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Razumikhin-type theorems on exponential stability of impulsive infinite delay differential systems. (English) Zbl 1179.34079

The authors consider the impulsive FDE of the form

x ' (t)=f(t,x(·)),tt k ,Δ t=t k =x(t k )-x(t k - )=I k (t k ,x(t k - )),k=1,2,...,

0t 0 <t 1 <, x ' denotes the right-hand derivative, f is a continuous functional defined in the appropriate space, f(t,0)=I k (t k ,0)=0. It is supposed that the IVP has a unique solution x(t,σ,φ) which can be continued to . The initial function φ is piecewise continuous.

The zero solution is said to be weak exponentially stable if for any ε>0 and σt 0 δ>0 such that φ<δ implies α(x(t,σ,φ))<εe -λ(t-σ) for tσ and some λ>0 and a strictly increasing α: + + . If α(s)=s we obtain the exponential stability.

Two theorems on the weak exponential stability are proved. The paper ends with two illustrative examples. There are many misprints.

34K20Stability theory of functional-differential equations
34K45Functional-differential equations with impulses