Existence of periodic solution for a higher-order p-Laplacian differential equation with multiple deviating arguments

By applying Mawhin's continuation theorem, theory of Fourier series, Bernoulli numbers theory and some new inequalities, we study the higher-order $p$-Laplacian differential equation with multiple deviating arguments of the form \[(\varphi_{p}(x^{(m)}(t)))^{(m)}= f(x(t))x'(t)+g(t,x(t),x(t-\tau_{1}(t)),\ldots,x(t-\tau_{k}(t)))+e(t).\] Some new results on the existence of periodic solutions for the previous equation are obtained.