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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 \(\| \Phi\| < \delta\) 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
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