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On stability with respect to the first approximation of systems with delay linearly dependent on the time. (Russian) Zbl 0698.34062
The system $$\dot x(t)=A(t)x(t)+B(t)x(\mu t)+f(t,x(t),x(\mu t))$$ is considered, where $$\mu=$$const., $$0<\mu<1$$, A(t) and B(t) are bounded matrix functions on $$R_+$$, and f(t,u,v) is Lipschitzian in (u,v) with a sufficiently small Lipschitz constant. A stability theorem in the first approximation is proved provided that the zero solution of the system $$\dot y=A(t)y$$ is exponentially stable, and that of the system $$\dot z=A(t)z(t)+B(t)z(\mu t)$$ is asymptotically stable so that the solutions z(t) of the system go to zero in the order of a power as $$t\to \infty$$.
Reviewer: L.Hatvani

##### MSC:
 34K20 Stability theory of functional-differential equations 34D20 Stability of solutions to ordinary differential equations
##### Keywords:
asymptotic stability; exponential stability