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Cím:On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals.
Szerző:Hatvani, L\'aszl\'o
Forrás:Trans. Am. Math. Soc. 354, No.9, 3555-3571 (2002). [ISSN 0002-9947; ISSN 1088-6850
Nyelv:English
Absztrakt:The paper has been written by a famous scientist in the area of functional-differential equations. A nonautonomous system of functional-differential equations with delay $$ \frac{dx(t)}{dt}=F(t,x_t),\quad F(t,0)\equiv 0, \tag 1 $$ is considered, and asymptotic stability and uniform asymptotic stability of its solution $$ x(t)\equiv 0\eqno(2) $$ are studied. It is assumed that the right-hand sides of equation (1) are unbounded functions in $t$. Sufficient conditions for the asymptotic stability and uniform asymptotic stability of solution (2) to equation (1) with finite delays are formulated by the method of Lyapunov functionals. The derivatives of the functionals with respect to the equations are negative semidefinite in terms of either $|x(t)|$ or $L_2$-norm of segment $x_t$ and may depend explicitly on time $t$. The theorems are applied to linear and nonlinear retarded functional-differential equations with one delay and with distributed delays.} \RV{Alexander Olegovich Ignatyev (Donetsk) 
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