Existence of periodic solutions for a Liénard type p-Laplacian differential equation with a deviating argument

By means of Mawhin’s continuation theorem, a class of p-Laplacian type differential equation with a deviating argument of the form (φp(x′(t)))′+f(x(t))x′(t)+β(t)g(t,x(t−τ(t,|x|∞)))=e(t)(φp(x′(t)))′+f(x(t))x′(t)+β(t)g(t,x(t−τ(t,|x|∞)))=e(t) is studied. A new result, related to β(t)β(t) and the deviating argument τ(t,|x|∞)τ(t,|x|∞), is obtained. It is significant that the growth degree with respect to the variable xx in g(t,x)g(t,x) is allowed to be greater than p−1p−1, which could be achieved infrequently in previous papers.