# American Institute of Mathematical Sciences

March  2018, 23(2): 809-836. doi: 10.3934/dcdsb.2018044

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

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

Received  December 2016 Revised  August 2017 Published  March 2018 Early access  December 2017

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.

Citation: István Győri, Ferenc Hartung, Nahed A. Mohamady. Boundedness of positive solutions of a system of nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 809-836. doi: 10.3934/dcdsb.2018044
##### References:

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##### References:
Numerical solution of the System (27).
Numerical solution of the System (40).
Numerical solution of the System (28)
 $k$ $\underline{x}_1^{(k)}$ $\underline{x}_2^{(k)}$ $\underline{x}_3^{(k)}$ $0$ $0$ $0$ $0$ $1$ $0.3761$ $1.7105$ $1.9834$ $2$ $1.8185$ $4.8060$ $3.7077$ $3$ $3.6353$ $7.5553$ $5.9214$ $4$ $4.0406$ $7.9252$ $6.4602$ $5$ $4.4130$ $8.1962$ $6.9628$ $6$ $4.5364$ $8.2765$ $7.1294$ $7$ $4.5767$ $8.3023$ $7.1836$ $8$ $4.5958$ $8.3146$ $7.2092$ $9$ $4.5960$ $8.3147$ $7.2095$ $10$ $4.5960$ $8.3147$ $7.2095$
 $k$ $\underline{x}_1^{(k)}$ $\underline{x}_2^{(k)}$ $\underline{x}_3^{(k)}$ $0$ $0$ $0$ $0$ $1$ $0.3761$ $1.7105$ $1.9834$ $2$ $1.8185$ $4.8060$ $3.7077$ $3$ $3.6353$ $7.5553$ $5.9214$ $4$ $4.0406$ $7.9252$ $6.4602$ $5$ $4.4130$ $8.1962$ $6.9628$ $6$ $4.5364$ $8.2765$ $7.1294$ $7$ $4.5767$ $8.3023$ $7.1836$ $8$ $4.5958$ $8.3146$ $7.2092$ $9$ $4.5960$ $8.3147$ $7.2095$ $10$ $4.5960$ $8.3147$ $7.2095$
Numerical solution of the System (30)
 $k$ $\overline{x}_1^{(k)}$ $\overline{x}_2^{(k)}$ $\overline{x}_3^{(k)}$ $0$ $0$ $0$ $0$ $1$ $0.6849$ $2.0198$ $2.8145$ $2$ $2.9151$ $5.9799$ $5.0354$ $3$ $5.5288$ $9.7858$ $7.5194$ $4$ $6.4086$ $10.7362$ $8.3557$ $5$ $6.6740$ $11.0053$ $8.6081$ $6$ $6.7520$ $11.0838$ $8.6822$ $7$ $6.7747$ $11.1067$ $8.7038$ $8$ $6.7839$ $11.1159$ $8.7125$ $9$ $6.7840$ $11.1161$ $8.7126$ $10$ $6.7840$ $11.1161$ $8.7126$
 $k$ $\overline{x}_1^{(k)}$ $\overline{x}_2^{(k)}$ $\overline{x}_3^{(k)}$ $0$ $0$ $0$ $0$ $1$ $0.6849$ $2.0198$ $2.8145$ $2$ $2.9151$ $5.9799$ $5.0354$ $3$ $5.5288$ $9.7858$ $7.5194$ $4$ $6.4086$ $10.7362$ $8.3557$ $5$ $6.6740$ $11.0053$ $8.6081$ $6$ $6.7520$ $11.0838$ $8.6822$ $7$ $6.7747$ $11.1067$ $8.7038$ $8$ $6.7839$ $11.1159$ $8.7125$ $9$ $6.7840$ $11.1161$ $8.7126$ $10$ $6.7840$ $11.1161$ $8.7126$
Numerical solution of the System (41)
 $k$ $\underline{x}_1^{(k)}$ $\underline{x}_2^{(k)}$ $0$ $0$ $0$ $1$ $3.4641$ $1.0831$ $2$ $4.7663$ $2.5795$ $3$ $5.2031$ $3.7627$ $4$ $5.4246$ $4.8659$ $5$ $5.4659$ $5.1549$ $6$ $5.4721$ $5.2008$ $7$ $5.4751$ $5.2419$ $8$ $5.4777$ $5.2429$ $9$ $5.4778$ $5.2430$ $10$ $5.4778$ $5.2430$
 $k$ $\underline{x}_1^{(k)}$ $\underline{x}_2^{(k)}$ $0$ $0$ $0$ $1$ $3.4641$ $1.0831$ $2$ $4.7663$ $2.5795$ $3$ $5.2031$ $3.7627$ $4$ $5.4246$ $4.8659$ $5$ $5.4659$ $5.1549$ $6$ $5.4721$ $5.2008$ $7$ $5.4751$ $5.2419$ $8$ $5.4777$ $5.2429$ $9$ $5.4778$ $5.2430$ $10$ $5.4778$ $5.2430$
Numerical solution of the System (43)
 $k$ $\overline{x}_1^{(k)}$ $\overline{x}_2^{(k)}$ $0$ $0$ $0$ $1$ $4.4721$ $4.6552$ $2$ $6.3445$ $7.3850$ $3$ $7.0199$ $8.5877$ $4$ $7.2586$ $9.0666$ $5$ $7.3436$ $9.2503$ $6$ $7.3744$ $9.3198$ $7$ $7.3918$ $9.3608$ $8$ $7.3920$ $9.3615$ $9$ $7.3921$ $9.3616$ $10$ $7.3921$ $9.3616$
 $k$ $\overline{x}_1^{(k)}$ $\overline{x}_2^{(k)}$ $0$ $0$ $0$ $1$ $4.4721$ $4.6552$ $2$ $6.3445$ $7.3850$ $3$ $7.0199$ $8.5877$ $4$ $7.2586$ $9.0666$ $5$ $7.3436$ $9.2503$ $6$ $7.3744$ $9.3198$ $7$ $7.3918$ $9.3608$ $8$ $7.3920$ $9.3615$ $9$ $7.3921$ $9.3616$ $10$ $7.3921$ $9.3616$
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