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)${\left({\phi }_p\left(x^{\prime }\left(t\right)\right)\right)}^{\prime }+f\left(x\left(t\right)\right)x^{\prime }\left(t\right)+\beta \left(t\right)g(t,x(t-\tau (t,{\left|x\right|}_{\infty })))=e\left(t\right)$

is studied. A new result, related to β(t)$\beta \left(t\right)$ and the deviating argument τ(t,_|x|∞)$\tau (t,{\left|x\right|}_{\infty })$, is obtained. It is significant that the growth degree with respect to the variable x$x$ in g(t,x)$g(t,x)$ is allowed to be greater than p−1$p-1$, which could be achieved infrequently in previous papers.