Periodic solution; High order; pp-Laplacian equation; Coincidence degree

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). Turn MathJax on 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.