Effectiveness of matrix measure in finding periodic solutions for nonlinear systems of differential and integro-differential equations with delays

Mouataz Billah Mesmouli; Amir Abdel Menaem; Taher S. Hassan; Mouataz Billah Mesmouli; Amir Abdel Menaem; Taher S. Hassan

doi:10.3934/math.2024693

AIMS Mathematics

2024, Volume 9, Issue 6: 14274-14287. doi: 10.3934/math.2024693

Previous Article Next Article

Research article Special Issues

Effectiveness of matrix measure in finding periodic solutions for nonlinear systems of differential and integro-differential equations with delays

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Automated Electrical Systems, Ural Power Engineering Institute, Ural Federal University, Yekaterinburg 620002, Russia; a.a.abdelmenaem@urfu.ru
3.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt; tshassan@mans.edu.eg
4.
Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II 39, Roma 00186, Italy

Received: 04 March 2024 Revised: 03 April 2024 Accepted: 08 April 2024 Published: 19 April 2024
MSC : 34A34, 34K13, 34K40, 47H10

In this manuscript, under the matrix measure and some sufficient conditions, we will overcame all difficulties and challenges related to the fundamental matrix for a generalized nonlinear neutral functional differential equations in matrix form with multiple delays. The periodicity of solutions, as well as the uniqueness under the considered conditions has been proved employing the fixed point theory. Our approach expanded and generalized certain previously published findings for example, we studied the uniqueness of a solution that was absent in some literature. Moreover, an example was given to confirm the main results.
- Arzela-Ascoli's theorem,
- fixed point theorem,
- matrix measure,
- system of differential equations,
- delay,
- periodic solutions
Citation: Mouataz Billah Mesmouli, Amir Abdel Menaem, Taher S. Hassan. Effectiveness of matrix measure in finding periodic solutions for nonlinear systems of differential and integro-differential equations with delays[J]. AIMS Mathematics, 2024, 9(6): 14274-14287. doi: 10.3934/math.2024693

Related Papers:

Abstract

In this manuscript, under the matrix measure and some sufficient conditions, we will overcame all difficulties and challenges related to the fundamental matrix for a generalized nonlinear neutral functional differential equations in matrix form with multiple delays. The periodicity of solutions, as well as the uniqueness under the considered conditions has been proved employing the fixed point theory. Our approach expanded and generalized certain previously published findings for example, we studied the uniqueness of a solution that was absent in some literature. Moreover, an example was given to confirm the main results.

References

[1]	K. Gu, V. Kharitonov, J. Chen, Stability of time-delay systems, Birkhauser, 2003. https://doi.org/10.1007/978-1-4612-0039-0
[2]	C. J. Guo, G. Q. Wang, S. S. Cheng, Periodic solutions for a neutral functional differential equation with multiple variable lags, Arch. Math., 42 (2006), 1–10.
[3]	M. N. Islam, Y. N. Raffoul, Periodic solutions of neutral nonlinear system of differential equations with functional delay, J. Math. Anal. Appl., 331 (2007), 1175–1186. https://doi.org/10.1016/j.jmaa.2006.09.030 doi: 10.1016/j.jmaa.2006.09.030
[4]	M. Pinto, Dichotomy and existence of periodic solutions of quasilinear functional differential equations, Nonlinear Anal. Theory Methods Appl., 72 (2010), 1227–1234. https://doi.org/10.1016/j.na.2009.08.007 doi: 10.1016/j.na.2009.08.007
[5]	C. Tunç, O. Tunç, J. C. Yao, On the stability, integrability and boundedness analysis of systems of integro-differential equations with time-delay, Fixed Point Theory, 24 (2023), 753–774. https://doi.org/10.24193/fpt-ro.2023.2.19 doi: 10.24193/fpt-ro.2023.2.19
[6]	O. Tunç, C. Tunç, On Ulam stabilities of delay Hammerstein integral equation, Symmetry, 15 (2023), 1736. https://doi.org/10.3390/sym15091736 doi: 10.3390/sym15091736
[7]	W. W. Mohammed, F. M. Al-Askar, C. Cesarano, On the dynamical behavior of solitary waves for coupled stochastic Korteweg-de Vries equations, Mathematics, 11 (2023), 3506. https://doi.org/10.3390/math11163506 doi: 10.3390/math11163506
[8]	H. E. Khochemane, A. Ardjouni, S. Zitouni, Existence and Ulam stability results for two orders neutral fractional differential equations, Afr. Mat., 33 (2022), 35. https://doi.org/10.1007/s13370-022-00970-5 doi: 10.1007/s13370-022-00970-5
[9]	P. S. Ngiamsunthorn, Existence of periodic solutions for differential equations with multiple delays under dichotomy condition, Adv. Differ. Equations, 2015 (2015), 259. https://doi.org/10.1186/s13662-015-0598-0 doi: 10.1186/s13662-015-0598-0
[10]	M. B. Mesmouli, A. A. Attiya, A. A. Elmandouha, A. Tchalla, T. S. Hassan, Dichotomy condition and periodic solutions for two nonlinear neutral systems, J. Funct. Spaces, 2022 (2022), 6319312. https://doi.org/10.1155/2022/6319312 doi: 10.1155/2022/6319312
[11]	M. B. Mesmouli, M. Alesemi, W. W. Mohammed, Periodic solutions for a neutral system with two Volterra terms, Mathematics, 11 (2023), 2204. https://doi.org/10.3390/math11092204 doi: 10.3390/math11092204
[12]	Y. Luo, W. Wang, J. H. Shen, Existence of positive periodic solutions for two kinds of neutral functional differential equations, Appl. Math. Lett., 21 (2008), 581–587. https://doi.org/10.1016/j.aml.2007.07.009 doi: 10.1016/j.aml.2007.07.009
[13]	J. Luo, J. Yu, Global asymptotic stability of nonautonomous mathematical ecological equations with distributed deviating arguments, Acta Math. Sin., 41 (1998), 1273–1282. https://doi.org/10.12386/A1998sxxb0191 doi: 10.12386/A1998sxxb0191
[14]	P. Weng, M. Liang, The existence and behavior of periodic solution of Hematopoiesis model, Math. Appl., 1995,434–439.
[15]	W. S. C. Gurney, S. P. Blythe, R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17–20. https://doi.org/10.1038/287017a0 doi: 10.1038/287017a0
[16]	W. Joseph, H. So, J. Yu, Global attractivity and uniform persistence in Nicholson's blowflies, Differ. Equations Dyn. Syst., 1 (1994), 11–18.
[17]	K. Gopalsamy, Stability and oscillation in delay differential equations of population dynamics, Kluwer Academic Press, 1992. https://doi.org/10.1007/978-94-015-7920-9
[18]	M. Vidyasagar, Nonlinear system analysis, Prentice Hall Inc., 1978.
[19]	R. Reissig, G. Sasone, R. Conti, Non-linear differential equations of higher order, Springer Dordrecht, 1974.
[20]	T. A. Burton, Stability by fixed point theory for functional differential equations, Dover Publications, 2006.
[21]	D. R. Smart, Fixed point theorems, Cambridge University Press, 1980.
[22]	F. V. Difonzo, P. Przybyłowicz, Y. Wu, Existence, uniqueness and approximation of solutions to Carathéodory delay differential equations, J. Comput. Appl. Math., 436 (2024), 115411. https://doi.org/10.1016/j.cam.2023.115411 doi: 10.1016/j.cam.2023.115411
[23]	F. V. Difonzo, P. Przybyłowicz, Y. Wu, X. Xie, A randomized Runge-Kutta method for time-irregular delay differential equations, arXiv, 2024. https://doi.org/10.48550/arXiv.2401.11658
[24]	L. Dieci, C. Elia, D. Pi, Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity, Discrete Contin. Dyn. Syst., 22 (2017), 3091–3112. https://doi.org/10.3934/dcdsb.2017165 doi: 10.3934/dcdsb.2017165

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)