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Nonuniform stability for a nonautonomous differential equation with “maxima”. (English) Zbl 0788.34076

The function \(\overline x:(-\infty,a) \to R\), \(\sigma<a\), is a solution of the initial value problem for the differential equation with maxima (1) \(x'(t)=f(t, \max_{g(t) \leq s \leq t} x(s))\), (2) \(x_ \sigma=\varphi \in E\) if \(\overline x \in C[\sigma,a)\), \(\overline x_ \sigma=\varphi\) and \(\overline x\) satisfies (1) for \(t \in[\sigma,a)\). \(E\) is a linear space of functions defined on \((-\infty,0]\) with certain properties which is supplied with a seminorm. In the paper several sufficient conditions are given which guarantee the stability of the zero solution to (1) and a necessary and sufficient condition is established for the uniform stability of that solution.

MSC:

34K20 Stability theory of functional-differential equations
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