American Institute of Mathematical Sciences

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

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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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