New results of periodic solutions for a Rayleigh equation with a deviating argument (Q834570)

The author investigates the existence and uniqueness of a \(T\)-periodic solution of the equation \[ x''(t) + f(t, x'(t))+ g(t, x(t-\tau(t))) = e(t), \] where the functions \(f\), \(g\), \(\tau\) and \(e\) are continuous and \(T\)-periodic in \(t\) with \(f(t, 0) \equiv 0\). It is proved that the equation has a unique \(T\)-periodic solution under the following assumptions: (i) \(\forall t, u, v \in \mathbb{R}\), \((g(t, u)-g(t, v))(u-v) >0\); (ii) \(\forall t, u, v\in \mathbb{R}\), \(|f(t, u)-f(t, v)| \leq C_1|u-v|\); (iii) \(\forall t, u, v\in \mathbb{R}\), \(|g(t, u)-g(t, v)| \leq C_2|u-v|\); (iv) \((2C_1+C_2T)T<4\pi\); (v) \(\forall t, x \in\mathbb{R}\) with \(|x|\geq D_1\), \(x(g(t, x) - e(t))>0\). It is also shown that the inequality in (i) or in (v) or in both can be reversed without affecting the existence and uniqueness of the \(T\)-periodic solution.