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

  • Syed AbbasEmail author
  • Lakshman Mahto


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.


Stepanov-like almost automorphic functions Composition theorem Impulsive differential equations Fixed point method Asymptotic stability 



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


  1. 1.
    S. Abbas, A note on Weyl pseudo almost automorphic functions and their properties. Math. Sci. (Springer), 6, 5 (2012). Art. 29Google Scholar
  2. 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)MathSciNetzbMATHGoogle Scholar
  3. 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)CrossRefGoogle Scholar
  4. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 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)CrossRefGoogle Scholar
  7. 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)CrossRefGoogle Scholar
  8. 8.
    D.D. Bainov, P.S. Simeonov, Systems with Impulsive Effects (Ellis Horwood Limited/John Wiley & Sons, Chichester, 1989)Google Scholar
  9. 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. 10.
    S. Bochner, Continuous mappings of almost automorphic and almost periodic functions. Proc. Nat. Acad. Sci. U.S.A. 52, 907–910 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    H. Bohr, Almost-Periodic Functions (Chelsea Publishing Company, New York City, 1947)zbMATHGoogle Scholar
  12. 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)MathSciNetGoogle Scholar
  13. 13.
    T. Diagana, Pseudo Almost Periodic Functions in Banach Spaces (Nova Science, Hauppauge, 2007)Google Scholar
  14. 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. 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)MathSciNetzbMATHGoogle Scholar
  16. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 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. 18.
    L. Mahto, S. Abbas, PC-almost automorphic solution of impulsive fractional differential equations. Mediterr. J. Math. 12(3), 771–790 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    G.M. N’Guérékata, Topics in Almost Automorphy (Springer, New York, 2005)zbMATHGoogle Scholar
  22. 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)MathSciNetzbMATHGoogle Scholar
  23. 23.
    A. Pankov, Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations (Kluwer, Dordrecht, 1990)zbMATHCrossRefGoogle Scholar
  24. 24.
    A.M. Samoilenko, N.A. Perestyuk, Differential Equations with Impulse Effects (Viska Skola, Kiev, 1987) (in Russian)Google Scholar
  25. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    M. Sannay, Exponential stability in Hopfield-type neural networks with impulses. Chaos, Solitons Fractals 32(2), 456–467 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 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. 28.
    G.T. Stamov, Almost Periodic Solutions of Impulsive Differential Equations. Lecture Notes in Mathematics, vol. 2047 (Springer, Heidelberg, 2012), pp. xx+217. ISBN: 978-3-642-27545-6Google Scholar
  29. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 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)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Basic SciencesIndian Institute of Technology MandiMandiIndia
  2. 2.Department of MathematicsIndian Institute of Information Technology DharwadHubliIndia

Personalised recommendations