Weighted piecewise pseudo double-almost periodic solution for impulsive evolution equations

Volume 10, Issue 7, pp 3863--3886 http://dx.doi.org/10.22436/jnsa.010.07.41

Publication Date: July 27, 2017 Submission Date: January 04, 2017

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Authors

Chao Wang - Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, People’s Republic of China. - Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., TX 78363-8202, Kingsville, USA. Ravi P. Agarwal - Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., TX 78363-8202, Kingsville, USA. Donal O'Regan - School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland.

Abstract

In this paper, based on the concept and properties of almost-complete closedness time scales (ACCTS), we investigate the existence of weighted pseudo double-almost periodic mild solutions for non-autonomous impulsive evolution equations. We also consider the exponential stability of weighted pseudo double-almost periodic solutions. Finally, we conclude our paper by providing several illustrative applications to different types of dynamic equations and mathematical models. These applications justify the practical usefulness of the established theoretical results.

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Keywords

• Time scales
• weighted pseudo double-almost periodic solution
• impulsive evolution equations.

MSC

•  34N05
•  43A60
•  34A37
•  34C27

References

• [1] R. P. Agarwal, M. Bohner, Basic calculus on time scales and some of its applications, Results Math., 35 (1999), 3–22.
  • View Article
  • MathSciNet
  • Google Scholar


• [2] R. P. Agarwal, M. Bohner, D. O’Regan, A. Peterson, Dynamic equations on time scales: a survey, Dynamic equations on time scales, J. Comput. Appl. Math., 141 (2002), 1–26.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] R. P. Agarwal, D. O’Regan, Some comments and notes on almost periodic functions and changing-periodic time scales, Electr. J. Math. Anal. Appl., 6 (2018), 125–136.
  • MathSciNet
  • Google Scholar
  • MATH


• [4] M. U. Akhmet, M. Turan, The differential equations on time scales through impulsive differential equations, Nonlinear Anal., 65 (2006), 2043–2060.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] S. Bochner, Beiträge zur Theorie der fastperiodischen Funktionen, (German) Math. Ann., 96 (1927), 119–147.
  • View Article
  • MathSciNet
  • Google Scholar


• [6] M. Bohner, G. S. Guseinov, Double integral calculus of variations on time scales, Comput. Math. Appl., 54 (2007), 45–57.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] M. Bohner, A. Peterson, Dynamic equations on time scales, An introduction with applications, Birkhäuser Boston, Inc., Boston, MA (2001)
  • Google Scholar


• [8] T. Caraballo, D. Cheban, Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard’s separation condition, I, J. Differential Equations, 246 (2009), 108–128.
  • View Article
  • MATH


• [9] T. Caraballo, D. Cheban, Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard’s separation condition, II, J. Differential Equations, 246 (2009), 1164–1186.
  • View Article
  • Google Scholar
  • MATH


• [10] C. Cuevas, E. Hernández, M. Rabelo , The existence of solutions for impulsive neutral functional differential equations, Comput. Math. Appl., 58 (2009), 744–757.
  • View Article
  • MathSciNet
  • Google Scholar


• [11] C. Cuevas, A. Sepúlveda, H. Soto, Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations, Appl. Math. Comput., 218 (2011), 1735–1745.
  • View Article
  • MathSciNet
  • Google Scholar


• [12] T. Diagana, Weighted pseudo almost periodic functions and applications, C. R. Math. Acad. Sci. Paris, 343 (2006), 643–646.
  • View Article
  • Google Scholar


• [13] T. Diagana, Existence of weighted pseudo-almost periodic solutions to some classes of nonautonomous partial evolution equations, Nonlinear Anal., 74 (2011), 600–615.
  • MathSciNet
  • Google Scholar
  • MATH


• [14] A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York (1974)
  • Google Scholar


• [15] J. Gao, Q.-R. Wang, L.-W. Zhang, Existence and stability of almost-periodic solutions for cellular neural networks with time-varying delays in leakage terms on time scales, Appl. Math. Comput., 237 (2014), 639–649.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] E. Hernández, M. Rabello, H. R. Henríquez, Existence of solutions for impulsive partial neutral functional differential equations, J. Math. Anal. Appl., 331 (2007), 1135–1158.
  • View Article
  • MathSciNet
  • Google Scholar


• [17] S. Hilger, Ein Maßkettenkalkäul mit anwendung auf zentrumsmannigfaltigkeiten, Ph.D thesis, Universität Wäurzburg (1988)
  • Google Scholar


• [18] S. Hilger, Analysis on measure chains–a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18–56.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] B. Jackson, Partial dynamic equations on time scales, J. Comput. Appl. Math., 186 (2006), 391–415.
  • View Article
  • Google Scholar


• [20] E. R. Kaufmann, Y. N. Raffoul, Periodic solutions for a neutral nonlinear dynamical equation on a time scale, J. Math. Anal. Appl., 319 (2006), 315–325.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] J. Liang, J. Zhang, T.-J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl., 340 (2008), 1493–1499.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] C. Pötzsche, Exponential dichotomies for dynamic equations on measure chains, Proceedings of the Third World Congress of Nonlinear Analysts, Part 2, Catania, (2000), Nonlinear Anal., 47 (2001), 873–884.
  • View Article
  • MathSciNet
  • Google Scholar


• [23] A. M. Samoĭlenko, N. A. Perestyuk, Impulsive differential equations, With a preface by Yu. A. Mitropolskiĭand a supplement by S. I. Trofimchuk, Translated from the Russian by Y. Chapovsky, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, World Scientific Publishing Co., Inc., River Edge, NJ (1995)
  • Google Scholar


• [24] G. T. Stamov, Almost periodic solutions of impulsive differential equations, Lecture Notes in Mathematics, Springer, Heidelberg (2012)
  • View Article
  • Google Scholar


• [25] I. Stamova, Stability analysis of impulsive functional differential equations, De Gruyter Expositions in Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin (2009)
  • View Article
  • MathSciNet
  • Google Scholar


• [26] C. C. Tisdell, A. Zaidi, Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling, Nonlinear Anal., 68 (2008), 3504–3524.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] C. Wang, Almost periodic solutions of impulsive BAM neural networks with variable delays on time scales, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2828–2842.
  • View Article
  • MathSciNet
  • Google Scholar


• [28] C. Wang, R. P. Agarwal, A further study of almost periodic time scales with some notes and applications, Abstr. Appl. Anal., 2014 (2014), 11 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [29] C. Wang, R. P. Agarwal, Relatively dense sets, corrected uniformly almost periodic functions on time scales, and generalizations, Adv. Difference Equ., 2015 (2015), 9 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [30] C. Wang, R. P. Agarwal, A classification of time scales and analysis of the general delays on time scales with applications, Math. Methods Appl. Sci., 39 (2016), 1568–1590.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] C. Wang, R. P. Agarwal, Almost periodic dynamics for impulsive delay neural networks of a general type on almost periodic time scales , Commun. Nonlinear Sci. Numer. Simul., 36 (2016), 238–251.
  • View Article
  • MathSciNet
  • Google Scholar


• [32] C. Wang, R. P. Agarwal, D. O’Regan, Compactness criteria and new impulsive functional dynamic equations on time scales, Adv. Difference Equ., 2016 (2016), 41 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [33] C. Wang, R. P. Agarwal, D. O’Regan, Periodicity, almost periodicity for time scales and related functions, Nonauton. Dyn. Syst., 3 (2016), 24–41.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] C. Wang, R. P. Agarwal, D. O’Regan, Piecewise double-almost periodic functions on almost-complete closedness time scales with generalizations, , (submitted),


• [35] C. Y. Zhang, Pseudo-almost-periodic solutions of some differential equations, J. Math. Anal. Appl., 181 (1994), 62–76.
  • View Article
  • Google Scholar


• [36] C. Y. Zhang, Pseudo almost periodic solutions of some differential equations, II, J. Math. Anal. Appl., 192 (1995), 543–561.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH