This article concerns first-order linear nonhomogeneous neutral delay differential equations with periodic coefficients and constant delays, where the coefficients share a common period and the delays are multiples of this period. First, we obtain periodic solutions of linear nonhomogeneous neutral delay differential equations using the variation of parameters method. These periodic solutions are expressed analytically. Two examples demonstrating the applicability of our results are also included. Second, we investigate the asymptotic behavior and estimation of solutions to linear nonhomogeneous neutral delay differential equations. The results are obtained using an appropriate real root of the relevant characteristic equation. Three examples are given to illustrate our results. Finally, we present the special case of first-order linear nonhomogeneous neutral delay differential equations with constant coefficients and constant delays, and provide an interesting example.
Citation: Ali Fuat Yeniçerioğlu, Vildan Yazıcı, Cüneyt Yazıcı. Periodic solutions and asymptotic properties of first order linear nonhomogeneous neutral delay differential equations[J]. AIMS Mathematics, 2026, 11(2): 3811-3838. doi: 10.3934/math.2026155
This article concerns first-order linear nonhomogeneous neutral delay differential equations with periodic coefficients and constant delays, where the coefficients share a common period and the delays are multiples of this period. First, we obtain periodic solutions of linear nonhomogeneous neutral delay differential equations using the variation of parameters method. These periodic solutions are expressed analytically. Two examples demonstrating the applicability of our results are also included. Second, we investigate the asymptotic behavior and estimation of solutions to linear nonhomogeneous neutral delay differential equations. The results are obtained using an appropriate real root of the relevant characteristic equation. Three examples are given to illustrate our results. Finally, we present the special case of first-order linear nonhomogeneous neutral delay differential equations with constant coefficients and constant delays, and provide an interesting example.
| [1] |
Q. Xu, G. Stepan, Z. Wang, Balancing a wheeled inverted pendulum with a single accelerometer in the presence of time delay, J. Vib. Control, 23 (2017), 604–614. https://doi.org/10.1177/1077546315583400 doi: 10.1177/1077546315583400
|
| [2] |
B. Kim, J. Kwon, S. Choi, J. Yang, Feedback stabilization of first order neutral delay systems using the Lambert W function, Appl. Sci., 9 (2019), 3539, 1–12. https://doi.org/10.3390/app9173539 doi: 10.3390/app9173539
|
| [3] |
Y. N. Kyrychko, K. B. Blyuss, A. Gonzalez-Buelga, S. J. Hogan, D. J. Wagg, Stability switches in a neutral delay differential equation with application to real-time dynamic substructuring, Appl. Mech. Mater., 5 (2006), 79–84. https://doi.org/10.4028/www.scientific.net/AMM.5-6.79 doi: 10.4028/www.scientific.net/AMM.5-6.79
|
| [4] | C. Baker, G. Bocharov, C. Paul, F. Rihan, Modelling and analysis of time-lags in some basic patterns of cell proliferation, J. Math. Biol., 37 (1998), 341–371. |
| [5] | V. Kolmanovski, A. Myshkis, Applied Theory of Functional Differential Equations, Kluver Academic: Dordrecht, The Netherlands, 1992. |
| [6] | J. K. Hale, S. M. Verduyn Lunel, Introduction to functional differential equations volume 99 of Applied Mathematical Sciences, Springer-Verlag, New York, 1993. |
| [7] |
M. Frasson, P. H. Tacuri, Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients, Commun. Pure Appl Anal., 13 (2014), 1105–1117. https://doi.org/10.3934/cpaa.2014.13.1105 doi: 10.3934/cpaa.2014.13.1105
|
| [8] |
Ch. G. Philos, I. K. Purnaras, Periodic first order linear neutral delay differential equations, Appl. Math. Comput., 117 (2001), 203–222. https://doi.org/10.1016/S0096-3003(99)00174-5 doi: 10.1016/S0096-3003(99)00174-5
|
| [9] | Ch. G. Philos, I. K. Purnaras, On periodic linear neutral delay differential and difference equations, Electron. J. Differ. Equ., 2006 (2006), 1–25. |
| [10] |
A. F. Yeniçerioğlu, C. Yazıcı, On the behaviors of solutions in linear nonhomogeneous delay differential equations with periodic coefficients, Proc. Inter. Math. Sci., 5 (2023), 1–13. https://doi.org/10.47086/pims.1363924 doi: 10.47086/pims.1363924
|
| [11] | M. Farkas, J. R. Graef, C. Qian, Asymptotic periodicity of delay differential equations, J. Math. Anal. Appl., 226 (1998), 150–165. |
| [12] | H. Li, N. Jin, Y. Zhang, Existence of nonoscillatory solutions for higher order nonlinear mixed neutral differential equations, Math. Model. Control, 4 (2024), 417–423. https://doi.org/10.3934/mmc.2024033 |
| [13] |
Ch. G. Philos, I. K. Purnaras, On the behavior of the solutions for certain first order linear autonomous functional differential equations, Rocky Mountain J. Math., 36 (2006), 1999–2019. https://doi.org/10.1216/rmjm/1181069357 doi: 10.1216/rmjm/1181069357
|
Figures(4)
Ali Fuat Yeniçerioğlu, Vildan Yazıcı, Cüneyt Yazıcı. Periodic solutions and asymptotic properties of first order linear nonhomogeneous neutral delay differential equations[J]. AIMS Mathematics, 2026, 11(2): 3811-3838. doi: 10.3934/math.2026155
DownLoad: