*(English)*Zbl 1112.34054

Solutions to integro-differential equations with time-varying delay are constructed via Banach’s fixed-point theorem, using an exponentially weighted norm. The method also gives results on stability of the zero solution (the contracting operator maps functions ${\Phi}$ with $\parallel {\Phi}\parallel <\delta $ to functions $\psi $ with $\parallel \psi \parallel <\epsilon $).

Results on asymptotic stability are obtained under conditions which ensure that the contracting operator induces a self-map on a space of functions converging to zero. It is shown that the constant 2 appearing in the stability conditions can, in general, not be improved (some of the arguments are elaborated very much). The paper contains a list of about 6 interesting concrete examples.