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Perron-type stability theorems for neutral equations. (English) Zbl 1044.34028

This article deals with asymptotic stability results for a neutral functional-differential equation of the form \[ x'(t)= Sx(t)+ Px(t-r)+ \frac{d}{dt} Q(t,x_t)+ G(t,x_t), \tag{1} \] with an asymptotically stable linear part \(x'(t)= Sx(t)+ Px(t-r)\). As is well known, the problem of the asymptotical stability of the zero solution to (1) is reduced to the problem of finding a fixed point for some nonlinear integral operator \(A\). The author describes conditions under which the operator \(A\) satisfies conditions of the following Krasnosel’skij fixed-point principle: if \(A= A_1+ A_2\), \(A_1\) is a contraction of a closed convex nonempty subset \(M\) in a Banach space \(S\), \(A_2\) is continuous and \(A_2M\) compact, and \(A_1M+ M_2M\subseteq M\), then there exists a fixed point of \(A\). The use of this theorem allows the author to consider the case when the functions \(Q\) and \(G\) are unbounded with respect to \(t\) and \(G\) is not differentiable or Lipschitzian.

MSC:

34K20 Stability theory of functional-differential equations
34K40 Neutral functional-differential equations
47H10 Fixed-point theorems
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References:

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