Positive periodic solutions of impulsive delay differential equations with sign-changing coefficient

Yuji Liu, Zhanbing Bai, Zhanjie Gui and Weigao Ge

Abstract: Consider the existence and nonexistence of positive periodic solutions of the non-autonomous delay differential equation
$$ x'(t)=-a(t)x(t)+\lambda h(t)f\bigl(x(t-\tau(t))\bigr), t\not=t_k, k\in Z, (*) $$
where $a$ and $h$ may change sign, with the impulses as follows
$$ x(t_k)=(1+b_k)x(t_k^-), k\in Z. (**) $$
It is shown that the system (*)--(**) has positive periodic solutions under certain reasonable conditions and no positive periodic solutions under some other conditions. Some applications and examples are given to illustrate our Theorems.

Keywords: impulses; functional differential equation; positive periodic solution; cone; fixed point theorem.

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