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Globally attractive periodic solution of a perturbed functional differential equation. (English) Zbl 1109.34051

The authors discuss the functional-differential equation
\[ x'(t)= \delta\left\{ \sum^m_{i=0}a_i(t)x(t- \tau_i(t))\right\}\tag{*} \]
and its perturbed equation
\[ x'(t)=\delta\left\{\sum^m_{i=0}a_i(t)x(t-\tau_i (t))\right\}+p(t)\tag{**} \]
for \(t\geq-\tau\), where \(\delta\) is \(+1\) or \(-1\), \(\tau_0(t)=0\) and \(\tau=\max_{1\leq i\leq m}\max_{t\in[0,T]}\tau_i (t)\), \(a_0,a_1,\dots,a_m\), \(\tau_1\), \(\tau_2,\dots,\tau_m\) are real continuous \(T\)-periodic functions defined on \([0,\infty)\) such that \(\sum^m_{i=0} a_i(t)\) is not the trivial function, and that \(\tau_1,\tau_2,\dots,\tau_m\) are nonnegative. \(p(t)\) is a real continuous \(T\)-periodic and nonnegative function defined on \([0,\infty)\). The main results are as follows: If \(\sum^m_{i=0}\int^t_0|a_i(t)|\,dt\leq 4\), then:
(I) the only \(T\)-periodic solution of (*) on \([-\tau,\infty)\) is the trivial one;
(II) equation (**) has a unique \(T\)-periodic solution.
Furthermore, the authors also discuss the equation
\[ x'(t)= -a_1(t) x(t)-a_2(t) x(t-\mu(t))+r(t),\tag{***} \]
where \(r,a_1,a_2\), and \(\mu\) are positive continuous \(T\)-periodic functions, \(T\) is a fixed positive number. Under the change of variable \(N(t)=e^{x(t)}\), equation (***) is transformed into the logarithmic model for a single population
\[ N'(t)=N(t) \left\{r(t)-a_1(t)\ln N(t)-a_2(t)\ln(N(t-\mu(t)))\right\}.\tag{****} \]
The authors show that the much weeker condition \(\int_0^T(a_1(s)+a_2 (s))ds<4\) is sufficient for equation (***) or (****) to have a unique \(T\)-periodic solution, and that the additional conditions \(a_2(t)\leq a_1(t)\) and \(\tau'(t)\leq 1\) for \(t\geq 0\) imply the global attractivity of this unique solution. It is improved the result of Y. K. Li [Appl. Math., Ser. A (Chin. Ed.) 12, 279–282 (1997; Zbl 0883.92023)].

MSC:

34K13 Periodic solutions to functional-differential equations

Citations:

Zbl 0883.92023
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References:

[1] Cao, X. B., On the existence of periodic solutions for nonlinear system with multiple delays, Appl. Math. Mech., 24, 1, 105-110 (2003), (in Chinese)
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[5] Li, Y. K., Attractivity of a positive periodic solution for all other positive solutions in a delay population model, Appl. Math. JCU, 12A, 3, 279-282 (1997), (in Chinese) · Zbl 0883.92023
[6] Tang, X. H., Asymptotic behavior of delay differential equations with instantaneously terms, J. Math. Anal. Appl., 302, 342-359 (2005) · Zbl 1068.34076
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