This paper examines the distribution of zeros of all solutions to first-order delay differential equations with several increasing delay arguments (DDEs). By advancing and generalizing the existing frameworks of the oscillation theory of delay and neutral differential equations, new explicit upper bounds are established for the separation between successive zeros of solutions. These bounds are constructed by analyzing the properties of positive solutions over a closed interval, providing sufficient conditions to prevent the persistence of such positivity. This approach provides a general analytical framework for identifying conditions under which all nontrivial solutions oscillate. Moreover, we determine the locations of the zeros of solutions to equations with several delays and oscillatory coefficients. Furthermore, the developed approach allows future extensions to functional and difference equations with non-monotone or distributed delays. Several illustrative examples and comparisons with existing criteria are included to demonstrate the accuracy and effectiveness of the proposed results.
Citation: Emad Attia. Explicit upper bound approximations for the distance between successive zeros of all solutions to first-order differential equations with several monotone delays[J]. AIMS Mathematics, 2026, 11(3): 5841-5852. doi: 10.3934/math.2026241
This paper examines the distribution of zeros of all solutions to first-order delay differential equations with several increasing delay arguments (DDEs). By advancing and generalizing the existing frameworks of the oscillation theory of delay and neutral differential equations, new explicit upper bounds are established for the separation between successive zeros of solutions. These bounds are constructed by analyzing the properties of positive solutions over a closed interval, providing sufficient conditions to prevent the persistence of such positivity. This approach provides a general analytical framework for identifying conditions under which all nontrivial solutions oscillate. Moreover, we determine the locations of the zeros of solutions to equations with several delays and oscillatory coefficients. Furthermore, the developed approach allows future extensions to functional and difference equations with non-monotone or distributed delays. Several illustrative examples and comparisons with existing criteria are included to demonstrate the accuracy and effectiveness of the proposed results.
| [1] | R. P. Agarwal, S. R. Grace, D. O'Regan, Oscillation theory for difference and functional differential equations, Springer, New York, 2013. |
| [2] |
E. R. Attia, On the upper bounds for the distance between zeros of solutions of a first-order linear neutral differential equation with several delays, AIMS Math., 9 (2024), 23564–23583. https://doi.org/10.3934/math.20241145 doi: 10.3934/math.20241145
|
| [3] |
E. R. Attia, O. N. Al-Masarer, I. Jadlovská, On the distribution of adjacent zeros of solutions to first-order neutral differential equations, Turkish J. Math., 47 (2023), 195–212. https://doi.org/10.55730/1300-0098.3354 doi: 10.55730/1300-0098.3354
|
| [4] |
E. R. Attia, P. Chatzarakis, Upper bounds for distances between adjacent zeros of first-order linear differential equations with several delays, Mathematics, 10 (2022), 648. https://doi.org/10.3390/math10040648 doi: 10.3390/math10040648
|
| [5] |
E. R. Attia, B. M. El-Matary, New results for the upper bounds of the distance between adjacent zeros of first-order differential equations with several variable delays, J. Inequal. Appl., 2023 (2023), 103. https://doi.org/10.1186/s13660-023-03017-w doi: 10.1186/s13660-023-03017-w
|
| [6] |
F. A. Baker, H. A. El-Morshedy, The distribution of zeros of all solutions of first-order neutral differential equations, Appl. Math. Comput., 259 (2015), 777–789. https://doi.org/10.1016/j.amc.2015.03.004 doi: 10.1016/j.amc.2015.03.004
|
| [7] |
F. A. Baker, H. A. El-Morshedy, On the distribution of zeros of solutions of a first-order neutral differential equation, Sci. J. Damietta Fac. Sci., 4 (2015), 1–9. https://doi.org/10.21608/sjdfs.2015.194323 doi: 10.21608/sjdfs.2015.194323
|
| [8] |
T. H. Barr, Oscillations in linear delay differential equations, J. Math. Anal. Appl., 195 (1995), 261–277. https://doi.org/10.1006/jmaa.1995.1354 doi: 10.1006/jmaa.1995.1354
|
| [9] |
L. Berezansky, Y. Domshlak, Can a solution of a linear delay differential equation have an infinite number of isolated zeros on a finite interval? Appl. Math. Lett., 19 (2006), 587–589. https://doi.org/10.1007/s10806-006-9009-6 doi: 10.1007/s10806-006-9009-6
|
| [10] | Y. Domshlak, A. I. Aliev, On oscillatory properties of first-order differential equations with one or two retarded arguments, Hiroshima Math. J., 18 (1988), 31–46. |
| [11] |
H. A. El-Morshedy, On the distribution of zeros of solutions of first-order delay differential equations, Nonlinear Anal., 74 (2011), 3353–3362. https://doi.org/10.1016/j.na.2011.02.011 doi: 10.1016/j.na.2011.02.011
|
| [12] |
H. A. El-Morshedy, E. R. Attia, On the distance between adjacent zeros of solutions of first-order differential equations with distributed delays, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), 8. https://doi.org/10.14232/ejqtde.2016.1.8 doi: 10.14232/ejqtde.2016.1.8
|
| [13] | L. H. Erbe, Q. K. Kong, B. G. Zhang, Oscillation theory for functional differential equations, CRC Press, Boca Raton, FL, 1994. |
| [14] |
R. G. Koplatadze, T. A. Chanturija, Oscillating and monotone solutions of first-order differential equations with deviating argument, Differ. Equ., 18 (1982), 1463–1465. https://doi.org/10.1109/TMAG.1982.1062044 doi: 10.1109/TMAG.1982.1062044
|
| [15] |
R. G. Koplatadze, G. N. Kvinikadze, On the oscillation of solutions of first-order delay differential inequalities and equations, Georgian Math. J., 1 (1994), 675–689. https://doi.org/10.1002/j.2040-4603.1994.tb01176.x doi: 10.1002/j.2040-4603.1994.tb01176.x
|
| [16] | K. Gopalsamy, Stability and oscillation in delay differential equations of population dynamics, Springer, Dordrecht, 1992. |
| [17] |
J. R. Graef, S. R. Grace, I. Jadlovská, E. Tunç, Some new oscillation results for higher-order nonlinear differential equations with a nonlinear neutral term, Mathematics, 10 (2022), 2997. https://doi.org/10.3390/math10162997 doi: 10.3390/math10162997
|
| [18] | A. Halanay, Differential equations: Stability, oscillations, time lags, Academic Press, New York, 1966. |
| [19] |
S. R. Grace, J. R. Graef, T. Li, E. Tunç, Oscillatory behavior of second-order nonlinear noncanonical neutral differential equations, Acta Univ. Sapientiae Math., 15 (2023), 259–271. https://doi.org/10.2478/ausm-2023-0014 doi: 10.2478/ausm-2023-0014
|
| [20] | I. Győri, G. Ladas, Oscillation theory of delay differential equations with applications, Clarendon Press, Oxford, 1991. |
| [21] |
I. Jadlovská, G. E. Chatzarakis, E. Tunç, Kneser-type oscillation theorems for second-order functional differential equations with unbounded neutral coefficients, Math. Slovaca, 74 (2024), 637–664. https://doi.org/10.1515/ms-2024-0049 doi: 10.1515/ms-2024-0049
|
| [22] |
Z. Jiao, I. Jadlovská, T. Li, Large-time behavior in a two-species chemotaxis–competition system with nonlocal nonlinear growth terms, Math. Nachr., 299 (2026), 5–34. https://doi.org/10.1002/mana.70041 doi: 10.1002/mana.70041
|
| [23] | V. Kolmanovskii, A. Myshkis, Applied theory of functional differential equations, Kluwer Acad. Publ., Boston, 1992. |
| [24] |
T. Li, D. Acosta-Soba, A. Columbu, G. Viglialoro, Dissipative gradient nonlinearities prevent $\delta$-formations in local and nonlocal attraction–repulsion chemotaxis models, Stud. Appl. Math., 154 (2025), e70018. https://doi.org/10.1111/sapm.70018 doi: 10.1111/sapm.70018
|
| [25] |
Q. Liu, S. R. Grace, E. Tunç, T. Li, Oscillation of noncanonical fourth-order dynamic equations, Appl. Math. Sci. Eng., 2023. https://doi.org/10.1080/27690911.2023.2239435 doi: 10.1080/27690911.2023.2239435
|
| [26] |
C. McCalla, Zeros of the solutions of first-order functional differential equations, SIAM J. Math. Anal., 9 (1978), 843–847. https://doi.org/10.1137/0509066 doi: 10.1137/0509066
|
| [27] | A. D. Myshkis, Linear homogeneous differential equations of first order with deviating arguments, Uspekhi Mat. Nauk, 5 (1950), 160–162. |
| [28] |
X. Tang, J. S. Yu, Distribution of zeros of solutions of first-order delay differential equations, Appl. Math. J. Chinese Univ. Ser. B, 14 (1999), 375–380. https://doi.org/10.1007/s11766-999-0066-2 doi: 10.1007/s11766-999-0066-2
|
| [29] |
H. W. Wu, L. Erbe, A. Peterson, Upper bounds for the distances between consecutive zeros of solutions of first-order delay differential equations, Appl. Math. Comput., 429 (2015), 562–575. https://doi.org/10.1016/j.jmaa.2015.04.049 doi: 10.1016/j.jmaa.2015.04.049
|
| [30] |
B. G. Zhang, Y. Zhou, The distribution of zeros of solutions of differential equations with a variable delay, J. Math. Anal. Appl., 256 (2001), 216–228. https://doi.org/10.1006/jmaa.2000.7309 doi: 10.1006/jmaa.2000.7309
|
Figures(1) / Tables(1)
Emad Attia. Explicit upper bound approximations for the distance between successive zeros of all solutions to first-order differential equations with several monotone delays[J]. AIMS Mathematics, 2026, 11(3): 5841-5852. doi: 10.3934/math.2026241
DownLoad: