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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

34K20 Stability theory of functional-differential equations
34D20 Stability of solutions to ordinary differential equations