March  2014, 13(2): 929-947. doi: 10.3934/cpaa.2014.13.929

Quasilinear retarded differential equations with functional dependence on piecewise constant argument

1. 

Department of Mathematics, Middle East Technical University, 06531, Ankara

Received  December 2012 Revised  July 2013 Published  October 2013

We introduce a new class of differential equations, retarded differential equations with functional dependence on piecewise constant argument, $RFDEPCA$ and focus on quasilinear systems. Formulation of the initial value problem, bounded solutions, periodic and almost periodic solutions, their stability are under investigation. Illustrating examples are provided.
Citation: Marat Akhmet. Quasilinear retarded differential equations with functional dependence on piecewise constant argument. Communications on Pure & Applied Analysis, 2014, 13 (2) : 929-947. doi: 10.3934/cpaa.2014.13.929
References:
[1]

A. Alonso, J. Hong and R. Obaya, Almost periodic type solutions of differential equations with piecewise constant argument via almost periodic type sequences,, Appl. Math. Lett., 13 (2000), 131. doi: 10.1016/S0893-9659(99)00176-7.

[2]

M. U. Akhmet, On the integral manifolds of the differential equations with piecewise constant argument of generalized type,, in, (2005), 1.

[3]

M. U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type,, Nonlinear Anal., 66 (2007), 367. doi: 10.1016/j.na.2005.11.032.

[4]

M. U. Akhmet, On the reduction principle for differential equations with piecewise constant argument of generalized type,, J. Math. Anal. Appl., 336 (2007), 646. doi: 10.1016/j.jmaa.2007.03.010.

[5]

M. U. Akhmet, "Nonlinear Hybrid Continuous/Discrete Time Models,", Amsterdam, (2011). doi: 10.2991/978-94-91216-03-9.

[6]

M. U. Akhmet, Stability of differential equations with piecewise constant argument of generalized type,, Nonlinear Analysis: TMA, 68 (2008), 794. doi: 10.1016/j.na.2006.11.037.

[7]

M. U. Akhmet, Exponentially dichotomous linear systems of differential equations with piecewise constant argument,, Discontinuity, 1 (2012), 337.

[8]

M. U. Akhmet and D. Aruğaslan, Lyapunov-Razumikhin method for differential equations with piecewise constant argument,, Discrete Contin. Dyn. Syst., 25 (2009), 457. doi: 10.3934/dcds.2009.25.457.

[9]

M. U. Akhmet, D. Aruğaslan and E. Yilmaz, Method of Lyapunov functions for differential equations with piecewise constant delay,, J. Comput. Appl. Math., 235 (2011), 4554. doi: 10.1016/j.cam.2010.02.043.

[10]

M. U. Akhmet, D. Aruğaslan and E. Yilmaz, Stability analysis of recurrent neural networks with piecewise constant argument of generalized type,, Neural Networks, 23 (2010), 805.

[11]

M. U. Akhmet, D. Aruğaslan and E. Yilmaz, Method of Lyapunov functions for differential equations with piecewise constant delay,, J. Comput. Appl. Math., 235 (2011), 4554. doi: 10.1016/j.cam.2010.02.043.

[12]

M. U. Akhmet and C. Buyukadali, Differential equations with a state-dependent piecewise constant argument,, Nonlinear Analysis: TMA, 72 (2010), 4200. doi: 10.1016/j.na.2010.01.050.

[13]

M. U. Akhmet and C. Buyukadali, Periodic solutions of the system with piecewise constant argument in the critical case,, Comput. Math. Appl., 56 (2008), 2034. doi: 10.1016/j.camwa.2008.03.031.

[14]

M. U. Akhmetov, N. A. Perestyuk and A. M. Samoilenko, Almost-periodic solutions of differential equations with impulse action,, (Russian) Akad. Nauk Ukrain. SSR Inst., 26 (1983).

[15]

G. Bao, S. Wen and Zh. Zeng, Robust stability analysis of interval fuzzy CohenGrossberg neural networks with piecewise constant argument of generalized type,, Neural Networks, 33 (2012), 32.

[16]

S. Busenberg and K. L. Cooke, Models of vertically transmitted diseases with sequential-continuous dynamics,, Nonlinear Phenomena in Mathematical Sciences, (1982), 179.

[17]

T. A. Burton, "Stability and Periodic Solutions of Ordinary and Functional Differential Equations,", Academic Press, (1985).

[18]

K. L. Cooke and J. Wiener, Neutral differential equations with piecewise constant argument,, Boll. Un. Mat. Ital., 7 (1987), 321.

[19]

L. Dai, "Nonlinear Dynamics of Piecewise Constant Systems and Implementation of Piecewise Constant Arguments,", World Scientific, (2008). doi: 10.1142/9789812818515.

[20]

A. Halanay and D. Wexler, "Qualitative Theory of Impulsive Systems,", (Russian), (1971).

[21]

J. Hale, "Functional Differential Equations,", Springer, (1971).

[22]

A. M. Fink, "Almost-periodic Differential Equations,", Lecture Notes in Mathematics, (1974).

[23]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Academic Press, (1993).

[24]

M. Pinto, Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments,, Math. Comput. Modelling, 49 (2009), 1750. doi: 10.1016/j.mcm.2008.10.001.

[25]

A. Samoilenko and N. Perestyuk, "Impulsive Differential Equations,", World Scientific, (1995). doi: 10.1142/9789812798664.

[26]

S. M. Shah and J. Wiener, Advanced differential equations with piecewise constant argument deviations,, Int. J. Math. Math. Sci., 6 (1983), 671. doi: 10.1155/S0161171283000599.

[27]

G. Seifert, Second-order neutral delay-differential equations with piecewise constant time dependence,, J. Math. Anal. Appl., 281 (2003), 1. doi: 10.1016/S0022-247X(02)00303-7.

[28]

G. Seifert, Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence,, J. Differential Equations, 164 (2000), 451. doi: 10.1006/jdeq.1999.3747.

[29]

G. Wang, Periodic solutions of a neutral differential equation with piecewise constant arguments,, J. Math. Anal. Appl., 326 (2007), 736. doi: 10.1016/j.jmaa.2006.02.093.

[30]

G. Q. Wang and S. S. Cheng, Note on the set of periodic solutions of a delay differential equation with piecewise constant argument,, Int. J. Pure Appl. Math., 9 (2003), 139.

[31]

L. Wang, R. Yuan and C. Zhang, A spectrum relation of almost periodic solution of second order scalar functional differential equations with piecewise constant argument,, Acta Mathematica Sinica, 27 (2011), 2275. doi: 10.1007/s10114-011-8392-8.

[32]

Y. Wang and J. Yan, A necessary and sufficient condition for the oscillation of a delay equation with continuous and piecewise constant arguments,, Acta Math. Hungar., 79 (1998), 229. doi: 10.1023/A:1006510024909.

[33]

Y. Wang and J. Yan, Necessary and sufficient condition for the global attractivity of the trivial solution of a delay equation with continuous and piecewise constant arguments,, Appl. Math. Lett., 10 (1997), 91. doi: 10.1016/S0893-9659(97)00089-X.

[34]

D. Wexler, Solutions périodiques et presque-périodiques des systémes d'équations différetielles linéaires en distributions,, J. Differential Equations., 2 (1966), 12.

[35]

J. Wiener, "Generalized Solutions of Functional Differential Equations,", World Scientific, (1993).

[36]

R. Yuan, The existence of almost periodic solutions of retarded differential equations with piecewise argument,, Nonlinear Analysis, 48 (2002), 1013. doi: 10.1016/S0362-546X(00)00231-5.

[37]

R. Yuan, On the spectrum of almost periodic solution of second order scalar functional differential equations with piecewise constant argument,, J. Math. Anal. Appl., 303 (2005), 103. doi: 10.1016/j.jmaa.2004.06.057.

show all references

References:
[1]

A. Alonso, J. Hong and R. Obaya, Almost periodic type solutions of differential equations with piecewise constant argument via almost periodic type sequences,, Appl. Math. Lett., 13 (2000), 131. doi: 10.1016/S0893-9659(99)00176-7.

[2]

M. U. Akhmet, On the integral manifolds of the differential equations with piecewise constant argument of generalized type,, in, (2005), 1.

[3]

M. U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type,, Nonlinear Anal., 66 (2007), 367. doi: 10.1016/j.na.2005.11.032.

[4]

M. U. Akhmet, On the reduction principle for differential equations with piecewise constant argument of generalized type,, J. Math. Anal. Appl., 336 (2007), 646. doi: 10.1016/j.jmaa.2007.03.010.

[5]

M. U. Akhmet, "Nonlinear Hybrid Continuous/Discrete Time Models,", Amsterdam, (2011). doi: 10.2991/978-94-91216-03-9.

[6]

M. U. Akhmet, Stability of differential equations with piecewise constant argument of generalized type,, Nonlinear Analysis: TMA, 68 (2008), 794. doi: 10.1016/j.na.2006.11.037.

[7]

M. U. Akhmet, Exponentially dichotomous linear systems of differential equations with piecewise constant argument,, Discontinuity, 1 (2012), 337.

[8]

M. U. Akhmet and D. Aruğaslan, Lyapunov-Razumikhin method for differential equations with piecewise constant argument,, Discrete Contin. Dyn. Syst., 25 (2009), 457. doi: 10.3934/dcds.2009.25.457.

[9]

M. U. Akhmet, D. Aruğaslan and E. Yilmaz, Method of Lyapunov functions for differential equations with piecewise constant delay,, J. Comput. Appl. Math., 235 (2011), 4554. doi: 10.1016/j.cam.2010.02.043.

[10]

M. U. Akhmet, D. Aruğaslan and E. Yilmaz, Stability analysis of recurrent neural networks with piecewise constant argument of generalized type,, Neural Networks, 23 (2010), 805.

[11]

M. U. Akhmet, D. Aruğaslan and E. Yilmaz, Method of Lyapunov functions for differential equations with piecewise constant delay,, J. Comput. Appl. Math., 235 (2011), 4554. doi: 10.1016/j.cam.2010.02.043.

[12]

M. U. Akhmet and C. Buyukadali, Differential equations with a state-dependent piecewise constant argument,, Nonlinear Analysis: TMA, 72 (2010), 4200. doi: 10.1016/j.na.2010.01.050.

[13]

M. U. Akhmet and C. Buyukadali, Periodic solutions of the system with piecewise constant argument in the critical case,, Comput. Math. Appl., 56 (2008), 2034. doi: 10.1016/j.camwa.2008.03.031.

[14]

M. U. Akhmetov, N. A. Perestyuk and A. M. Samoilenko, Almost-periodic solutions of differential equations with impulse action,, (Russian) Akad. Nauk Ukrain. SSR Inst., 26 (1983).

[15]

G. Bao, S. Wen and Zh. Zeng, Robust stability analysis of interval fuzzy CohenGrossberg neural networks with piecewise constant argument of generalized type,, Neural Networks, 33 (2012), 32.

[16]

S. Busenberg and K. L. Cooke, Models of vertically transmitted diseases with sequential-continuous dynamics,, Nonlinear Phenomena in Mathematical Sciences, (1982), 179.

[17]

T. A. Burton, "Stability and Periodic Solutions of Ordinary and Functional Differential Equations,", Academic Press, (1985).

[18]

K. L. Cooke and J. Wiener, Neutral differential equations with piecewise constant argument,, Boll. Un. Mat. Ital., 7 (1987), 321.

[19]

L. Dai, "Nonlinear Dynamics of Piecewise Constant Systems and Implementation of Piecewise Constant Arguments,", World Scientific, (2008). doi: 10.1142/9789812818515.

[20]

A. Halanay and D. Wexler, "Qualitative Theory of Impulsive Systems,", (Russian), (1971).

[21]

J. Hale, "Functional Differential Equations,", Springer, (1971).

[22]

A. M. Fink, "Almost-periodic Differential Equations,", Lecture Notes in Mathematics, (1974).

[23]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Academic Press, (1993).

[24]

M. Pinto, Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments,, Math. Comput. Modelling, 49 (2009), 1750. doi: 10.1016/j.mcm.2008.10.001.

[25]

A. Samoilenko and N. Perestyuk, "Impulsive Differential Equations,", World Scientific, (1995). doi: 10.1142/9789812798664.

[26]

S. M. Shah and J. Wiener, Advanced differential equations with piecewise constant argument deviations,, Int. J. Math. Math. Sci., 6 (1983), 671. doi: 10.1155/S0161171283000599.

[27]

G. Seifert, Second-order neutral delay-differential equations with piecewise constant time dependence,, J. Math. Anal. Appl., 281 (2003), 1. doi: 10.1016/S0022-247X(02)00303-7.

[28]

G. Seifert, Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence,, J. Differential Equations, 164 (2000), 451. doi: 10.1006/jdeq.1999.3747.

[29]

G. Wang, Periodic solutions of a neutral differential equation with piecewise constant arguments,, J. Math. Anal. Appl., 326 (2007), 736. doi: 10.1016/j.jmaa.2006.02.093.

[30]

G. Q. Wang and S. S. Cheng, Note on the set of periodic solutions of a delay differential equation with piecewise constant argument,, Int. J. Pure Appl. Math., 9 (2003), 139.

[31]

L. Wang, R. Yuan and C. Zhang, A spectrum relation of almost periodic solution of second order scalar functional differential equations with piecewise constant argument,, Acta Mathematica Sinica, 27 (2011), 2275. doi: 10.1007/s10114-011-8392-8.

[32]

Y. Wang and J. Yan, A necessary and sufficient condition for the oscillation of a delay equation with continuous and piecewise constant arguments,, Acta Math. Hungar., 79 (1998), 229. doi: 10.1023/A:1006510024909.

[33]

Y. Wang and J. Yan, Necessary and sufficient condition for the global attractivity of the trivial solution of a delay equation with continuous and piecewise constant arguments,, Appl. Math. Lett., 10 (1997), 91. doi: 10.1016/S0893-9659(97)00089-X.

[34]

D. Wexler, Solutions périodiques et presque-périodiques des systémes d'équations différetielles linéaires en distributions,, J. Differential Equations., 2 (1966), 12.

[35]

J. Wiener, "Generalized Solutions of Functional Differential Equations,", World Scientific, (1993).

[36]

R. Yuan, The existence of almost periodic solutions of retarded differential equations with piecewise argument,, Nonlinear Analysis, 48 (2002), 1013. doi: 10.1016/S0362-546X(00)00231-5.

[37]

R. Yuan, On the spectrum of almost periodic solution of second order scalar functional differential equations with piecewise constant argument,, J. Math. Anal. Appl., 303 (2005), 103. doi: 10.1016/j.jmaa.2004.06.057.

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