Positive periodic solution for higher-order p-Laplacian neutral singular Rayleigh equation with variable coefficient

where p > , φp(x) = |x|p–x for x =  and φp() = , c ∈ Cn(R,R) and c(t + T) ≡ c(t), f is a continuous function defined in R and periodic in t with f (t, ·) = f (t + T , ·) and f (t, ) = , g(t,x) = g(x) + g(t,x), where g : R × (, +∞) → R is an L-Carathéodory function, g(t, ·) = g(t + T , ·), g ∈ C((,∞);R) has a singularity at x = , e : R→ R is a continuous periodic function with e(t+T)≡ e(t) and ∫ T  e(t)dt = , T is a positive constant, and n and m are positive integers. In recent years, there are many works on periodic solutions for high-order neutral differential equations (see [–] and the references therein).Wang and Lu [] in  investigated the existence of periodic solution for the following high-order neutral functional differential equation with distributed delay: