Discontinuity, Nonlinearity, and Complexity
Exponentially Dichotomous Linear systems of Differential Equations with Piecewise Constant Argument
Discontinuity, Nonlinearity, and Complexity 1(4) (2012) 337352  DOI:10.5890/DNC.2012.09.001
M. Akhmet
Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, 06531 Ankara, Turkey
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Abstract
We consider differential equations with piecewise constant argument of generalized type. It is the first time, an attention is given to the exponential dichotomy of linear systems. Bounded, almost periodic and periodic solutions and their stability are discussed. The study is made in such a way that further construction of the theory will follow for ordinary differential equations. The results are illustrated by examples.
References

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

[2]  Cooke, K.L. andWiener, J. (1984), Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl., 99, 265297. 

[3]  Aftabizadeh, A.R. and Wiener, J. (1988), Oscillatory and periodic solutions for systems of two first order linear differential equations with piecewise constant arguments, Applicable Anal., 26, 327333. 

[4]  Aftabizadeh, A.R. and Wiener, J. (1985), Oscillatory properties of first order linear functionaldifferential equations, Applicable Anal., 20, 165187. 

[5]  Alonso, A., Hong, J., and Obaya, R. (2000),Almost periodic type solutions of differential equationswith piecewise constant argument via almost periodic type sequences, Appl. Math. Lett., 13, 131137. 

[6]  Cooke, K.L. and Wiener, J. (1991), A survey of differential equation with piecewise continuous argument, Lecture Notes in Math., 1475, Springer, Berlin, 115. 

[7]  Cooke, K.L. and Wiener, J. (1987), An equation alternately of retarded and advanced type, Proc. Amer. Math. Soc., 99, 726732. 

[8]  Cooke, K.L. and Wiener, J (1987), Neutral differential equations with piecewise constant argument, Boll. Un. Mat. Ital., 7, 321346. 

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

[10]  Dai, L. and Singh, M. C. (1994), On oscillatory motion of springmass systems subjected to piecewise constant forces, J. Sound Vibration, 173, 217232. 

[11]  Gopalsamy, K. (1992), Stability and Oscillations in Delay Differential Equations, Kluwer Academic Publishers Group, Dordrecht. 

[12]  Györi, I. and Ladas, G. (1991), Oscillation Theory of Delay Differential Equations: With Applications, Oxford University Press, New York. 

[13]  Küpper, T. and Yuan, R. (2002), On quasiperiodic solutions of differential equations with piecewise constant argument, J. Math. Anal. Appl., 267, 173193. 

[14]  Muroya, Yoshiaki, (2002), Persistence, contractivity and global stability in logistic equations with piecewise constant delays, J. Math. Anal. Appl., 270, 602635. 

[15]  Papaschinopoulos, G. (1994), Some results concerning a class of differential equations with piecewise constant argument, Math. Nachr., 166, 193206. 

[16]  Shah, S.M. and Wiener, J. (1983), Advanced differential equations with piecewise constant argument deviations, Int. J. Math. Math. Sci., 6, 671703. 

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

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

[19]  Wiener, J. (1984), Pointwise initial value problems for functionaldifferential equations, Differential Equations, NorthHolland, Amsterdam, 571580. 

[20]  Wiener, J. (1983), Differential equations with piecewise constant delays, Trends in the Theory and Practice of Nonlinear Differential Equations,Marcel Dekker, New York, 547552. 

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

[22]  Yuan, R. (2005), On the spectrum of almost periodic solution of second order scalar functional differential equations with piecewise constant argument, J. Math. Anal. Appl., 303, 103118. 

[23]  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. 

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

[25]  Akhmet, M.U. (2007), Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal: TMA, 66, 367383. 

[26]  Akhmet, M.U. (2007), On the reduction principle for differential equations with piecewise constant argument of generalized type, J. Math. Anal. Appl., 336, 646663. 

[27]  Akhmet M.U. (2008), Stability of differential equations with piecewise constant argument of generalized type, Nonlinear Analysis: TMA, 68, 794803. 

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

[29]  Akhmet, M.U. (2008), Asymptotic behavior of solutions of differential equations with piecewise constant arguments, Appl. Math. Lett., 21, 951956. 

[30]  Akhmet, M.U. (2009), Almost periodic solutions of the linear differential equation with piecewise constant argument, Discrete and Impulsive Systems, Series A, Mathematical Analysis, 16, 743753. 

[31]  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. 

[32]  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. 

[33]  Akhmet, M.U., Aruğaslan, D., and Yılmaz, E. (2011), Method of Lyapunov functions for differential equations with piecewise constant delay, Journal of Computational and Applied Mathematics, 235, 45544560. 

[34]  Akhmetov,M.U., Perestyuk, N.A., and Samoilenko, A.M. (1983), Almostperiodic Solutions of Differential equations with impulse action, (Russian) Akad. Nauk Ukrain. SSR Inst., Mat. Preprint, no. 26, 49 pp. 

[35]  Akhmet, M.U. and Buyukadali, C. (2010), Differential equations with a state dependent piecewise constant argu ment, Nonlinear Analysis: TMA, 72, 42004210. 

[36]  Akhmet, M.U. and Buyukadali, C. (2008), Periodic solutions of the system with piecewise constant argument in the critical case,Comput. Math. Appl., 56, 20342042. 

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

[38]  Pinto, M. (2009), Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments, Math. Comput. Modeling, 49, 17501758. 

[39]  Pinto, M. (2011), Cauchy and Green matrices type and stability in alternately advanced and delayed differential systems, J. Difference Equations and Applications, 17, 235254. 

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

[41]  Hale, J. (1971), Functional Differential Equations, Springer, NewYork. 

[42]  Coppel,W.A. (1978), Dichotomies in Stability Theory, Lecture notes in mathematics, SpringerVerlag, New York. 

[43]  Wexler, D. (1966), Solutions périodiques et presque périodiques des systémes d'équations différetielles linéaires en distributions, J. Differential Equations, 2, 1232. 

[44]  Halanay, A. andWexler, D. (1971), Qualitative Theory of Impulsive Systems, (Russian), Moscow, Mir. 

[45]  Bohr, H. (1932), Fastperiodische Functionen, SpringerVerlag, Berlin. 

[46]  Corduneanu, C. (1961), Almost Periodic Functions, Interscience Publishers, New York. 

[47]  Fink, A.M. (1974),Almostperiodic differential quations, Lecture Notes in Mathematics, SpringerVerlag,Berlin, Heidelberg, New York. 

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