Existence and global attractivity of positive periodic solutions of an impulsive delay differential equation. (English) Zbl 1065.34068
Summary: Sufficient conditions are obtained for the existence and global attractivity of periodic positive solutions of an impulsive delay differential equation of the form
\[
\begin{cases} x'(t)+\alpha(t)x(t)=p(t)f\biggl(x \bigl(t-\sigma(t) \bigr)\biggr),\quad\text{a.e. }t>0,\;t\neq t_k,\\ x(t_k^+) -x(t_k)=b_kx(t_k),\quad k=1,2,\dots\end{cases}
\]
Our theorems generalize some known results on special delay population models. It is shown that under appropriate linear periodic impulsive perturbations, the impulsive delay differential equation preserves the original periodicity and global attractivity properties of the nonimpulsive delay differential equation.
MSC:
| 34K13 | Periodic solutions to functional-differential equations |
| 34K20 | Stability theory of functional-differential equations |
| 34K45 | Functional-differential equations with impulses |
| 92D25 | Population dynamics (general) |
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