An asymptotic stability and a uniform asymptotic stability for functional-differential equations

Ko, Younhee

doi:10.1090/S0002-9939-1993-1169036-6

An asymptotic stability and a uniform asymptotic stability for functional-differential equations
HTML articles powered by AMS MathViewer

by Younhee Ko

Proc. Amer. Math. Soc. 119 (1993), 535-545

DOI: https://doi.org/10.1090/S0002-9939-1993-1169036-6

PDF | Request permission

Abstract:

We consider a system of functional differential equation ${x’}(t) = F(t,{x_t})$ and obtain conditions on a Liapunov functional to ensure the asymptotic stability and the uniform asymptotic stability of the zero solution.

References

T. A. Burton, Uniform asymptotic stability in functional differential equations, Proc. Amer. Math. Soc. 68 (1978), no. 2, 195–199. MR 481371, DOI 10.1090/S0002-9939-1978-0481371-5
Theodore Allen Burton, Volterra integral and differential equations, Mathematics in Science and Engineering, vol. 167, Academic Press, Inc., Orlando, FL, 1983. MR 715428
T. A. Burton, Stability and periodic solutions of ordinary and functional-differential equations, Mathematics in Science and Engineering, vol. 178, Academic Press, Inc., Orlando, FL, 1985. MR 837654
Theodore Burton and László Hatvani, Stability theorems for nonautonomous functional-differential equations by Liapunov functionals, Tohoku Math. J. (2) 41 (1989), no. 1, 65–104. MR 985304, DOI 10.2748/tmj/1178227868
Rodney D. Driver, Existence and stability of solutions of a delay-differential system, Arch. Rational Mech. Anal. 10 (1962), 401–426. MR 141863, DOI 10.1007/BF00281203
Jack Hale, Theory of functional differential equations, 2nd ed., Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977. MR 0508721, DOI 10.1007/978-1-4612-9892-2
V. Lakshmikantham and Xinzhi Liu, On asymptotic stability for nonautonomous differential systems, Nonlinear Anal. 13 (1989), no. 10, 1181–1189. MR 1020726, DOI 10.1016/0362-546X(89)90006-0
H. L. Royden, Real analysis, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1963. MR 0151555
L. Z. Wen, On the uniform asymptotic stability in functional-differential equations, Proc. Amer. Math. Soc. 85 (1982), no. 4, 533–538. MR 660599, DOI 10.1090/S0002-9939-1982-0660599-6
Taro Yoshizawa, Stability theory by Liapunov’s second method, Publications of the Mathematical Society of Japan, vol. 9, Mathematical Society of Japan, Tokyo, 1966. MR 0208086