### For REFEREES

 Volume 9, Number 2, 2019, Pages 501-525 Study on a kind of $p$-Laplacian neutral differential equation with multiple variable coefficients Zhibo Cheng,Zhonghua Bi Keywords:Neutral operator with multiple variable coefficients, $p$-Laplacian, periodic solution, extension of Mawhin's continuation theorem, singularity. Abstract: In this paper, we first discuss some properties of the neutral operator with multiple variable coefficients $(Ax)(t):=x(t)-\sum\limits_{i=1}^{n}c_i(t)x(t-\delta_i)$. Afterwards, by using an extension of Mawhin's continuation theorem, a kind of second order $p$-Laplacian neutral differential equation with multiple variable coefficients as follows $$\left(\phi_p\left(x(t)-\sum\limits_{i=1}^{n}c_i(t)x(t-\delta_i)\right)'\right)'=\tilde{f}(t,x(t),x'(t))$$ is studied. Finally, we consider the existence of periodic solutions for two kinds of second-order $p$-Laplacian neutral Rayleigh equations with singularity and without singularity. Some new results on the existence of periodic solutions are obtained. It is worth noting that $c_i$ ($i=1,\cdots,n$) are no longer constants which are different from the corresponding ones of past work. PDF      Download reader