关键词: 泛函微分方程, 无穷时滞, 中立型, 周期解, 重合度理论

Abstract: Discuss the periodic solutions of the following neutral functional differential equation with infinite delay
$$ \frac{\rm d}{{\rm d}t}\left(x(t)-\int_{-\infty}^{0}g(s,x(t+s)){\rm d}s\right)
=A(t,x(t))x(t)+f(t,x_t).
$$
Some results on the existence and uniqueness of periodic solutions are obtained by using continuation theorem in coincidence degree. Especially, when $g(s,x)\equiv 0$ and $A(t,x)=A(t)$, the conditions which guarantee the existence of the unique and stable periodic solution are derived.