Discontinuity, Nonlinearity, and Complexity
Almost Periodic Solutions of Second Order Neutral Differential Equations with Functional Response on Piecewise Constant Argument
Discontinuity, Nonlinearity, and Complexity 2(4) (2013) 369388  DOI:10.5890/DNC.2013.11.006
M. Akhmet
Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, 06531 Ankara, Turkey
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Abstract
We consider second order functional differential equations with generalized piecewise constant argument. Conditions for existence, uniqueness and stability of Bohr almost periodic solutions are defined. Appropriate examples which illustrate the results are provided.
References

[1]  Seifert, G. (2003), Second order neutral delaydifferential equations with piecewise constant time dependence, Journal of Mathematical Analysis and Applications, 281, 19. 

[2]  Seifert, G.(2003), Almost periodic solutions of certain neutral functional differential equations, Communications in Applied Analysis, 7, 437442. 

[3]  Wang, G. (2007), Periodic solutions of a neutral differential equation with piecewise constant arguments, Journal of Mathematical Analysis and Applications, 326, 736747. 

[4]  Wang, G.Q. and Cheng, S.S.(2005), Existence of periodic solutions for a neutral differential equation with piecewise constant argument, Funkcialaj Ekvacioj, 48, 299311. 

[5]  Wang, Y. and Yan, J. (1997), Necessary and sufficient condition for the global attractivity of the trivial solution of a delay equation with continuous and piecewise constant arguments, Applied Mathematics Letters, 10, 9196. 

[6]  Yuan, R. (2005), On the spectrum of almost periodic solution of second order scalar functional differential equations with piecewise constant argument, Journal of Mathematical Analysis and Applications, 303, 103118. 

[7]  S. Busenberg, K.L.(1982), Models of vertically transmitted diseases with sequentialcontinuous dynamics, Nonlinear Phenomena in Mathematical Sciences, Academic Press, New York, 179187. 

[8]  Cooke, K.L. and Wiener, J. (1984), Retarded differential equations with piecewise constant delays, Journal of Mathematical Analysis and Applications, 99, 265297. 

[9]  Shah, S.M. and Wiener,J. (1983), Advanced differential equations with piecewise constant argument deviations, The International Journal of Mathematics and Mathematical Sciences, 6, 671703. 

[10]  Alonso, A., Hong, J., and Obaya, R. (2000),Almost periodic type solutions of differential equationswith piecewise constant argument via almost periodic type sequences, Applied Mathematics Letters, 13, 131137. 

[11]  Muroya, Y. (2002), Persistence, contractivity and global stability in logistic equations with piecewise constant delays, Journal of Mathematical Analysis and Applications, 270, 602635. 

[12]  Nieto, J. and RodriguezLopez, R. (2012), Second order linear differential equations with piecewise constant arguments subject to nonlocal boundary conditions, Applied Mathematics and Computation, 218, 96479656. 

[13]  Seifert, G. (2000), Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence, Journal of Differential Equations, 164, 451458. 

[14]  Wiener, J. (1993), Generalized Solutions of Functional Differential Equations,World Scientific, Singapore. 

[15]  Wiener, J. and Lakshmikantham, V. (2000), A damped oscillator with piecewise constant time delay, Nonlinear Studies, 7, 7884. 

[16]  Akhmet,M. U. (2006), On the integral manifolds of the differential equations with piecewise constant argument of generalized type, Proceedings of the Conference on Differential and Difference Equations at the Florida Institute of Technology, August 15, 2005, , Melbourne, Florida, Editors: R.P. Agarval and K. Perera, Hindawi Publishing Corporation, 1120. 

[17]  Akhmet, M. U. (2011), Nonlinear Hybrid Continuous/Discrete Time Models, Amsterdam, Paris, Antlantis Press. 

[18]  Akhmet, M. U. (2007), Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Analysis, Theory, Methods & Applications, 66, 367383. 

[19]  Akhmet, M. U.(2007), On the reduction principle for differential equations with piecewise constant argument of generalized type, Journal of Mathematical Analysis and Applications, 336, 646663. 

[20]  Akhmet,M.U. and Aruğaslan, D. (2009), LyapunovRazumikhinmethod for differential equations with piecewise constant argument, Discrete and Continuous Dynamical Systems, 25, 457466. 

[21]  Akhmet, M.U. (2008), Almost periodic solutions of differential equations with piecewise constant argument of generalized type, Nonlinear Analysis: HS, 2, 456467. 

[22]  Akhmet,M.U. (2012), Exponentially dichotomous linear systems of differential equations with piecewise constant argument, Discontinuity, Nonlinearity and Complexity, 1, 337352. 

[23]  Akhmet, M.U., Buyukadali, C., and Ergenc, T. (2008), Periodic solutions of the hybrid system with small parameter, Nonlinear Analysis: HS , 2, 532543. 

[24]  Akhmet,M.U., Aruğaslan, D., and Yılmaz, E. (2010), Stability in cellular neural networks with piecewise constant argument, Journal of Computational and Applied Mathematics, 233, 23652373. 

[25]  Akhmet, M.U., Aruğaslan, D., and Yılmaz, E. (2010), Stability analysis of recurrent neural networks with piecewise constant argument of generalized type, Neural Networks, 23, 805811. 

[26]  Pinto, M. (2009), Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments, Mathematical and Computer Modelling, 49, 17501758. 

[27]  Bao, G., Wen, S. and Zeng, Zh. (2012), Robust stability analysis of interval fuzzy Cohen Grossberg neural networks with piecewise constant argument of generalized type, Neural Networks, 33, 3241. 

[28]  Abbas, S. and Bahuguna, D. (2008), Almost periodic solutions of neutral functional differential equations, Computers and Mathematics with Applications, 55, 25932601. 

[29]  Islam, M.N. and Raffoul, Y.N., Periodic solutions of neutral nonlinear system of differential equations with functional delay, Journal of Mathematical Analysis and Applications, 331, 11751186. 

[30]  Burton, T.A. (1985), Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, Orlando, Florida. 

[31]  Hale, J. (1971), Functional Differential Equations, Springer, New York. 

[32]  Dai, L.(2008), Nonlinear Dynamics of Piecewise Constant Systems and Implementation of Piecewise Constant Arguments, World Scientific, Hackensack, NJ. 

[33]  Fink, A.M. (1974), Almostperiodic Differential Equations, Lecture Notes in Mathematics, SpringerVerlag, Berlin, Heidelberg, New York. 

[34]  Halanay A. and Wexler, D. (1968), Teoria calitativǎ a sistemelor cu impulsuri, Editura Academiei Republicii Socialiste România, Bucuresti.(in Romanian). 

[35]  Samoilenko, A. and Perestyuk, N. (1995), Impulsive Differential Equations,World scientific, Singapore. 

[36]  Wexler, D.(1966), Solutions périodiques et presquepériodiques des systémes d'équations différetielles linéaires en distributions, Journal of Differential Equations, 2, 1232. 

[37]  Akhmetov, M. U. Perestyuk, N. A. and Samoilenko, A.M. (1983), Almostperiodic solutions of differential equations with impulse action, Akad. Nauk Ukrain. SSR Inst., Mat. Preprint, 26, 49(in Russian). 