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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
S. Abbas, A note on Weyl pseudo almost automorphic functions and their properties. Math. Sci. (Springer), 6, 5 (2012). Art. 29
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)
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)
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)
S. Ahmad, I.M. Stamova, Global exponential stability for impulsive cellular neural networks with time-varying delays. Nonlinear Anal. 69(3), 786–795 (2008)
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)
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)
D.D. Bainov, P.S. Simeonov, Systems with Impulsive Effects (Ellis Horwood Limited/John Wiley & Sons, Chichester, 1989)
D.D. Bainov, P.S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Its Applications (Longman Scientific and Technical Group, England, 1993)
S. Bochner, Continuous mappings of almost automorphic and almost periodic functions. Proc. Nat. Acad. Sci. U.S.A. 52, 907–910 (1964)
H. Bohr, Almost-Periodic Functions (Chelsea Publishing Company, New York City, 1947)
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)
T. Diagana, Pseudo Almost Periodic Functions in Banach Spaces (Nova Science, Hauppauge, 2007)
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)
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)
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)
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)
L. Mahto, S. Abbas, PC-almost automorphic solution of impulsive fractional differential equations. Mediterr. J. Math. 12(3), 771–790 (2014)
L. Mahto, S. Abbas, A. Favini, Analysis of Caputo impulsive fractional order differential equations with applications. Int. J. Differ. Equ. 2013, 1–11 (2013)
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)
G.M. N’Guérékata, Topics in Almost Automorphy (Springer, New York, 2005)
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)
A. Pankov, Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations (Kluwer, Dordrecht, 1990)
A.M. Samoilenko, N.A. Perestyuk, Differential Equations with Impulse Effects (Viska Skola, Kiev, 1987) (in Russian)
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)
M. Sannay, Exponential stability in Hopfield-type neural networks with impulses. Chaos, Solitons Fractals 32(2), 456–467 (2007)
G.T. Stamov, Impulsive cellular neural networks and almost periodicity. Proc. Jpn. Acad. Sci. 80, Ser. A, 10, 198–203 (2005)
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
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)
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)
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)
Acknowledgements
We would like to thank the anonymous referee for his/her constructive comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-030-15242-0_4
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-15241-3
Online ISBN: 978-3-030-15242-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)