Preview

Nanosystems: Physics, Chemistry, Mathematics

Advanced search

Some conditions for the existence of 4-periodic solutions in non-homogeneous differential equations involving piecewise alternately advanced and retarded arguments

https://doi.org/10.17586/2220-8054-2024-15-6-749-754

Abstract

The manuscript introduces a method to characterize 4-periodic solutions in first-order non-homogeneous differential equations involving piecewise alternately advanced and retarded argument. It systematically delineates the prerequisites for these solutions to exist and furnishes precise methodologies for their determination. Additionally, the paper includes the illustrative example, including scenarios with infinitely many solutions, to demonstrate the effectiveness of the proposed approach.

About the Authors

Kuo-Shou Chiu
Universidad Metropolitana de Ciencias de la Educaci´on
Chile

Kuo-Shou Chiu – Departamento de Matem´atica, Facultad de Ciencias B´asicas

Jos´e Pedro Alessandri 774, Santiago



F. Cordova-Lepe
Universidad Cat´olica del Maule
Chile

Fernando Cordova-Lepe – Departamento de Matem´atica, F´ısica y Estad´ıstica

Talca



References

1. Busenberg S., Cooke K.L. Models of vertically transmitted diseases with sequential-continuous dynamics. Nonlinear Phenomena in Mathematical Sciences, Academic Press, New York, 1982, P. 179–187.

2. Dai L. Nonlinear dynamics of piecewise constant systems and implementation of piecewise constant arguments, World Scientific, Singapore, 2008.

3. Chiu K.-S. Global exponential stability of bidirectional associative memory neural networks model with piecewise alternately advanced and retarded argument. Comp. Appl. Math., 2021, 40, P. 263.

4. Chiu K.-S., Li T. New stability results for bidirectional associative memory neural networks model involving generalized piecewise constant delay. Math. Comput. Simul., 2022, 194, P. 719–743.

5. Chiu K.-S. Existence and global exponential stability of periodic solution for Cohen-Grossberg neural networks model with piecewise constant argument. Hacettepe J. Math. Stat., 2022, 51(5), P. 1219–1236.

6. Karakoc F. Asymptotic behaviour of a population model with piecewise constant argument. Appl. Math. Lett., 2017, 70, P. 7–13.

7. Lafci M. Behavior of the solutions of a single-species population model with piecewise constant argument. Iran. J. Math. Sci. Inform., 2024, 19(2), P. 111–117.

8. Aftabizadeh A.R., Wiener J. Oscillatory and periodic solutions of an equation alternately of retarded and advanced type. Appl. Anal., 1986, 23, P. 219–231.

9. Chiu K.-S., Pinto M. Oscillatory and periodic solutions in alternately advanced and delayed differential equations. Carpathian J. Math., 2013, 29(2), P. 149–158.

10. Chiu K.-S. Green’s function for periodic solutions in alternately advanced and delayed differential systems. Acta Math. Appl. Sin. Engl. Ser., 2020, 36, P. 936–951.

11. 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), P. 15–37.

12. Chiu K.-S. Periodic solutions of impulsive differential equations with piecewise alternately advanced and retarded argument of generalized type. Rocky Mountain J. Math., 2022, 52, P. 87–103.

13. Chiu K.-S. Numerical-analytic successive approximation method for the investigation of periodic solutions of nonlinear integro-differential systems with piecewise constant argument of generalized type. Hacettepe J. Math. Stat., 2024, 53(5), P. 1272–1290.

14. Karakoc F., H. Bereketoglu H., Seyhan G. Oscillatory and periodic solutions of impulsive differential equations with piecewise constant argument. Acta Appl. Math., 2010, 110, P. 499–510.

15. Karakoc F., Unal A., Bereketoglu H. Oscillation of a nonlinear impulsive differential equation system with piecewise constant argument. Adv. Differ. Equ., 2018, 99.

16. 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, 2017, 25, P. 89–98.

17. Wang G.-Q. Periodic solutions of a neutral differential equation with piecewise constant arguments. J. Math. Anal. Appl., 2007, 326, P. 736–747.

18. Muminov M.I. On the method of finding periodic solutions of second-order neutral differential equations with piecewise constant arguments. Adv. Differ. Equ., 2017, 336.

19. Muminov M.I., Ali H.M. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments. Open Math., 2020, 18(1), P. 93–105.

20. Muminov M.I., Jumaev Z.Z. Exact periodic solutions of second-order differential equations with piecewise constant arguments. Advances in Mathematics: Scientific Journal, 2021, 10(9), P. 3113–3128.

21. Muminov M.I., Radjabov T.A. On existence conditions for periodic solutions to a differential equation with constant argument. Nanosystems: Phys. Chem. Math., 2022, 13(5), P. 491–497.

22. Muminov M.I., Radjabov T.A. Existence conditions for 2-periodic solutions to a non-homogeneous differential equations with piecewise constant argument. Examples and Counterexamples, 2024, 5, P. 100145.

23. 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., 2011, 27(4), P. 561–568.

24. Chiu K.-S., Berna I. Nonautonomous impulsive differential equations of alternately advanced and retarded type. Filomat, 2023, 37(23), P. 7813–7829.

25. Jayasree K.N., Deo S.G. Variation of parameters formula for the equation of Cooke and Wiener. Proc. Am. Math. Soc., 1991, 112(1), P. 75–80.


Review

For citations:


Chiu K., Cordova-Lepe F. Some conditions for the existence of 4-periodic solutions in non-homogeneous differential equations involving piecewise alternately advanced and retarded arguments. Nanosystems: Physics, Chemistry, Mathematics. 2024;15(6):749-754. https://doi.org/10.17586/2220-8054-2024-15-6-749-754

Views: 6


Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.


ISSN 2220-8054 (Print)
ISSN 2305-7971 (Online)