# 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 and 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), 131-137. 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 "Proceedings of the Conference on Differential and Difference Equations at the Florida Institute of Technology," August 1-5, 2005, Melbourne, Florida, Editors: R.P. Agarval and K. Perera, Hindawi Publishing Corporation, 2006, 11-20. [3] M. U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal., 66 (2007), 367-383. 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), 646-663. doi: 10.1016/j.jmaa.2007.03.010. [5] M. U. Akhmet, "Nonlinear Hybrid Continuous/Discrete Time Models," Amsterdam, Paris, Antlantis Press, 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), 794-803. doi: 10.1016/j.na.2006.11.037. [7] M. U. Akhmet, Exponentially dichotomous linear systems of differential equations with piecewise constant argument, Discontinuity, Nonlinearity and Complexity, 1 (2012), 337-352. [8] M. U. Akhmet and D. Aruğaslan, Lyapunov-Razumikhin method for differential equations with piecewise constant argument, Discrete Contin. Dyn. Syst., 25 (2009), 457-466. 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), 4554-4560. 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), 805-811. [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), 4554-4560. 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), 4200-4210. 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), 2034-2042. 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., Mat. Preprint, 26 (1983), 49. [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), 32-41. [16] S. Busenberg and K. L. Cooke, Models of vertically transmitted diseases with sequential-continuous dynamics, Nonlinear Phenomena in Mathematical Sciences, Academic Press, New York, (1982), 179-187. [17] T. A. Burton, "Stability and Periodic Solutions of Ordinary and Functional Differential Equations," Academic Press, Orlando, Florida, 1985. [18] K. L. Cooke and J. Wiener, Neutral differential equations with piecewise constant argument, Boll. Un. Mat. Ital., 7 (1987), 321-346. [19] L. Dai, "Nonlinear Dynamics of Piecewise Constant Systems and Implementation of Piecewise Constant Arguments," World Scientific, Hackensack, NJ, 2008. doi: 10.1142/9789812818515. [20] A. Halanay and D. Wexler, "Qualitative Theory of Impulsive Systems," (Russian), Moscow, Mir, 1971. [21] J. Hale , "Functional Differential Equations," Springer, New-York, 1971. [22] A. M. Fink, "Almost-periodic Differential Equations," Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, 1974. [23] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, Boston, New Yorke, 1993. [24] M. Pinto, Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments, Math. Comput. Modelling, 49 (2009), 1750-1758. doi: 10.1016/j.mcm.2008.10.001. [25] A. Samoilenko and N. Perestyuk, "Impulsive Differential Equations," World Scientific, Singapore, 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), 671-703. doi: 10.1155/S0161171283000599. [27] G. Seifert, Second-order neutral delay-differential equations with piecewise constant time dependence, J. Math. Anal. Appl., 281 (2003), 1-9. 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), 451-458. 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), 736-747. 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), 139-143. [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, English Series, 27 (2011), 2275-284. 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), 229-235. 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), 91-96. 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), 12-32. [35] J. Wiener, "Generalized Solutions of Functional Differential Equations," World Scientific, Singapore, 1993. [36] R. Yuan, The existence of almost periodic solutions of retarded differential equations with piecewise argument, Nonlinear Analysis, Theory, Methods and Applications, 48 (2002), 1013-1032. 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), 103-118. 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), 131-137. 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 "Proceedings of the Conference on Differential and Difference Equations at the Florida Institute of Technology," August 1-5, 2005, Melbourne, Florida, Editors: R.P. Agarval and K. Perera, Hindawi Publishing Corporation, 2006, 11-20. [3] M. U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal., 66 (2007), 367-383. 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), 646-663. doi: 10.1016/j.jmaa.2007.03.010. [5] M. U. Akhmet, "Nonlinear Hybrid Continuous/Discrete Time Models," Amsterdam, Paris, Antlantis Press, 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), 794-803. doi: 10.1016/j.na.2006.11.037. [7] M. U. Akhmet, Exponentially dichotomous linear systems of differential equations with piecewise constant argument, Discontinuity, Nonlinearity and Complexity, 1 (2012), 337-352. [8] M. U. Akhmet and D. Aruğaslan, Lyapunov-Razumikhin method for differential equations with piecewise constant argument, Discrete Contin. Dyn. Syst., 25 (2009), 457-466. 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), 4554-4560. 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), 805-811. [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), 4554-4560. 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), 4200-4210. 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), 2034-2042. 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., Mat. Preprint, 26 (1983), 49. [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), 32-41. [16] S. Busenberg and K. L. Cooke, Models of vertically transmitted diseases with sequential-continuous dynamics, Nonlinear Phenomena in Mathematical Sciences, Academic Press, New York, (1982), 179-187. [17] T. A. Burton, "Stability and Periodic Solutions of Ordinary and Functional Differential Equations," Academic Press, Orlando, Florida, 1985. [18] K. L. Cooke and J. Wiener, Neutral differential equations with piecewise constant argument, Boll. Un. Mat. Ital., 7 (1987), 321-346. [19] L. Dai, "Nonlinear Dynamics of Piecewise Constant Systems and Implementation of Piecewise Constant Arguments," World Scientific, Hackensack, NJ, 2008. doi: 10.1142/9789812818515. [20] A. Halanay and D. Wexler, "Qualitative Theory of Impulsive Systems," (Russian), Moscow, Mir, 1971. [21] J. Hale , "Functional Differential Equations," Springer, New-York, 1971. [22] A. M. Fink, "Almost-periodic Differential Equations," Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, 1974. [23] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, Boston, New Yorke, 1993. [24] M. Pinto, Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments, Math. Comput. Modelling, 49 (2009), 1750-1758. doi: 10.1016/j.mcm.2008.10.001. [25] A. Samoilenko and N. Perestyuk, "Impulsive Differential Equations," World Scientific, Singapore, 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), 671-703. doi: 10.1155/S0161171283000599. [27] G. Seifert, Second-order neutral delay-differential equations with piecewise constant time dependence, J. Math. Anal. Appl., 281 (2003), 1-9. 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), 451-458. 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), 736-747. 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), 139-143. [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, English Series, 27 (2011), 2275-284. 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), 229-235. 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), 91-96. 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), 12-32. [35] J. Wiener, "Generalized Solutions of Functional Differential Equations," World Scientific, Singapore, 1993. [36] R. Yuan, The existence of almost periodic solutions of retarded differential equations with piecewise argument, Nonlinear Analysis, Theory, Methods and Applications, 48 (2002), 1013-1032. 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), 103-118. doi: 10.1016/j.jmaa.2004.06.057.
 [1] Marat Akhmet, Duygu Aruğaslan. Lyapunov-Razumikhin method for differential equations with piecewise constant argument. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 457-466. doi: 10.3934/dcds.2009.25.457 [2] Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure and Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291 [3] Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048 [4] Min Zhu, Panpan Ren, Junping Li. Exponential stability of solutions for retarded stochastic differential equations without dissipativity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2923-2938. doi: 10.3934/dcdsb.2017157 [5] Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167 [6] Xianhua Huang. Almost periodic and periodic solutions of certain dissipative delay differential equations. Conference Publications, 1998, 1998 (Special) : 301-313. doi: 10.3934/proc.1998.1998.301 [7] Tomás Caraballo, David Cheban. Almost periodic and almost automorphic solutions of linear differential equations. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1857-1882. doi: 10.3934/dcds.2013.33.1857 [8] Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113 [9] Yoshihiro Hamaya. Stability properties and existence of almost periodic solutions of volterra difference equations. Conference Publications, 2009, 2009 (Special) : 315-321. doi: 10.3934/proc.2009.2009.315 [10] Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703 [11] Yong Liu, Jing Tian, Xuelin Yong. On the even solutions of the Toda system: A degree argument approach. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1895-1916. doi: 10.3934/cpaa.2021075 [12] Denis Pennequin. Existence of almost periodic solutions of discrete time equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 51-60. doi: 10.3934/dcds.2001.7.51 [13] Gaston N'Guerekata. On weak-almost periodic mild solutions of some linear abstract differential equations. Conference Publications, 2003, 2003 (Special) : 672-677. doi: 10.3934/proc.2003.2003.672 [14] José Luis Bravo, Manuel Fernández, Antonio Tineo. Periodic solutions of a periodic scalar piecewise ode. Communications on Pure and Applied Analysis, 2007, 6 (1) : 213-228. doi: 10.3934/cpaa.2007.6.213 [15] Jianqing Chen. A variational argument to finding global solutions of a quasilinear Schrödinger equation. Communications on Pure and Applied Analysis, 2008, 7 (1) : 83-88. doi: 10.3934/cpaa.2008.7.83 [16] Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385 [17] Marcelo Marchesin. The mass dependence of the period of the periodic solutions of the Sitnikov problem. Discrete and Continuous Dynamical Systems - S, 2008, 1 (4) : 597-609. doi: 10.3934/dcdss.2008.1.597 [18] Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6425-6462. doi: 10.3934/dcdsb.2021026 [19] Paolo Perfetti. Hamiltonian equations on $\mathbb{T}^\infty$ and almost-periodic solutions. Conference Publications, 2001, 2001 (Special) : 303-309. doi: 10.3934/proc.2001.2001.303 [20] Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

2021 Impact Factor: 1.273