Asymptotic theory for a class of functional differential equations with state-dependent delays (Q1916809)

The authors study the asymptotic behaviour of the scalar valued functional differential equation with state dependent delay \(\dot x(t)=f(x[g(x(t))])+ h(t)\), \(t\geq 0\), \(x(t)=\varphi(t)\), \(t\leq 0\). The delay is considered to grow linearly or faster than its argument, \(f\) being monotone increasing. It is proved that, depending on the initial condition, some curves \(t-g(t)=\text{const}\), locally attract exact solutions. A study is made for the discretization of the given equation using a forward difference scheme.