On the existence of periodic solutions for a kind of second order neutral functional differential equation (Q2581157)

The neutral functional-differential equation \[ \frac{d^2}{dt^2}\big(u(t)-ku(t-\tau)\big)=f(u(t))\,u'(t)+\alpha(t)\,g(u(t))+\sum_{j=1}^n \beta_i(t)\,g(u(t-\nu_i(t)))+p(t) \] is considered, where \(f,g\in C(\mathbb{R};\mathbb{R})\), \(\alpha,p,\beta_i,\nu_i\), \(i=1,\dots,n\), are continuous periodic functions defined on \(\mathbb{R}\) with period \(T>0\), \(k,\tau\in \mathbb{R}\), \(| k| \neq 1\). It is proved that the equation has at last one \(T\)-periodic solution.