Positive periodic solutions of a leukopoiesis model with iterative terms. (English) Zbl 1537.34079
In this paper, a first-order differential equation as a leukopoiesis model with a harvesting strategy that the production and harvesting terms include iterative terms is investigated:
\[
\varphi'(t) = -a(t)\varphi(t) + \sum^N_{i=1}b_i(t)\frac{(\varphi^{[i]}(t))^m}{1+(\varphi^{[i]}(t))^n} - h(t, \varphi(t), \varphi^{[2]}(t), \ldots, \varphi^{[N]}(t)),
\]
where \(\varphi^{[2]}(t)= \varphi(\varphi(t))\) and \(\varphi^{[i]}(t)\) denotes the compound of \(\varphi\) with itself \(i\) times, \(a\) and \(b_i\) are continuous \(\omega\)-periodic functions, and \(h\) is the harvesting that is \(\omega\)-periodic with respect to the first argument and globally Lipschitz with respect to the other arguments. Applying the Krasnoselskii fixed point theorem with the aid of some Green’s function properties the authors prove the existence of at least one positive periodic solution. Under another condition, the Banach fixed point theorem is used to prove the existence of a unique positive periodic solution. Moreover, the continuous dependence on parameters is also guaranteed.
Reviewer: Zhanyuan Hou (London)
MSC:
| 34K13 | Periodic solutions to functional-differential equations |
| 47H10 | Fixed-point theorems |
| 92C37 | Cell biology |
Keywords:
fixed point theorem; Green’s function; iterative differential equation; leukopoiesis model; periodic solutionReferences:
| [1] | Alzabut, J.; Khuddush, M.; Selvam, AGM, Second order iterative dynamic boundary value problems with mixed derivative operators with applications, Qual. Theory Dyn. Syst., 22, 1, 32 (2023) · Zbl 1512.34153 · doi:10.1007/s12346-022-00736-1 |
| [2] | Ben Fredj, H.; Chérif, F., Positive pseudo almost periodic solutions to a class of hematopoiesis model: oscillations and dynamics, J. Appl. Math. Comput., 63, 479-500 (2020) · Zbl 1475.34035 · doi:10.1007/s12190-020-01326-7 |
| [3] | Bohner, M.; Streipert, S., Optimal harvesting policy for the Beverton-Holt model, Math. Biosci. Eng., 13, 4, 673-695 (2016) · Zbl 1348.39008 · doi:10.3934/mbe.2016014 |
| [4] | Bohner, M.; Streipert, S., Optimal harvesting policy for the Beverton-Holt quantum difference model, Math. Morav., 20, 2, 39-57 (2016) · Zbl 1460.91174 · doi:10.5937/MatMor1602039B |
| [5] | Bouakkaz, A.; Ardjouni, A.; Djoudi, A., Existence of positive periodic solutions for a second-order nonlinear neutral differential equation by the Krasnoselskii’s fixed point theorem, Nonlinear Dyn. Syst. Theory, 17, 3, 230-238 (2017) · Zbl 1377.34091 |
| [6] | Bouakkaz, A.; Ardjouni, A.; Khemis, R., Periodic solutions of a class of third-order functional differential equations with iterative source terms, Bol. Soc. Mat. Mex., 26, 443-458 (2020) · Zbl 1458.34121 · doi:10.1007/s40590-019-00267-x |
| [7] | Bouakkaz, A., Bounded solutions to a three-point fourth-order iterative boundary value problem, Rocky Mt. J. Math., 52, 3, 793-803 (2022) · Zbl 1502.34075 · doi:10.1216/rmj.2022.52.793 |
| [8] | Bouakkaz, A., Positive periodic solutions for a class of first-order iterative differential equations with an application to a hematopoiesis model, Carpathian J. Math., 38, 2, 347-355 (2022) · Zbl 1538.34268 · doi:10.37193/CJM.2022.02.07 |
| [9] | Bouakkaz, A.; Khemis, R., Existence, uniqueness and stability of solutions to a delay hematopoiesis model, J. Innov. Appl. Math. Comput. Sci., 2, 2, 23-30 (2022) · doi:10.58205/jiamcs.v2i2.21 |
| [10] | Bouakkaz, A.; Khemis, R., Positive periodic solutions for a class of second-order differential equations with state-dependent delays, Turk. J. Math., 44, 4, 1412-1426 (2020) · Zbl 1459.34157 · doi:10.3906/mat-2004-52 |
| [11] | Bouakkaz, A.; Khemis, R., Positive periodic solutions for revisited Nicholson’s blowflies equation with iterative harvesting term, J. Math. Anal. Appl., 494, 2 (2021) · Zbl 1462.34111 · doi:10.1016/j.jmaa.2020.124663 |
| [12] | Cheraiet, S.; Bouakkaz, A.; Khemis, R., Some new findings on bounded solution of a third order iterative boundary-value problem, J. Interdiscip. Math., 25, 4, 1153-1162 (2022) · doi:10.1080/09720502.2021.1995215 |
| [13] | Chouaf, S.; Bouakkaz, A.; Khemis, R., On bounded solutions of a second-order iterative boundary value problem, Turk. J. Math., 46, 2, 453-464 (2022) · Zbl 1495.34092 |
| [14] | Chouaf, S.; Khemis, R.; Bouakkaz, A., Some existence results on positive solutions for an iterative second-order boundary-value problem with integral boundary conditions, Bol. Soc. Parana. Mat., 3, 40, 1-10 (2022) · Zbl 07801906 |
| [15] | Diagana, T.; Zhou, H., Existence of positive almost periodic solutions to the hematopoiesis model, Appl. Math. Comput., 274, 644-648 (2016) · Zbl 1410.34206 |
| [16] | Ding, HS; Liu, QL; Nieto, JJ, Existence of positive almost periodic solutions to a class of hematopoiesis model, Appl. Math. Model., 40, 4, 3289-3297 (2016) · Zbl 1452.92006 · doi:10.1016/j.apm.2015.10.020 |
| [17] | Faria, T.; Oliveira, JJ, Global asymptotic stability for a periodic delay hematopoiesis model with impulses, Appl. Math. Model., 79, 843-864 (2020) · Zbl 1481.92018 · doi:10.1016/j.apm.2019.10.063 |
| [18] | Guerfi, A.; Ardjouni, A., Periodic solutions for second order totally nonlinear iterative differential equations, J. Anal., 30, 1, 353-367 (2021) · Zbl 1483.34096 · doi:10.1007/s41478-021-00347-0 |
| [19] | Han, X.; Lei, C., Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays, Open Math., 20, 1, 1380-1393 (2022) · Zbl 1503.34046 · doi:10.1515/math-2022-0491 |
| [20] | Khemis, M.; Bouakkaz, A., Existence and uniqueness results for a neutral erythropoiesis model with iterative production and harvesting terms, Bol. Soc. Parana. Mat., 3, 41, 1-9 (2023) · Zbl 07805687 |
| [21] | Khemis, M.; Bouakkaz, A.; Khemis, R., Existence, uniqueness and stability results of an iterative survival model of red blood cells with a delayed nonlinear harvesting term, J. Math. Model., 10, 3, 515-528 (2022) · Zbl 1524.34210 |
| [22] | Khemis, M.; Bouakkaz, A.; Khemis, R., Positive periodic solutions for a delay model of erythropoiesis with iterative terms, Appl. Anal. (2023) · Zbl 1539.34076 · doi:10.1080/00036811.2023.2186862 |
| [23] | Khemis, R.; Ardjouni, A.; Bouakkaz, A.; Djoudi, A., Periodic solutions of a class of third-order differential equations with two delays depending on time and state, Comment Math. Univ. Carol., 60, 3, 379-399 (2019) · Zbl 1474.34471 |
| [24] | Khemis, R.; Bouakkaz, A.; Chouaf, S., On the existence of periodic solutions of a second order iterative differential equation, Acta Math. Univ. Comen. (N.S.), 92, 1, 9-22 (2023) · Zbl 1539.34076 |
| [25] | Khemis, R., Existence, uniqueness and stability of positive periodic solutions for an iterative Nicholson’s blowflies equation, J. Appl. Math. Comput., 69, 1903-1916 (2023) · Zbl 1518.34009 · doi:10.1007/s12190-022-01820-0 |
| [26] | Khuddush, M.; Prasad, KR, Nonlinear two-point iterative functional boundary value problems on time scales, J. Appl. Math. Comput., 68, 6, 4241-4251 (2022) · Zbl 1499.34419 · doi:10.1007/s12190-022-01703-4 |
| [27] | Liu, G.; Yan, J.; Zhang, F., Existence and global attractivity of unique positive periodic solution for a model of hematopoiesis, J. Math. Anal. Appl., 334, 1, 157-171 (2007) · Zbl 1155.34041 · doi:10.1016/j.jmaa.2006.12.015 |
| [28] | Liu, B., New results on the positive almost periodic solutions for a model of hematopoiesis, Nonlinear Anal. Real World Appl., 17, 252-264 (2014) · Zbl 1316.34087 · doi:10.1016/j.nonrwa.2013.12.003 |
| [29] | Liu, X.; Jia, M., A class of iterative functional fractional differential equation on infinite interval, Appl. Math. Lett., 136 (2023) · Zbl 1512.34150 · doi:10.1016/j.aml.2022.108473 |
| [30] | Mackey, MC; Glass, L., Oscillation and chaos in physiological control system, Science, 197, 287-289 (1977) · Zbl 1383.92036 · doi:10.1126/science.267326 |
| [31] | Mezghiche, L.; Khemis, R., On periodic solutions of a recruitment model with iterative terms and a nonlinear harvesting, Bol. Soc. Parana. Mat., 3, 41, 1-9 (2023) · Zbl 07805685 |
| [32] | Rihan, FA, Delay Differential Equations and Applications to Biology (2021), Berlin: Springer, Berlin · Zbl 07332853 · doi:10.1007/978-981-16-0626-7 |
| [33] | Song, Q.; Hao, X., Positive solutions for fractional iterative functional differential equation with a convection term, Electron. Res. Arch., 31, 4, 1863-1875 (2023) · doi:10.3934/era.2023096 |
| [34] | Wang, X.; Zhang, H., A new approach to the existence, nonlinear and uniqueness of positive almost periodic solution for a model of hematopoiesis, Nonlinear Anal. Real World Appl., 11, 60-66 (2010) · Zbl 1225.47072 · doi:10.1016/j.nonrwa.2008.10.015 |
| [35] | Wu, XM; Li, JW; Zhou, HQ, A necessary and sufficient condition for the existence of positive periodic solutions of a model of hematopoiesis, Comput. Math. Appl., 54, 840-849 (2007) · Zbl 1137.34333 · doi:10.1016/j.camwa.2007.03.004 |
| [36] | Xu, C.; Liao, M.; Li, P.; Guo, Y.; Liu, Z., Bifurcation properties for fractional order delayed BAM neural networks, Cogn. Comput., 13, 322-356 (2021) · doi:10.1007/s12559-020-09782-w |
| [37] | Xu, C.; Liu, Z.; Aouiti, C.; Li, P.; Yao, L.; Yan, J., New exploration on bifurcation for fractional-order quaternion-valued neural networks involving leakage delays, Cogn. Neurodyn., 16, 1233-1248 (2022) · doi:10.1007/s11571-021-09763-1 |
| [38] | Yang, X., Existence and global attractivity of unique positive almost periodic solution for a model of hematopoiesis, Appl. Math. J. Chin. Univ. Ser. B, 25, 1, 25-34 (2010) · Zbl 1224.34150 · doi:10.1007/s11766-010-2111-6 |
| [39] | Zhao, HY; Fečkan, M., Periodic solutions for a class of differential equations with delays depending on state, Math. Commun., 23, 29-42 (2018) · Zbl 1454.34101 |
| [40] | Zhao, HY; Liu, J., Periodic solutions of an iterative functional differential equation with variable coefficients, Math. Methods Appl. Sci., 40, 286-292 (2017) · Zbl 1356.34071 · doi:10.1002/mma.3991 |
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