Periodic solutions for a kind of high-order pp-Laplacian differential equation with sign-changing coefficient ahead of the non-linear term

In this paper, by using the theory of Fourier series, Bernoulli number theory and the continuation theorem of coincidence degree theory, we study a kind of high-order pp-Laplacian differential equation as follows: (φp(y(m)(t)))(m)=f(y(t))y′(t)+h(y(t))+β(t)g(y(t−τ(t)))+e(t).(φp(y(m)(t)))(m)=f(y(t))y′(t)+h(y(t))+β(t)g(y(t−τ(t)))+e(t). Some new results on the existence of periodic solutions are obtained. The interesting thing is that the coefficient β(t)β(t) is allowed to change sign. But, the methods used to estimate a priori bounds of periodic solutions are different from the corresponding ones used in the past.