Periodic solutions of impulsive differential equations with piecewise alternately advanced and retarded argument of generalized type. (English) Zbl 1500.34057
The paper investigates the nonlinear vector impulsive system
\[
\dot{y}(t)=A(t)y(t)+f(t,y(t), y(\gamma(t))),
\]
\[ \Delta y(t)=J_k(y(t_k^-)), t=t_k, k\in Z,
\]
where \(\gamma(t)\) is a piecewise constant argument of mixed type (alternatively delayed or advanced).
The existence and uniqueness of periodic and subharmonic solutions are obtained by applying the Poincaré operator and fixed point theory.
The existence and uniqueness of periodic and subharmonic solutions are obtained by applying the Poincaré operator and fixed point theory.
Reviewer: Leonid Berezanski (Be’er Sheva)
MSC:
| 34K13 | Periodic solutions to functional-differential equations |
| 34K45 | Functional-differential equations with impulses |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 47N20 | Applications of operator theory to differential and integral equations |
Keywords:
impulsive differential equation; piecewise constant argument (delayed or advanced); existence and uniqueness of periodic solutions; fixed point theorems; Gronwall’s inequalityReferences:
| [1] | A. R. Aftabizadeh, J. Wiener, and J.-M. Xu, “Oscillatory and periodic solutions of delay differential equations with piecewise constant argument”, Proc. Amer. Math. Soc. 99:4 (1987), 673-679. · Zbl 0631.34078 · doi:10.2307/2046474 |
| [2] | M. Akhmet, Nonlinear hybrid continuous/discrete-time models, Atlantis Studies in Mathematics for Engineering and Science 8, Atlantis Press, Paris, 2011. · Zbl 1328.93001 · doi:10.2991/978-94-91216-03-9 |
| [3] | M. S. Alwan and X. Liu, “Stability of singularly perturbed switched systems with time delay and impulsive effects”, Nonlinear Anal. 71:9 (2009), 4297-4308. · Zbl 1181.34084 · doi:10.1016/j.na.2009.02.131 |
| [4] | H. Bereketoglu, G. Seyhan, and A. Ogun, “Advanced impulsive differential equations with piecewise constant arguments”, Math. Model. Anal. 15:2 (2010), 175-187. · Zbl 1218.34095 · doi:10.3846/1392-6292.2010.15.175-187 |
| [5] | L. E. J. Brouwer, “Über Abbildung von Mannigfaltigkeiten”, Math. Ann. 71:1 (1911), 97-115. · JFM 42.0417.01 · doi:10.1007/BF01456931 |
| [6] | T. A. Burton, Stability and periodic solutions of ordinary and functional-differential equations, Mathematics in Science and Engineering 178, Academic Press, Orlando, FL, 1985. · Zbl 0635.34001 |
| [7] | S. Busenberg and K. L. Cooke, “Models of vertically transmitted diseases with sequential-continuous dynamics”, pp. 179-187 in Nonlinear phenomena in mathematical sciences, edited by V. Lakshmikantham, Academic Press, New York, 1982. · Zbl 0512.92018 · doi:10.1016/B978-0-12-434170-8.50028-5 |
| [8] | S. Castillo, M. Pinto, and R. Torres, “Asymptotic formulae for solutions to impulsive differential equations with piecewise constant argument of generalized type”, Electron. J. Differential Equations 2019 (2019), art. id. 40. · Zbl 1411.34109 |
| [9] | K.-S. Chiu, “Stability of oscillatory solutions of differential equations with a general piecewise constant argument”, Electron. J. Qual. Theory Differ. Equ. (2011), art. id. 88. · Zbl 1340.34240 · doi:10.14232/ejqtde.2011.1.88 |
| [10] | K.-S. Chiu, “Existence and global exponential stability of equilibrium for impulsive cellular neural network models with piecewise alternately advanced and retarded argument”, Abstr. Appl. Anal. (2013), art. id. 196139. · Zbl 1298.34136 · doi:10.1155/2013/196139 |
| [11] | K.-S. Chiu, “Periodic solutions for nonlinear integro-differential systems with piecewise constant argument”, Sci. World J. (2014), art. id. 514854. |
| [12] | K.-S. Chiu, “On generalized impulsive piecewise constant delay differential equations”, Sci. China Math. 58:9 (2015), 1981-2002. · Zbl 1333.34118 · doi:10.1007/s11425-015-5010-8 |
| [13] | K.-S. Chiu, “Exponential stability and periodic solutions of impulsive neural network models with piecewise constant argument”, Acta Appl. Math. 151 (2017), 199-226. · Zbl 1373.92010 · doi:10.1007/s10440-017-0108-3 |
| [14] | K.-S. Chiu, “Asymptotic equivalence of alternately advanced and delayed differential systems with piecewise constant generalized arguments”, Acta Math. Sci. Ser. B 38:1 (2018), 220-236. · Zbl 1399.34183 · doi:10.1016/S0252-9602(17)30128-5 |
| [15] | K.-S. Chiu, “Green’s function for impulsive periodic solutions in alternately advanced and delayed differential systems and applications”, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat. 70:1 (2021), 15-37. · Zbl 1489.34031 · doi:10.31801/cfsuasmas.785502 |
| [16] | K.-S. Chiu and T. Li, “Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments”, Math. Nachr. 292:10 (2019), 2153-2164. · Zbl 1442.34104 · doi:10.1002/mana.201800053 |
| [17] | K.-S. Chiu and M. Pinto, “Periodic solutions of differential equations with a general piecewise constant argument and applications”, Electron. J. Qual. Theory Differ. Equ. (2010), art. id. 46. · Zbl 1211.34082 · doi:10.14232/ejqtde.2010.1.46 |
| [18] | K.-S. Chiu and M. Pinto, “Oscillatory and periodic solutions in alternately advanced and delayed differential equations”, Carpathian J. Math. 29:2 (2013), 149-158. · Zbl 1299.34225 · doi:10.37193/CJM.2013.02.15 |
| [19] | K.-S. Chiu, M. Pinto, and J.-C. Jeng, “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 (2014), 133-152. · Zbl 1315.34073 · doi:10.1007/s10440-013-9863-y |
| [20] | K. L. Cooke and J. Wiener, “Retarded differential equations with piecewise constant delays”, J. Math. Anal. Appl. 99:1 (1984), 265-297. · Zbl 0557.34059 · doi:10.1016/0022-247X(84)90248-8 |
| [21] | K. L. Cooke and J. Wiener, “An equation alternately of retarded and advanced type”, Proc. Amer. Math. Soc. 99:4 (1987), 726-732. · Zbl 0628.34074 · doi:10.2307/2046483 |
| [22] | W. S. C. Gurney, S. P. Blythe, and R. M. Nisbet, “Nicholson’s blowflies revisited”, Nature 287 (1980), 17-21. · doi:10.1038/287017a0 |
| [23] | K. N. Jayasree and S. G. Deo, “On piecewise constant delay differential equations”, J. Math. Anal. Appl. 169:1 (1992), 55-69. · Zbl 0913.34054 · doi:10.1016/0022-247X(92)90103-K |
| [24] | F. Karakoc, H. Bereketoglu, and G. Seyhan, “Oscillatory and periodic solutions of impulsive differential equations with piecewise constant argument”, Acta Appl. Math. 110:1 (2010), 499-510. · Zbl 1196.34104 · doi:10.1007/s10440-009-9458-9 |
| [25] | F. Karakoç and H. Bereketoğlu, “Some results for linear impulsive delay differetial equations”, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 16:3 (2009), 313-326. · Zbl 1180.34091 |
| [26] | F. Karakoç, A. Ogun Unal, and H. Bereketoglu, “Oscillation of nonlinear impulsive differential equations with piecewise constant arguments”, Electron. J. Qual. Theory Differ. Equ. (2013), art. id. 49. · Zbl 1340.34305 · doi:10.14232/ejqtde.2013.1.49 |
| [27] | V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of impulsive differential equations, Series in Modern Applied Mathematics 6, World Scientific, Singapore, 1989. · Zbl 0719.34002 · doi:10.1142/0906 |
| [28] | J.-l. Li and J.-h. Shen, “Periodic boundary value problems of impulsive differential equations with piecewise constant argument”, J. Nat. Sci. Hunan Norm. Univ. 25:3 (2002), 5-9. · Zbl 1047.34076 |
| [29] | X. Li and P. Weng, “Impulsive stabilization of two kinds of second-order linear delay differential equations”, J. Math. Anal. Appl. 291:1 (2004), 270-281. · Zbl 1047.34098 · doi:10.1016/j.jmaa.2003.11.002 |
| [30] | A. A. Martynyuk, J. H. Shen, and I. P. Stavroulakis, “Stability theorems in impulsive functional differential equations with infinite delay”, pp. 153-174 in Advances in stability theory at the end of the 20th century, Stability Control Theory Methods Appl. 13, Taylor & Francis, London, 2003. · Zbl 1047.34087 |
| [31] | A. D. Myshkis, “On certain problems in the theory of differential equations with deviating argument”, Uspehi Mat. Nauk 32:2 (1977), 173-202. In Russian; translated in Russian Math. Surveys 32:2 (1977), 181-213. · Zbl 0378.34052 · doi:10.1070/RM1977v032n02ABEH001623 |
| [32] | G. S. Oztepe, F. Karakoc, and H. Bereketoglu, “Oscillation and periodicity of a second order impulsive delay differential equation with a piecewise constant argument”, Commun. Math. 25:2 (2017), 89-98. · Zbl 1391.34105 · doi:10.1515/cm-2017-0009 |
| [33] | M. Pinto, “Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments”, Math. Comput. Modelling 49:9-10 (2009), 1750-1758. · Zbl 1171.34321 · doi:10.1016/j.mcm.2008.10.001 |
| [34] | F. J. Richards, “A flexible growth function for empirical use”, J. Experimental Botany 29:29 (1959), 290-300. · doi:10.1093/jxb/10.2.290 |
| [35] | R. Rosen, Anticipatory systems, IFSR International Series on Systems Science and Engineering 1, Pergamon Press, New York, 1985. · Zbl 1282.93002 |
| [36] | A. M. Samoĭlenko and N. A. Perestyuk, Impulsive differential equations, World Scientific Series on Nonlinear Science. Series A: monographs and treatises 14, World Scientific, Singapore, 1995. · Zbl 0837.34003 · doi:10.1142/9789812798664 |
| [37] | S. M. Shah and J. Wiener, “Advanced differential equations with piecewise constant argument deviations”, Internat. J. Math. Math. Sci. 6:4 (1983), 671-703. · Zbl 0534.34067 · doi:10.1155/S0161171283000599 |
| [38] | I. Stamova and G. Stamov, Applied impulsive mathematical models, Springer, Cham, 2016. · Zbl 1355.34004 · doi:10.1007/978-3-319-28061-5 |
| [39] | M. Ważewska-Czyżewska and A. Lasota, “Mathematical problems of the dynamics of a system of red blood cells”, Mat. Stos. (3) 6 (1976), 23-40. · Zbl 0363.92012 |
| [40] | J. Wiener, “Differential equations with piecewise constant delays”, pp. 547-580 in Trends in theory and practice of nonlinear differential equations, edited by V. Lakshmikantham, Lecture Notes in Pure and Applied Mathematics 90, Marcel Dekker, New York, 1984. |
| [41] | J. Wiener, Generalized solutions of functional-differential equations, World Scientific, Singapore, 1993. · Zbl 0874.34054 · doi:10.1142/1860 |
| [42] | J. Wiener and V. Lakshmikantham, “Differential equations with piecewise constant argument and impulsive equations”, Nonlinear Stud. 7:1 (2000), 60-69 · Zbl 0997.34076 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
