Abstract
We introduce a new class of functional differential equations with functional response on piecewise constant argument, \({ FDEPCA}\). It contains functional differential equations with continuous time [21, 25, 28, 31] as well as differential equations with piecewise constant argument [1, 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17, 22, 22, 23]. We concentrate only on retarded equations, but one can easily extend the discussion to any type of piecewise constant argument and functional differential equations. Nonlinear and quasilinear systems are under consideration. At the end of the chapter, we suggest how one can apply the systems for solution of real-world problems, provided more general systems for future investigations.
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References
Akhmet, M.U.: On the integral manifolds of the differential equations with piecewise constant argument of generalized type. In: Agarval, R.P., Perera, K. (eds.) Proceedings of the Conference on Differential and Difference Equations at the Florida Institute of Technology, pp. 11–20. Hindawi Publishing Corporation (2006)
Akhmet, M.U.: Integral manifolds of differential equations with piecewise constant argument of generalized type. Nonlinear Anal. 66, 367–383 (2007)
Akhmet, M.U.: On the reduction principle for differential equations with piecewise constant argument of generalized type. J. Math. Anal. Appl. 336, 646–663 (2007)
Akhmet, M.U.: Stability of differential equations with piecewise constant argument of generalized type. Nonlinear Anal. TMA 68, 794–803 (2008)
Akhmet, M.U.: Almost periodic solutions of differential equations with piecewise constant argument of generalized type. Nonlinear Anal. HS 2, 456–467 (2008)
Akhmet, M.U.: Asymptotic behavior of solutions of differential equations with piecewise constant arguments. Appl. Math. Lett. 21, 951–956 (2008)
Akhmet, M.U.: Almost periodic solutions of the linear differential equation with piecewise constant argument. Discret. Impuls. Syst. Ser. A, Math. Anal. 16, 743–753 (2009)
Akhmet, M.U.: Nonlinear Hybrid Continuous/Discrete Time Models. Antlantis Press, Amsterdam (2011)
Akhmet, M.U.: Exponentially dichotomous linear systems of differential equations with piecewise constant argument. Discontinuity, Nonlinearity Complex. 1, 337–352 (2012)
Akhmet, M.U.: Quasilinear retarded differential equations with functional dependence on piecewise constant argument. Commun. Pure Appl. Anal. 13, 929–947 (2014)
Akhmet, M.U., Aruğaslan, D.: Lyapunov–Razumikhin method for differential equations with piecewise constant argument. Discrete Contin. Dyn. Syst. 25, 457–466 (2009)
Akhmet, M.U., Aruğaslan, D., Yilmaz, E.: Stability in cellular neural networks with piecewise constant argument. J. Comput. Appl. Math. 233, 2365–2373 (2010)
Akhmet, M.U., Aruğaslan, D., Yilmaz, E.: Stability analysis of recurrent neural networks with piecewise constant argument of generalized type. Neural Netw. 23, 805–811 (2010)
Akhmet, M.U., Aruğaslan, D., Yilmaz, E.: Method of Lyapunov functions for differential equations with piecewise constant delay. J. Comput. Appl. Math. 235, 4554–4560 (2011)
Akhmet, M.U., Buyukadali, C.: Differential equations with a state-dependent piecewise constant argument. Nonlinear Anal. TMA 72, 4200–4210 (2010)
Akhmet, M.U., Buyukadali, C.: Periodic solutions of the system with piecewise constant argument in the critical case. Comput. Math. Appl. 56, 2034–2042 (2008)
Akhmet, M.U., Buyukadali, C., Ergenc, T.: Periodic solutions of the hybrid system with small parameter. Nonlinear Anal. HS 2, 532–543 (2008)
Akhmetov, M. U., Perestyuk, N.A., Samoilenko, A.M. : Almost-periodic solutions of differential equations with impulse action (Russian). Akad. Nauk Ukrain. SSR Inst. Mat. Preprint, no. 26. p. 49 (1983)
Alonso, A., Hong, J., Obaya, R.: Almost periodic type solutions of differential equations with piecewise constant argument via almost periodic type sequences. Appl. Math. Lett. 13, 131–137 (2000)
Bao, G., Wen, S., Zeng, Z.: Robust stability analysis of interval fuzzy CohenGrossberg neural networks with piecewise constant argument of generalized type. Neural Netw. 33, 32–41 (2012)
Burton, T.A.: Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press, Orlando (1985)
Busenberg, S., Cooke, K.L.: Models of vertically transmitted diseases with sequential-continuous dynamics. Nonlinear Phenomena in Mathematical Sciences. Academic Press, New York (1982)
Cooke, K.L., Wiener, J.: Retarded differential equations with piecewise constant delays. J. Math. Anal. Appl. 99, 265–297 (1984)
Dai, L.: Nonlinear Dynamics of Piecewise Constant Systems and Implementation of Piecewise Constant Arguments. World Scientific, Hackensack (2008)
Diekmann, O., van Gils, S.A., Verduyn, L., Sjoerd, M., Walther, H.-O.: Delay Equations. Functional, Complex, and Nonlinear Analysis. Applied Mathematical Sciences, vol. 110. Springer, New York (1995)
Dai, L., Singh, M.C.: On oscillatory motion of spring-mass systems subjected to piecewise constant forces. J. Sound Vibration 173, 217–232 (1994)
Dunkel, G.: Single-species model for population growth depending on past history. Seminar of Differential Equations and Dynamical Systems. Lecture Notes in Mathematics, vol. 60. Springer, Berlin (1968)
Elsgolts, L.E. : Introduction to the Theory of Differential Equations with Deviating Arguments. Holden-Day, Inc. (1966)
Gopalsamy, K., Zhang, B.G.: On a neutral delay logistic equation. Dyn. Stab. Syst. 2, 183–195 (1988)
Halanay, A., Wexler, D.: Qualitative Theory of Impulsive Systems (Russian). Mir, Moscow (1971)
Hale, J., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993). Wiley, New York (1964)
Fink, A.M.: Almost-Periodic Differential Equations. Lecture Notes in Mathematics. Springer, Berlin (1974)
Kolmanovskii, V.B.: Stability of Functional Differential Equations. Academic Press, Orlando (1986)
Krasovskii, N.N.: Certain problems in the theory of stability of motion. (Russian) Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow (1959)
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston (1993)
Küpper, T., Yuan, R.: On quasi-periodic solutions of differential equations with piecewise constant argument. J. Math. Anal. Appl. 267, 173–193 (2002)
Papaschinopoulos, G.: Some results concerning a class of differential equations with piecewise constant argument. Math. Nachr. 166, 193–206 (1994)
Pinto, M.: Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments. Math. Comput. Modelling 49, 1750–1758 (2009)
Samoilenko, A., Perestyuk, N.: Impulsive Differential Equations. World Scientific, Singapore (1995)
Seifert, G.: Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence. J. Differ. Equ. 164, 451–458 (2000)
Wang, G.: Periodic solutions of a neutral differential equation with piecewise constant arguments. J. Math. Anal. Appl. 326, 736–747 (2007)
Wang, G.Q., Cheng, S.S.: Note on the set of periodic solutions of a delay differential equation with piecewise constant argument. Int. J. Pure Appl. Math. 9, 139–143 (2003)
Wang, G.Q., Cheng, S.S.: Existence of periodic solutions for a neutral differential equation with piecewise constant argument. Funkcial. Ekvac. 48, 299–311 (2005)
Wang, L., Yuan, R., Zhang, C.: Corrigendum to: “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, by Yuan, R.]. J. Math. Anal. Appl. 349, 299 (2009)
Wang, Y., Yan, J.: Oscillation of a differential equation with fractional delay and piecewise constant argument. Comput. Math. Appl. 52, 1099–1106 (2006)
Wang, Z., Wu, J.: The stability in a logistic equation with piecewise constant arguments. Differ. Equ. Dyn. Syst. 14, 179–193 (2006)
Wiener, J.: Generalized Solutions of Functional Differential Equations. World Scientific, Singapore (1993)
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Akhmet, M.U. (2018). Functional Differential Equations with Piecewise Constant Argument. In: Volchenkov, D., Leoncini, X. (eds) Regularity and Stochasticity of Nonlinear Dynamical Systems. Nonlinear Systems and Complexity, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-319-58062-3_4
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