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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[35]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[35]

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

[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.  Google Scholar

[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.  Google Scholar

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