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Piecewise Continuous Stepanov-Like Almost Automorphic Functions with Applications to Impulsive Systems

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Current Trends in Mathematical Analysis and Its Interdisciplinary Applications

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Abstract

In this chapter, we discuss Stepanov-like almost automorphic function in the framework of impulsive systems. Next, we establish the existence and uniqueness of such solution of a very general class of delayed model of impulsive neural network. The coefficients and forcing term are assumed to be Stepanov-like almost automorphic in nature. Since the solution is no longer continuous, so we introduce the concept of piecewise continuous Stepanov-like almost automorphic function. We establish some basic and important properties of these functions and then prove composition theorem. Composition theorem is an important result from the application point of view. Further, we use composition result and fixed point theorem to investigate existence, uniqueness and stability of solution of the problem under consideration. Finally, we give a numerical example to illustrate our analytical findings.

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References

  1. S. Abbas, A note on Weyl pseudo almost automorphic functions and their properties. Math. Sci. (Springer), 6, 5 (2012). Art. 29

    Google Scholar 

  2. S. Abbas, Y.K. Chang, M. Hafayed, Stepanov type weighted pseudo almost automorphic sequences and their applications to difference equations. Nonlinear Stud. 21(1), 99–111 (2014)

    MathSciNet  MATH  Google Scholar 

  3. S. Abbas, L. Mahto, M. Hafayed, A.M. Alimi, Asymptotic almost automorphic solutions of impulsive neural network with almost automorphic coefficients. Neurocomputing 144, 326–334 (2014)

    Article  Google Scholar 

  4. S. Abbas, V. Kavitha, R. Murugesu, Stepanov-like weighted pseudo almost automorphic solutions to fractional order abstract integro-differential equations. Proc. Indian Acad. Sci. Math. Sci. 125(3), 323–351 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. S. Ahmad, I.M. Stamova, Global exponential stability for impulsive cellular neural networks with time-varying delays. Nonlinear Anal. 69(3), 786–795 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. W. Allegretto, D. Papini, M. Forti, Common asymptotic behavior of solutions and almost periodicity for discontinuous, delayed, and impulsive neural networks. IEEE Trans. Neural Netw. 21(7), 1110–1125 (2010)

    Article  Google Scholar 

  7. B. Ammar, F. Cherif, A.M. Alimi, Existence and uniqueness of pseudo almost-periodic solutions of recurrent neural networks with time-varying coefficients and mixed delays. IEEE Trans. Neural Netw. Learning Sys. 23(1), 109–118 (2012)

    Article  Google Scholar 

  8. D.D. Bainov, P.S. Simeonov, Systems with Impulsive Effects (Ellis Horwood Limited/John Wiley & Sons, Chichester, 1989)

    Google Scholar 

  9. D.D. Bainov, P.S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Its Applications (Longman Scientific and Technical Group, England, 1993)

    Google Scholar 

  10. S. Bochner, Continuous mappings of almost automorphic and almost periodic functions. Proc. Nat. Acad. Sci. U.S.A. 52, 907–910 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Bohr, Almost-Periodic Functions (Chelsea Publishing Company, New York City, 1947)

    MATH  Google Scholar 

  12. A. Chavez, S. Castiallo, M. Pinto, Discontinuous almost automorphic functions and almost solutions of differential equations with piecewise constant argument. Electron. J. Differ. Equ. 2014(56), 1–13 (2014)

    MathSciNet  Google Scholar 

  13. T. Diagana, Pseudo Almost Periodic Functions in Banach Spaces (Nova Science, Hauppauge, 2007)

    Google Scholar 

  14. T. Diagana, E. Hernndez, J.C. Santos, Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations. Nonlinear Anal. (71), 248–257 (2009)

    Google Scholar 

  15. V. Kavitha, S. Abbas, R. Murugesu, Existence of Stepanov-like weighted pseudo almost automorphic solutions of fractional integro-differential equations via measure theory. Nonlinear Stud. 24(4), 825–850 (2017)

    MathSciNet  MATH  Google Scholar 

  16. H.X. Li, L.L. Zhang, Stepanov-like pseudo-almost periodicity and semilinear differential equations with uniform continuity. Results Math. 59(1–2), 43–61 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Liu, C. Zhang, Composition of piecewise pseudo almost periodic functions and applications to abstract impulsive differential equations. Adv. Differ. Equ. 2013(11), 21 (2013)

    Google Scholar 

  18. L. Mahto, S. Abbas, PC-almost automorphic solution of impulsive fractional differential equations. Mediterr. J. Math. 12(3), 771–790 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. L. Mahto, S. Abbas, A. Favini, Analysis of Caputo impulsive fractional order differential equations with applications. Int. J. Differ. Equ. 2013, 1–11 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. G.M. Mophou, G.M. N’Guérékata, On some classes of almost automorphic functions and applications to fractional differential equations. Comput. Math. Appl. 59, 1310–1317 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. G.M. N’Guérékata, Topics in Almost Automorphy (Springer, New York, 2005)

    MATH  Google Scholar 

  22. G.M. N’Guérékata, A. Pankov, Integral operators in spaces of bounded, almost periodic and almost automorphic functions. Differ. Integral Equ. 21(11–12), 1155–1176 (2008)

    MathSciNet  MATH  Google Scholar 

  23. A. Pankov, Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations (Kluwer, Dordrecht, 1990)

    Book  MATH  Google Scholar 

  24. A.M. Samoilenko, N.A. Perestyuk, Differential Equations with Impulse Effects (Viska Skola, Kiev, 1987) (in Russian)

    Google Scholar 

  25. R. Samidurai, S.M. Anthoni, K. Balachandran, Global exponential stability of neutral-type impulsive neural networks with discrete and distributed delays. Nonlinear Anal. Hybrid Syst. 4(1), 103–112 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Sannay, Exponential stability in Hopfield-type neural networks with impulses. Chaos, Solitons Fractals 32(2), 456–467 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. G.T. Stamov, Impulsive cellular neural networks and almost periodicity. Proc. Jpn. Acad. Sci. 80, Ser. A, 10, 198–203 (2005)

    Google Scholar 

  28. G.T. Stamov, Almost Periodic Solutions of Impulsive Differential Equations. Lecture Notes in Mathematics, vol. 2047 (Springer, Heidelberg, 2012), pp. xx+217. ISBN: 978-3-642-27545-6

    Google Scholar 

  29. I.M. Stamova, R. Ilarionov, On global exponential stability for impulsive cellular neural networks with time-varying delays. Comput. Math. Appl. 59(11), 3508–3515 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. I.M. Stamova, G.T. Stamov, Impulsive control on global asymptotic stability for a class of impulsive bidirectional associative memory neural networks with distributed delays. Math. Comput. Model. 53(5–6), 824–831 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. G.T. Stamov, I.M. Stamova, J.O. Alzabut, Existence of almost periodic solutions for strongly stable nonlinear impulsive differential-difference equations. Nonlinear Anal. Hybrid Syst. 6, 818–823 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank the anonymous referee for his/her constructive comments and suggestions.

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Authors and Affiliations

  1. School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, H.P., India

    Syed Abbas

  2. Department of Mathematics, Indian Institute of Information Technology Dharwad, Hubli, Karnataka, India

    Lakshman Mahto

Authors
  1. Syed Abbas
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  2. Lakshman Mahto
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Correspondence to Syed Abbas .

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Editors and Affiliations

  1. Department of Mathematics, Gauhati University, Guwahati, Assam, India

    Hemen Dutta

  2. University of Nis, Faculty of Sciences and Mathematics, Aleksandrovac, Serbia

    Ljubiša D. R. Kočinac

  3. Dept. Mathematics and Statistics, University of Victoria, Victoria, BC, Canada

    Hari M. Srivastava

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Abbas, S., Mahto, L. (2019). Piecewise Continuous Stepanov-Like Almost Automorphic Functions with Applications to Impulsive Systems. In: Dutta, H., Kočinac, L.D.R., Srivastava, H.M. (eds) Current Trends in Mathematical Analysis and Its Interdisciplinary Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-15242-0_4

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