Article Contents
Article Contents

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

• 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.

Mathematics Subject Classification: Primary: 34K12.

 Citation:

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

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

Table 1.  Numerical solution of the System (28)

 $k$ $\underline{x}_1^{(k)}$ $\underline{x}_2^{(k)}$ $\underline{x}_3^{(k)}$ $0$ $0$ $0$ $0$ $1$ $0.3761$ $1.7105$ $1.9834$ $2$ $1.8185$ $4.8060$ $3.7077$ $3$ $3.6353$ $7.5553$ $5.9214$ $4$ $4.0406$ $7.9252$ $6.4602$ $5$ $4.4130$ $8.1962$ $6.9628$ $6$ $4.5364$ $8.2765$ $7.1294$ $7$ $4.5767$ $8.3023$ $7.1836$ $8$ $4.5958$ $8.3146$ $7.2092$ $9$ $4.5960$ $8.3147$ $7.2095$ $10$ $4.5960$ $8.3147$ $7.2095$

Table 2.  Numerical solution of the System (30)

 $k$ $\overline{x}_1^{(k)}$ $\overline{x}_2^{(k)}$ $\overline{x}_3^{(k)}$ $0$ $0$ $0$ $0$ $1$ $0.6849$ $2.0198$ $2.8145$ $2$ $2.9151$ $5.9799$ $5.0354$ $3$ $5.5288$ $9.7858$ $7.5194$ $4$ $6.4086$ $10.7362$ $8.3557$ $5$ $6.6740$ $11.0053$ $8.6081$ $6$ $6.7520$ $11.0838$ $8.6822$ $7$ $6.7747$ $11.1067$ $8.7038$ $8$ $6.7839$ $11.1159$ $8.7125$ $9$ $6.7840$ $11.1161$ $8.7126$ $10$ $6.7840$ $11.1161$ $8.7126$

Table 3.  Numerical solution of the System (41)

 $k$ $\underline{x}_1^{(k)}$ $\underline{x}_2^{(k)}$ $0$ $0$ $0$ $1$ $3.4641$ $1.0831$ $2$ $4.7663$ $2.5795$ $3$ $5.2031$ $3.7627$ $4$ $5.4246$ $4.8659$ $5$ $5.4659$ $5.1549$ $6$ $5.4721$ $5.2008$ $7$ $5.4751$ $5.2419$ $8$ $5.4777$ $5.2429$ $9$ $5.4778$ $5.2430$ $10$ $5.4778$ $5.2430$

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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