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ISSN 1088-6850(e) ISSN 0002-9947(p)

On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals

Author(s): László Hatvani
Journal: Trans. Amer. Math. Soc. 354 (2002), 3555-3571.
MSC (2000): Primary 34K20
Posted: April 30, 2002
MathSciNet review: 1911511
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Abstract: Sufficient conditions are given for the asymptotic stability and uniform asymptotic stability of the zero solution of the nonautonomous FDE's whose right-hand sides can be unbounded functions of the time. The theorems are based upon Lyapunov-Krasovski ${\u{\i}}\kern.15em$ functionals whose derivatives with respect to the equations are negative semidefinite and can vanish at long intervals. The functionals and their derivatives are estimated by either , the norm of the instantaneous value of the solutions or $\Vert x_t\Vert _2$ , the -norm of the phase segment of the solutions. Examples are given to show that the conditions are sharp, and the main theorems with the two different types of estimates are independent and improve earlier results. The theorems are applied to linear and nonlinear retarded FDE's with one delay and with distributed delays.

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