous generalized double-almost-periodic functions on almost-complete-closed time scales. Bound Value Probl. 165 (2019). https://doi.org/10.1186/s13661-019-1283-0

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Authors and Affiliations

  1. Department of Mathematics, Yunnan University, Kunming, Yunnan, China

    Chao Wang

  2. Department of Mathematics, Texas A&M University—Kingsville, Kingsville, TX, USA

    Ravi P. Agarwal

  3. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

    Donal O’Regan

  4. Department of Applied Mathematics, Bharathiar University, Coimbatore, Tamil Nadu, India

    Rathinasamy Sakthivel

Authors
  1. Chao Wang
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  2. Ravi P. Agarwal
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  3. Donal O’Regan
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  4. Rathinasamy Sakthivel
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Wang, C., Agarwal, R.P., O’Regan, D., Sakthivel, R. (2020). Piecewise Almost Periodic Functions and Generalizations on Translation Time Scales. In: Theory of Translation Closedness for Time Scales . Developments in Mathematics, vol 62. Springer, Cham. https://doi.org/10.1007/978-3-030-38644-3_4

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