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

Abstract

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 p -Laplacian differential equation as follows: ( φ p ( 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 ) 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.