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On stability properties for one-dimensional functional differential equations. (English) Zbl 0746.34045
The author considers the equation (1) x ' (t)=F(t,x t ), where F:[0,)×BCR (BC denotes the set of bounded, continuous functions mapping (,0] into R), F(·,0)0. If aR, ψC((-,a],R) and ta, then ψ t BC is defined by ψ t (s)=ψ(t+s), s0. For any ψC(R,R) with ψ t BC the function tF(t,ψ t ) is continuous on [0,). For given t 0 0, φBC, the function x(·)=x(·;t 0 ,φ)C((-,t 0 +ω),R) is a solution of (1) through (t 0 ,φ) on [t 0 ,t 0 +ω), ω>0, if x t 0 =φ and (1) holds on [t 0 ,t 0 +ω). It is assumed that additional conditions are satisfied for F such that x(·;t 0 ,φ) uniquely exists on [t 0 ,) for all t 0 0 and φBC. The main results of the paper give sufficient conditions for the uniform stability and for the uniform asymptotic stability of the zero solution of (1).

MSC:
34K20Stability theory of functional-differential equations