This paper deals with the existence of positive $\omega$-periodic solutions for the neutral functional differential equation with multiple delays $${(u\left(t\right)-cu(t-\delta ))}^{\text{'}}+a\left(t\right)u\left(t\right)=f(t,u(t-{\tau }_1),\cdots ,u(t-{\tau }_n)).$$ 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 $a\left(t\right)$, 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.

The author applies a generalized Leggett-Williams fixed point theorem to the study of the nonlinear functional differential equation $x^{\text{'}}\left(t\right)=-a(t,x\left(t\right))x\left(t\right)+f(t,x_t)$. Sufficient conditions are established for the existence of multiple positive periodic solutions.

In this paper, we employ some new techniques to study the existence of positive periodic solution of $n$-species neutral delay system $$N_i^{\text{'}}\left(t\right)=N_i\left(t\right)\left[a_i\left(t\right)-\sum _{j=1}^n{\beta }_{ij}\left(t\right)N_j\left(t\right)-\sum _{j=1}^n{b}_{ij}\left(t\right)N_j(t-{\tau }_{ij}\left(t\right))-\sum _{j=1}^n{c}_{ij}\left(t\right)N_j^{\text{'}}(t-{\tau }_{ij}\left(t\right))\right].$$ As a corollary, we answer an open problem proposed by Y. Kuang.