## Stability, fixed points and inverses of delays.(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 $$\| \Phi\| < \delta$$ to functions $$\psi$$ with $$\| \psi\| < \varepsilon$$).
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.

### MSC:

 34K20 Stability theory of functional-differential equations 34K05 General theory of functional-differential equations 47H10 Fixed-point theorems 47N20 Applications of operator theory to differential and integral equations
Full Text: