Continuability of solutions of the generalized Liénard system with time delay
The author considers the system of differential equations with time delay \[ (1)\quad x'(t)=y(t)-F(x(t)),\quad y'(t)=-g(t,x(t-r(t)), \] where \(F: {\mathbb{R}}\to {\mathbb{R}}\), g: [0,\(\infty)\in {\mathbb{R}}\to {\mathbb{R}}\), r: [0,\(\infty)\to [0,\infty)\) are continuous functions. It is proved that all solutions of (1) are continuable in the future if and only if \((A^+)\) and \((A^-)\) hold, where \((A^{\pm})\) is defined by \((A_ 1^{\pm})\) or \((A_ 2^{\pm})\), \((A_ 1^{\pm})\) means that \(\int^{\infty}_{0}(1+F_{\mp}(x))^{-1}dx=\pm \infty\) and \((A_ 2^{\pm})\) means that there exists a sequence \(\{x_ n\}\) such that \(x_ n\to \pm \infty\) and \(F(x_ n)\to \pm \infty\) as \(n\to \infty\); \(F_+(F_-)\) denotes a positive (negative) part of F.
scientific article; zbMATH DE number 3905036