Some notes to existence and stability of the positive periodic solutions for a delayed nonlinear differential equations. (English) Zbl 1352.34096
The authors consider the existence of positive \(\omega\)-periodic solutions for delay differential equations of the form
\[
\dot{x}(t)=-p(t)x(t)+\sum_{i=1}^nq_i(t)f(x(\tau_i(t))),\quad t\geq t_0,
\]
where
(i) \(p, q_i \in C([t_0,\infty),(0,\infty))\), \(i=1,...,n\), \(f\in C^1(\mathbb{R},\mathbb{R})\), \(f(x)>0\) for \(x>0\),
(ii) \(\tau_i \in C([t_0,\infty),(0,\infty))\), \(\tau_i(t)<t\) and \(\lim_{t\to\infty}\tau_i(t)=\infty\) for \(i=1,\dots,n\).
They provide sufficient conditions to assure the existence and exponential stability of such solutions. It seems that is very difficult to check in practice these conditions. But, in turn, it is not assumed that the functions \(p\), \(q_i\), \(\tau_i\) are periodic.
(i) \(p, q_i \in C([t_0,\infty),(0,\infty))\), \(i=1,...,n\), \(f\in C^1(\mathbb{R},\mathbb{R})\), \(f(x)>0\) for \(x>0\),
(ii) \(\tau_i \in C([t_0,\infty),(0,\infty))\), \(\tau_i(t)<t\) and \(\lim_{t\to\infty}\tau_i(t)=\infty\) for \(i=1,\dots,n\).
They provide sufficient conditions to assure the existence and exponential stability of such solutions. It seems that is very difficult to check in practice these conditions. But, in turn, it is not assumed that the functions \(p\), \(q_i\), \(\tau_i\) are periodic.
Reviewer: Adriana Buică (Cluj-Napoca)
MSC:
| 34K13 | Periodic solutions to functional-differential equations |
| 34K20 | Stability theory of functional-differential equations |
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