利用Mawhin拓展定理研究了一类带$p$-Laplace算子的非线性项前系数可变号的高阶Rayleigh型泛函微分方程$$({\phi }_p(y^{(m)}(t)){)}^{(m)}+f(y^{\mathrm{\prime }}(t))+\beta (t)g(y(t-\tau (t)))=e(t)$$周期解存在性问题,与已有工作不同的是此中允许非线性项前的系数可变号,引进有别于以往工作的一点估计的方法获得周期解先验界,
故估计先验界的方法和所获得的结果是新的.

We study a kind of high-order $p$-Laplacian Rayleigh functional differential equations as follows $$({\phi }_p(y^{(m)}(t)){)}^{(m)}+f(y^{\mathrm{\prime }}(t))+\beta (t)g(y(t-\tau (t)))=e(t).$$ Some new results on the existence of periodic solutions are obtained. Here the coefficient $\beta (t)$ is allowed to change sign, and the methods to estimate a priori bounds  of periodic solutions are different from the existing ones.