Positive periodic solutions of a neutral functional differential equation with multiple delays

summary:This paper deals with the existence of positive $\omega $-periodic solutions for the neutral functional differential equation with multiple delays $$(u(t)-cu(t-\delta ))'+a(t) u(t)=f(t, u(t-\tau _1), \cdots , u(t-\tau _n)).$$ The essential inequality conditions on the existence of positive periodic solutions are obtained. These inequality conditions concern with the relations of $c$ and the coefficient function $a(t)$, and the nonlinearity $f(t, x_1,\cdots , x_n)$. Our discussion is based on the perturbation method of positive operator and fixed point index theory in cones