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Green's function for impulsive periodic solutions in alternately advanced and delayed differential systems and applications

Yıl 2021, Cilt: 70 Sayı: 1, 15 - 37, 30.06.2021

Kuo-shou Chiu

https://doi.org/10.31801/cfsuasmas.785502
Cited By: 3

Öz

In this paper we investigate the existence of the periodic solutions of a nonlinear impulsive differential system with piecewise alternately advanced and retarded arguments, in short IDEPCAG, 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 system are obtained.
In the process we use the Green's function for impulsive periodic solutions and convert the given the IDEPCAG into an equivalent integral equation system.
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 impulsive differential systems. We also use the contraction mapping principle to show the existence of a unique impulsive periodic solution.
Appropriate examples are given to show the feasibility of our results.

Anahtar Kelimeler

Impulsive differential equation Piecewise constant arguments of generalized type Green's function Periodic solutions Fixed point theorems

Destekleyen Kurum

Universidad Metropolitana de Ciencias de la Educación

Proje Numarası

PGI 03-2020 DIUMCE

Kaynakça

  • Akhmet, M. U., Nonlinear hybrid continuous/discrete-time models, Atlantis Press, Paris, 2011.
  • Alwan, M. S., Liu, X., Stability of singularly perturbed switched systems with time delay and impulsive effects, Nonlinear Analysis: TMA., 71 (2009), 4297--4308.
  • Bereketoglu, H., Seyhan, G., Ogun, A., Advanced impulsive differential equations with piecewise constant arguments, Mathematical Modelling and Analysis, 15 (2) (2010) 175--187.
  • Burton, T. A., A fixed-point theorem of Krasnoselskii, Appl. Math. Lett., 11 (1998), 85--88.
  • Busenberg, S., Cooke, K. L., Models of vertically transmitted diseases with sequential-continuous dynamics, in: V. Lakshmikantham (Ed.), Nonlinear Phenomena in Mathematical Sciences, Academic Press, New York, 1982, 179--187.
  • 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 (2004), 690--697.
  • Castillo, S., Pinto, M., Torres, R., Asymptotic formulae for solutions to impulsive differential equations with piecewise constant argument of generalized type, Electron. J. Differential Equations, Vol. 2019 (2019), No. 40, pp. 1--22.
  • Cooke, K. L., Wiener, J., An equation alternately of retarded and advanced type, Proc. Amer. Math. Soc., 99 (1987), 726--732.
  • 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 (2010), 1--19. Chiu, K.-S., Stability of oscillatory solutions of differential equations with a general piecewise constant argument, E. J. Qualitative Theory of Diff. Equ., 88 (2011), 1--15. Chiu, K.-S., Periodic solutions for nonlinear integro-differential systems with piecewise constant argument, The Scientific World Journal, vol. 2014 (2014), Article ID 514854, 14 pages.
  • Chiu, K.-S., Existence and global exponential stability of equilibrium for impulsive cellular neural network models with piecewise alternately advanced and retarded argument, Abstract and Applied Analysis, vol. 2013 (2013),Article ID 196139, 13 pages.
  • Chiu, K.-S., Pinto, M., Jeng, J.-C., 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.
  • Chiu, K.-S., Pinto, M., Oscillatory and periodic solutions in alternately advanced and delayed differential equations, Carpathian J. Math., 29 (2) (2013), 149--158.
  • Chiu, K.-S., On generalized impulsive piecewise constant delay differential equations, Science China Mathematics, 58 (2015), 1981--2002.
  • K.-S., Exponential stability and periodic solutions of impulsive neural network models with piecewise constant argument, Acta Appl. Math., 151 (2017), 199--226.
  • Chiu, K.-S., Asymptotic equivalence of alternately advanced and delayed differential systems with piecewise constant generalized arguments, Acta Math. Sci., 38 (2018), 220--236.
  • Chiu, K.-S., Li, T., Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments, Math. Nachr., 292 (2019), 2153--2164.
  • Chiu, K.-S., Green's function for periodic solutions in alternately advanced and delayed differential systems, Acta Math. Appl. Sin. Engl. Ser. 36 (4) (2020), 936--951.
  • Jayasree, K. N., Deo, S. G., On piecewise constant delay differential equations, J. Math. Anal. Appl., 169 (1992), 55--69.
  • Karakoc, F., Bereketoglu, H., Seyhan, G., Oscillatory and periodic solutions of impulsive differential equations with piecewise constant argument, Acta Appl. Math., 110(1) (2009), 499--510.
  • Karakoc, F., Bereketoglu, H., Some results for linear impulsive delay differential equations, Dynamics of Cont. Discrete and Impulsive Systems Series A: Mathematical Analysis, 16 (2009), 313--326.
  • Karakoc, F., Ogun Unal, A., Bereketoglu,H., Oscillation of nonlinear differential equations with piecewise constant arguments. E. J. Qualitative Theory of Diff. Equ., 49 (2013), 1--12.
  • Li, X., Weng, P., Impulsive stabilization of two kinds of second-order linear delay differential equations, J. Math. Anal. Appl., 291 (2004), 270--281.
  • Martynyuk, A. A., Shen, J. H., Stavroulakis, I. P., Stability theorems in impulsive functional differential equations with infinite delay, Advances in stability theory at the end of the 20th century, Stability Control Theory Methods Appl., 13 (2003), 153--174.
  • Lakshmikantham, V., Bainov, D. D., Simeonov, P. S., Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
  • Myshkis, A.D., On certain problems in the theory of differential equations with deviating arguments, Uspekhi Mat. Nauk., 32 (1977), 173--202.
  • Nieto, J. J. and, Rodríguez-López, R., Green's function for second-order periodic boundary value problems with piecewise constant arguments, J. Math. Anal. Appl., 304 (2005), 33--57.
  • Oztepe, G. S., Karakoc, F., Bereketoglu, H., Oscillation and periodicity of a second order impulsive delay differential equation with a piecewise constant argument, Communications in Mathematics, 25 (2017), 89--98.
  • Pinto, M., Cauchy and Green matrices type and stability in alternately advanced and delayed differential systems, J. Difference Equ. Appl., 17 (2) (2011), 235--254.
  • Samoilenko, A. M., Perestyuk, N. A., Impulsive Differential Equations, World Scientific Singapore, 1995.
  • Raffoul, Y. N., Periodic solutions for neutral nonlinear differential equations with functional delay, Electron. J. Differential Equations, Vol. 2003 (2003), no. 102, 1--7.
  • Shah, S. M., Wiener, J., Advanced differential equations with piecewise constant argument deviations, Internat, J. Math. and Math. Sci., 6 (1983), 671--703.
  • 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., 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.
  • Wiener, J., Lakshmikantham, V., Differential equations with piecewise constant argument and impulsive equations, Nonlinear Stud., 7 (2000), 60--69.
  • 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 (2007), 798--816.

Yıl 2021, Cilt: 70 Sayı: 1, 15 - 37, 30.06.2021

Kuo-shou Chiu

https://doi.org/10.31801/cfsuasmas.785502
Cited By: 3

Öz

Proje Numarası

PGI 03-2020 DIUMCE

Kaynakça

  • Akhmet, M. U., Nonlinear hybrid continuous/discrete-time models, Atlantis Press, Paris, 2011.
  • Alwan, M. S., Liu, X., Stability of singularly perturbed switched systems with time delay and impulsive effects, Nonlinear Analysis: TMA., 71 (2009), 4297--4308.
  • Bereketoglu, H., Seyhan, G., Ogun, A., Advanced impulsive differential equations with piecewise constant arguments, Mathematical Modelling and Analysis, 15 (2) (2010) 175--187.
  • Burton, T. A., A fixed-point theorem of Krasnoselskii, Appl. Math. Lett., 11 (1998), 85--88.
  • Busenberg, S., Cooke, K. L., Models of vertically transmitted diseases with sequential-continuous dynamics, in: V. Lakshmikantham (Ed.), Nonlinear Phenomena in Mathematical Sciences, Academic Press, New York, 1982, 179--187.
  • 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 (2004), 690--697.
  • Castillo, S., Pinto, M., Torres, R., Asymptotic formulae for solutions to impulsive differential equations with piecewise constant argument of generalized type, Electron. J. Differential Equations, Vol. 2019 (2019), No. 40, pp. 1--22.
  • Cooke, K. L., Wiener, J., An equation alternately of retarded and advanced type, Proc. Amer. Math. Soc., 99 (1987), 726--732.
  • 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 (2010), 1--19. Chiu, K.-S., Stability of oscillatory solutions of differential equations with a general piecewise constant argument, E. J. Qualitative Theory of Diff. Equ., 88 (2011), 1--15. Chiu, K.-S., Periodic solutions for nonlinear integro-differential systems with piecewise constant argument, The Scientific World Journal, vol. 2014 (2014), Article ID 514854, 14 pages.
  • Chiu, K.-S., Existence and global exponential stability of equilibrium for impulsive cellular neural network models with piecewise alternately advanced and retarded argument, Abstract and Applied Analysis, vol. 2013 (2013),Article ID 196139, 13 pages.
  • Chiu, K.-S., Pinto, M., Jeng, J.-C., 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.
  • Chiu, K.-S., Pinto, M., Oscillatory and periodic solutions in alternately advanced and delayed differential equations, Carpathian J. Math., 29 (2) (2013), 149--158.
  • Chiu, K.-S., On generalized impulsive piecewise constant delay differential equations, Science China Mathematics, 58 (2015), 1981--2002.
  • K.-S., Exponential stability and periodic solutions of impulsive neural network models with piecewise constant argument, Acta Appl. Math., 151 (2017), 199--226.
  • Chiu, K.-S., Asymptotic equivalence of alternately advanced and delayed differential systems with piecewise constant generalized arguments, Acta Math. Sci., 38 (2018), 220--236.
  • Chiu, K.-S., Li, T., Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments, Math. Nachr., 292 (2019), 2153--2164.
  • Chiu, K.-S., Green's function for periodic solutions in alternately advanced and delayed differential systems, Acta Math. Appl. Sin. Engl. Ser. 36 (4) (2020), 936--951.
  • Jayasree, K. N., Deo, S. G., On piecewise constant delay differential equations, J. Math. Anal. Appl., 169 (1992), 55--69.
  • Karakoc, F., Bereketoglu, H., Seyhan, G., Oscillatory and periodic solutions of impulsive differential equations with piecewise constant argument, Acta Appl. Math., 110(1) (2009), 499--510.
  • Karakoc, F., Bereketoglu, H., Some results for linear impulsive delay differential equations, Dynamics of Cont. Discrete and Impulsive Systems Series A: Mathematical Analysis, 16 (2009), 313--326.
  • Karakoc, F., Ogun Unal, A., Bereketoglu,H., Oscillation of nonlinear differential equations with piecewise constant arguments. E. J. Qualitative Theory of Diff. Equ., 49 (2013), 1--12.
  • Li, X., Weng, P., Impulsive stabilization of two kinds of second-order linear delay differential equations, J. Math. Anal. Appl., 291 (2004), 270--281.
  • Martynyuk, A. A., Shen, J. H., Stavroulakis, I. P., Stability theorems in impulsive functional differential equations with infinite delay, Advances in stability theory at the end of the 20th century, Stability Control Theory Methods Appl., 13 (2003), 153--174.
  • Lakshmikantham, V., Bainov, D. D., Simeonov, P. S., Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
  • Myshkis, A.D., On certain problems in the theory of differential equations with deviating arguments, Uspekhi Mat. Nauk., 32 (1977), 173--202.
  • Nieto, J. J. and, Rodríguez-López, R., Green's function for second-order periodic boundary value problems with piecewise constant arguments, J. Math. Anal. Appl., 304 (2005), 33--57.
  • Oztepe, G. S., Karakoc, F., Bereketoglu, H., Oscillation and periodicity of a second order impulsive delay differential equation with a piecewise constant argument, Communications in Mathematics, 25 (2017), 89--98.
  • Pinto, M., Cauchy and Green matrices type and stability in alternately advanced and delayed differential systems, J. Difference Equ. Appl., 17 (2) (2011), 235--254.
  • Samoilenko, A. M., Perestyuk, N. A., Impulsive Differential Equations, World Scientific Singapore, 1995.
  • Raffoul, Y. N., Periodic solutions for neutral nonlinear differential equations with functional delay, Electron. J. Differential Equations, Vol. 2003 (2003), no. 102, 1--7.
  • Shah, S. M., Wiener, J., Advanced differential equations with piecewise constant argument deviations, Internat, J. Math. and Math. Sci., 6 (1983), 671--703.
  • 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., 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.
  • Wiener, J., Lakshmikantham, V., Differential equations with piecewise constant argument and impulsive equations, Nonlinear Stud., 7 (2000), 60--69.
  • 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 (2007), 798--816.
Toplam 35 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Uygulamalı Matematik
Bölüm Research Article
Yazarlar

Kuo-shou Chiu 0000-0002-3823-5898

Proje Numarası PGI 03-2020 DIUMCE
Yayımlanma Tarihi 30 Haziran 2021
Gönderilme Tarihi 25 Ağustos 2020
Kabul Tarihi 17 Aralık 2020
Yayımlandığı Sayı Yıl 2021 Cilt: 70 Sayı: 1

Kaynak Göster

APA Chiu, K.-s. (2021). Green’s function for impulsive periodic solutions in alternately advanced and delayed differential systems and applications. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70(1), 15-37. https://doi.org/10.31801/cfsuasmas.785502
AMA Chiu Ks. 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. Haziran 2021;70(1):15-37. doi:10.31801/cfsuasmas.785502
Chicago Chiu, Kuo-shou. “Green’s Function for Impulsive Periodic Solutions in Alternately Advanced and Delayed Differential Systems and Applications”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70, sy. 1 (Haziran 2021): 15-37. https://doi.org/10.31801/cfsuasmas.785502.
EndNote Chiu K-s (01 Haziran 2021) Green’s function for impulsive periodic solutions in alternately advanced and delayed differential systems and applications. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70 1 15–37.
IEEE 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., c. 70, sy. 1, ss. 15–37, 2021, doi: 10.31801/cfsuasmas.785502.
ISNAD Chiu, Kuo-shou. “Green’s Function for Impulsive Periodic Solutions in Alternately Advanced and Delayed Differential Systems and Applications”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70/1 (Haziran 2021), 15-37. https://doi.org/10.31801/cfsuasmas.785502.
JAMA Chiu K-s. 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. 2021;70:15–37.
MLA Chiu, Kuo-shou. “Green’s Function for Impulsive Periodic Solutions in Alternately Advanced and Delayed Differential Systems and Applications”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, c. 70, sy. 1, 2021, ss. 15-37, doi:10.31801/cfsuasmas.785502.
Vancouver Chiu K-s. 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. 2021;70(1):15-37.

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