We consider first order neutral functional differential equations with multiple deviating arguments of the form ${(x\left(t\right)+Bx(t-\delta ))}^{\text{'}}=g₀(t,x\left(t\right))+{\sum }_{k=1}^n{g}_k(t,x(t-{\tau }_k\left(t\right)))+p\left(t\right)$. By using coincidence degree theory, we establish some sufficient conditions on the existence and uniqueness of periodic solutions for the above equation. Moreover, two examples are given to illustrate the effectiveness of our results.

By using the coincidence degree theory, we study a type of $p$-Laplacian neutral Rayleigh functional differential equation with deviating argument to establish new results on the existence of $T$-periodic solutions.

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.

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.