Sufficient conditions for the existence of periodic solutions to some second order differential equations with a deviating argument

By means of Mawhin's continuation theorem, we study some second order differential equations with a deviating argument: x″(t)=f(t,x(t),x(t−τ(t)),x′(t))+e(t). Some new results on the existence of periodic solutions are obtained. The interest is that we allow the degree with respect to the variables x0,x1,x2 of f(t,x0,x1,x2) to be greater than 1; and also the result (Theorem 3.2) is related to the deviating argument τ(t). Meanwhile, we give an example to demonstrate our result.