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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 $\Vert \Phi\Vert < \delta$ to functions $ \psi$ with $\Vert \psi\Vert < \varepsilon$). 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.

34K20Stability theory of functional-differential equations
34K05General theory of functional-differential equations
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47N20Applications of operator theory to differential and integral equations
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