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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)+d dtQ(t,x t )+G(t,x t ),(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 2 M compact, and A 1 M+M 2 MM, 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.

34K20Stability theory of functional-differential equations
34K40Neutral functional-differential equations
47H10Fixed point theorems for nonlinear operators on topological linear spaces