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

Author:
László Hatvani

Journal:
Trans. Amer. Math. Soc. **354** (2002), 3555-3571

MSC (2000):
Primary 34K20

DOI:
https://doi.org/10.1090/S0002-9947-02-03029-5

Published electronically:
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 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 , 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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Additional Information

**László Hatvani**

Affiliation:
Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary

Email:
hatvani@math.u-szeged.hu

DOI:
https://doi.org/10.1090/S0002-9947-02-03029-5

Received by editor(s):
November 5, 2001

Published electronically:
April 30, 2002

Additional Notes:
The author was supported by the Hungarian National Foundation for Scientific Research (OTKA T/029188).

Article copyright:
© Copyright 2002
American Mathematical Society