Existence and Lyapunov stability of periodic solutions for generalized higher-order neutral differential equations (Q610843)

The higher-order neutral functional differential equation \[ [\varphi_p(x(t)-cx(t-\sigma)^{(l)}]^{(n- l)} = F(t, x(t), x'(t), \dots, x^{(l-1)}(t)) \] is considered, where \(\varphi_p(s) = |s|^{p-2}s\) with a constant \(p\geq 2\), \(F\) is continuous and \(T\)-periodic in \(t\) with \(F(t, a, 0, \dots, 0)\not\equiv 0\) for all real \(a\), and \(c\), \(\sigma\) are constants. The problem of periodic solutions has been studied in the literature when \(|c| = 1\). This paper deals with the case of \(|c|\not=1\). By first transforming the equation into a first-order system and then applying Mawhin's continuation theory, sufficient conditions are obtained for the existence of periodic solutions. Moreover, the Lyapunov stability of the periodic solutions is investigated.