On the asymptotic behavior of the solutions of a state-dependent delay equation
The initial value problem (1) \(\dot x(t)= ax(t- b|x(t)|)\), \(t\geq 0\), \(x(t)= \phi(t)\), \(t\leq 0\), for \(a> 0\), \(b> 0\) is considered. It is shown that 1. if there exists \(T> 0\) such that the solution of (1) satisfies \(\dot x(T)= {1\over b}\), then the solution has the form \(x(t)= {1\over b}(t- T)+ x(T)\), \(t\geq T\); 2. there exist a constant \(\alpha\) and a function \(\beta(t)\) such that the solution has the form \(x(t)= {1\over b}(t+ \alpha+ \beta(t))\), \(t\geq 0\), where \(\lim_{t\to \infty} \beta(t)= 0\) and \(\lim_{t\to \infty} \dot\beta(t)= 0\) if and only if either i) \(\phi(- bx_0)\leq {1\over ab}\) and \(t- b|x(t)|\geq - bx_0\), \(t\geq 0\), or ii) \(\phi(- bx_0)> {1\over ab}\) and there exists a constant \(L\geq bx_0\) such that \(\phi(- L)= {1\over ab}\). In this case \(- L< t- b|x(t)|\leq - bx_0\), \(t\geq 0\).
scientific article; zbMATH DE number 775654