Abstract
In this paper we investigate the existence of the periodic solutions of a nonlinear differential equation with a general piecewise constant argument, in short DEPCAG, that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use the Green’s function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii’s fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results.
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Aftabizadeh, A.R., Wiener, J. Oscillatory and periodic solutions of an equation alternately of retarded and advanced types. Appl. Anal., 23: 219–231 (1986)
A.R. Aftabizadeh, J. Wiener, J.M. Xu. Oscillatory and periodic solutions of delay differential equations with piecewise constant argument. Proc. Amer. Math. Soc., 99: 673–679 (1987)
Akhmet, M.U. Integral manifolds of differential equations with piecewise constant argument of generalized type. Nonlinear Anal. TMA., 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., Buyukadali, C., Ergenc, T. Periodic solutions of the hybrid system with small parameter. Nonlinear Anal. Hybrid Syst., 2: 532–543 (2008)
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)
Burton, T.A. Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press Inc., New York, 1985
Burton, T.A. Krasnoselskii’s inversion principle and fixed points. Nonlinear Anal. TMA., 30: 3975–3986 (1997)
Burton, T.A. A fixed-point theorem of Krasnoselskii. Appl. Math. Lett., 11: 85–88 (1998)
Burton, T.A., Furumochi, T. Periodic and asymptotically periodic solutions of neutral integral equations. E.J. Qualitative Theory of Diff. Equ., 10: 1–19 (2000)
Burton, T.A., Furumochi, T. Existence theorems and periodic solutions of neutral integral equations. Nonlinear Anal. TMA., 43: 527–546 (2001)
Cabada, A., Ferreiro, J.B., Nieto, J.J. Green’s function and comparison principles for first order periodic differential equations with piecewise constant arguments. J. Math. Anal. Appl., 291: 690–697 (2004)
Cooke, K.L., Wiener, J. Retarded differential equations with piecewise constant delays. J. Math. Anal. Appl., 99: 265–297 (1984)
Cooke, K.L., Wiener, J. An equation alternately of retarded and advanced type. Proc. Amer. Math. Soc., 99: 726–732 (1987)
Chiu, K.S., Pinto, M. Oscillatory and periodic solutions in alternately advanced and delayed differential equations. Carpathian J. Math., 29(2): 149–158 (2013)
Chiu, K.S. Stability of oscillatory solutions of differential equations with a general piecewise constant argument. E.J. Qualitative Theory of Diff. Equ., 88: 1–15 (2011)
Chiu, K.S., Pinto, M. Variation of parameters formula and Gronwall inequality for differential equations with a general piecewise constant argument. Acta Math. Appl. Sin. Engl. Ser, 27(4): 561–568 (2011)
Chiu, K.S., Pinto, M. Periodic solutions of differential equations with a general piecewise constant argument and applications. E.J. Qualitative Theory of Diff. Equ., 46: 1–19 (2010)
Chiu, K.S., Pinto, M., Jeng, J.Ch. Existence and global convergence of periodic solutions in recurrent neural network models with a general piecewise alternately advanced and retarded argument. Acta Appl. Math., 133: 133–152 (2014)
Chiu, K.S. On generalized impulsive piecewise constant delay differential equations. Science China Mathematics., 58: 1981–2002 (2015)
Chiu, K.S., Jeng, J.Ch. Stability of oscillatory solutions of differential equations with general piecewise constant arguments of mixed type. Math. Nachr., 288: 1085–1097 (2015)
Chiu, K.S. Exponential stability and periodic solutions of impulsive neural network models with piecewise constant argument. Acta Appl. Math., 151: 199–226 (2017)
Chiu, K.S. Asymptotic equivalence of alternately advanced and delayed differential systems with piecewise constant generalized arguments. Acta Math. Sci., 38: 220–236 (2018)
Chiu, K.S., Li, T. Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments. Math. Nachr., 292: 2153–2164 (2019)
Dai, L.M. Nonlinear Dynamics of Piecewise of Constant Systems and Implememtation of Piecewise Constants Arguments. World Scientific, Singapore, 2008
Karakoc, F., Bereketoglu, H., Seyhan, G. Oscillatory and periodic solutions of impulsive differential equations with piecewise constant argument. Acta Appl. Math., 110: 499–510 (2009)
Liu, Y., Yang, P., Ge, W. Periodic solutions of higher-order delay differential equations. Nonlinear Anal. TMA., 63: 136–152 (2005)
Liu, X.L., Li, W.T. Periodic solutions for dynamic equations on time scales. Nonlinear Anal. TMA., 67: 1457–1463 (2007)
Myshkis, A.D. On certain problems in the theory of differential equations with deviating arguments. Uspekhi Mat. Nauk., 32: 173–202 (1977)
Nieto, J.J., Rodríguez-López, R. Green’s function for second-order periodic boundary value problems with piecewise constant arguments. J. Math. Anal. Appl., 304: 33–57 (2005)
Pinto, M. Asymptotic equivalence of nonlinear and quasilinear differential equations with piecewise constant arguments. Math. Comput. Model., 49: 1750–1758 (2009)
Pinto, M. Dichotomy and existence of periodic solutions of quasilinear functional differential equations. Nonlinear Anal., TMA., 72: 1227–1234 (2010)
Pinto, M. Cauchy and Green matrices type and stability in alternately advanced and delayed differential systems. J. Difference Equ. Appl., 17(2): 235–254 (2011)
Pinto, M., Sepúlveda, D. h-asymptotic stability by fixed point in neutral nonlinear differential equations with delay. Nonlinear Anal. TMA., 74: 3926–3933 (2011)
Raffoul, Y.N. Periodic solutions for neutral nonlinear differential equations with functional delay. Electron. J. Differential Equations, 2003(102): 1–7 (2003)
Shah, S.M., Wiener, J. Advanced differential equations with piecewise constant argument deviations. Internat. J. Math. and Math. Sci., 6: 671–703 (1983)
Smart, D.R. Fixed Points Theorems. Cambridge University Press, Cambridge, 1980
Wiener, J. Generalized Solutions of Functional Differential Equations. World Scientific, Singapore, 1993
Wang, G.Q. Existence theorem of periodic solutions for a delay nonlinear differential equation with piecewise constant arguments. J. Math. Anal. Appl., 298: 298–307 (2004)
Wang, G.Q., Cheng, S.S. Periodic solutions of discrete Rayleigh equations with deviating arguments. Taiwanese J. Math., 13(6B): 2051–2067 (2009)
Wang, G.Q., Cheng, S.S. Existence and uniqueness of periodic solutions for a second-order nonlinear differential equation with piecewise constant argument. Int. J. Math. Math. Sci., 2009, Art. ID 950797, 14 pp
Xia, Y.H., Huang, Z., Han, M. Existence of almost periodic solutions for forced perturbed systems with piecewise constant argument. J. Math. Anal. Appl., 333: 798–816 (2007)
Yang, P., Liu, Y., Ge, W. Green’s function for second order differential equations with piecewise constant argument. Nonlinear Anal. TMA., 64: 1812–1830 (2006)
Yuan, R. The existence of almost periodic solutions of retarded differential equations with piecewise constant argument. Nonlinear Anal. TMA., 48: 1013–1032 (2002)
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The author thanks the referees very much for their valuable suggestions which made this paper much improved. This research was in part supported by FGI 05-16 DIUMCE.
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This research was in part supported by FGI 05-16 DIUMCE.
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Chiu, KS. Green’s Function for Periodic Solutions in Alternately Advanced and Delayed Differential Systems. Acta Math. Appl. Sin. Engl. Ser. 36, 936–951 (2020). https://doi.org/10.1007/s10255-020-0975-7
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DOI: https://doi.org/10.1007/s10255-020-0975-7