Boundedness of positive solutions of a system of nonlinear delay differential equations

István Győri; Ferenc Hartung; Nahed A. Mohamady

doi:10.3934/dcdsb.2018044

Article Contents

2018, Volume 23, Issue 2: 809-836. Doi: 10.3934/dcdsb.2018044

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Boundedness of positive solutions of a system of nonlinear delay differential equations

Department of Mathematics, University of Pannonia, Veszprém, H-8200, Hungary

Received: November 30, 2016

Revised: July 31, 2017

Early access: November 2017

Published: June 2018

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Abstract

In this manuscript the system of nonlinear delay differential equations $\dot{x}_i(t) =\sum\limits_{j =1}^{n}\sum\limits_{\ell =1}^{n_0}α_{ij\ell} (t) h_{ij}(x_j(t-τ_{ij\ell}(t)))$$-β_i(t)f_i(x_i(t))+ρ_i(t)$, $t≥0$, $1≤i ≤n$ is considered. Sufficient conditions are established for the uniform permanence of the positive solutions of the system. In several particular cases, explicit formulas are given for the estimates of the upper and lower limit of the solutions. In a special case, the asymptotic equivalence of the solutions is investigated.

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Mathematics Subject Classification: Primary: 34K12.

Citation:

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Figure 1. Numerical solution of the System (27).

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Figure 2. Numerical solution of the System (40).

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Table 1. Numerical solution of the System (28)

$k$	$\underline{x}_1^{(k)}$	$\underline{x}_2^{(k)}$	$\underline{x}_3^{(k)}$

$0$	$0$	$0$	$0$
$1$	$0.3761$	$1.7105$	$1.9834$
$2$	$1.8185$	$4.8060$	$3.7077$
$3$	$ 3.6353$	$7.5553$	$5.9214$
$4$	$ 4.0406$	$7.9252$	$6.4602$
$5$	$4.4130$	$8.1962$	$6.9628$
$6$	$4.5364$	$8.2765$	$7.1294$
$7$	$ 4.5767$	$8.3023$	$7.1836$
$8$	$ 4.5958$	$8.3146$	$7.2092$
$9$	$4.5960$	$ 8.3147$	$7.2095$
$10$	$4.5960$	$ 8.3147$	$7.2095$

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Table 2. Numerical solution of the System (30)

$k$	$\overline{x}_1^{(k)}$	$\overline{x}_2^{(k)}$	$\overline{x}_3^{(k)}$

$0$	$0$	$0$	$0$
$1$	$0.6849$	$2.0198$	$ 2.8145$
$2$	$2.9151$	$5.9799$	$ 5.0354$
$3$	$5.5288$	$9.7858$	$7.5194$
$4$	$6.4086$	$10.7362$	$8.3557$
$5$	$6.6740$	$11.0053$	$8.6081$
$6$	$6.7520$	$11.0838$	$8.6822$
$7$	$6.7747$	$11.1067$	$8.7038$
$8$	$ 6.7839$	$11.1159$	$ 8.7125$
$9$	$6.7840$	$11.1161$	$8.7126$
$10$	$6.7840$	$11.1161$	$8.7126$

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Table 3. Numerical solution of the System (41)

$k$	$\underline{x}_1^{(k)}$	$\underline{x}_2^{(k)}$

$0$	$0$	$0$
$1$	$3.4641$	$1.0831$
$2$	$4.7663$	$2.5795$
$3$	$5.2031$	$3.7627$
$4$	$5.4246$	$4.8659$
$5$	$5.4659$	$5.1549$
$6$	$5.4721$	$5.2008$
$7$	$5.4751$	$5.2419$
$8$	$5.4777$	$5.2429$
$9$	$ 5.4778$	$5.2430$
$10$	$ 5.4778$	$5.2430$

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Table 4. Numerical solution of the System (43)

$k$	$\overline{x}_1^{(k)}$	$\overline{x}_2^{(k)}$

$0$	$0$	$0$
$1$	$ 4.4721$	$4.6552$
$2$	$6.3445$	$7.3850$
$3$	$7.0199$	$8.5877$
$4$	$7.2586$	$9.0666$
$5$	$7.3436$	$9.2503$
$6$	$7.3744$	$9.3198$
$7$	$7.3918$	$9.3608$
$8$	$7.3920$	$9.3615$
$9$	$7.3921$	$9.3616$
$10$	$7.3921$	$9.3616$

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