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Global attractivity in differential equations with variable delays

Published online by Cambridge University Press:  17 February 2009

J. R. Graef  and
C. Qian
J. R. Graef
Affiliation:
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA.
C. Qian
Affiliation:
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA.
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Abstract

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Consider the forced differential equation with variable delay

where

We establish a sufficient condition for every solution to tend to zero. We also obtain a sharper condition for every solution to tend to zero when is asymptotically constant.

Type
Research Article
Information
The ANZIAM Journal , Volume 41 , Issue 4 , April 2000 , pp. 568 - 579
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Burton, T. A. and Haddock, J. R., “On the delay-differential equations x′(t)+a(t)f(x(t−r(t))) = 0 and x″(t) + af (x(t − r(t))) = 0”, J. Math. Anal. Appl. 54 (1976) 37–48.CrossRefGoogle Scholar
[2]Cooke, K. L., “Asymptotic theory for the delay-differential equation u′(t) = −au(t − r(u(t)))”, J. Math. Anal. Appl. 19 (1967) 160–173.CrossRefGoogle Scholar
[3]Cooke, K. L., “Linear functional differential equations of asymptotically autonomous type”, J. Differential Eq. 7 (1970) 154–174.CrossRefGoogle Scholar
[4]Driver, R. D., Ordinary and Delay Differential Equations (Springer, New York, 1977).CrossRefGoogle Scholar
[5]Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics (Kluwer Academic Publishers, Dordrecht, 1992).CrossRefGoogle Scholar
[6]Graef, J. R. and Qian, C., “Asymptotic behavior of forced delay equations with periodic coefficients”, Comm. Appl. Anal. 2 (1998) 551–564.Google Scholar
[7]Györi, I., “Necessary and sufficient conditions in an asymptotically ordinary delay differential equation”, Differential Integral Eq. 6 (1993) 225–239.Google Scholar
[8]Kolmanovskii, V. B., Torelli, L. and Vermiglio, R., “Stability of some test equations with delay”, SIAM J. Math. Anal. 25 (1994) 948–961.CrossRefGoogle Scholar
[9]Ladas, G., Sficas, Y. G. and Stavroulakis, I. P., “Asymptotic behavior of solutions of retarded differential equations”, Proc. Amer. Math. Soc. 88 (1983) 247–253.CrossRefGoogle Scholar
[10]Lim, Beg-Bin, “Asymptotic bounds of solution of functional differential equation x′(t) = ax(λt) + bx(t) + f(t), 0 < λ < 1”, SIAM J. Math. Anal. 9 (1978) 915–920.CrossRefGoogle Scholar
[11]Yoneyama, T., “On the 3/2 stability theorem for one-dimensional delay-differential equations”, J. Math. Anal. Appl. 125 (1987) 161–173.CrossRefGoogle Scholar
[12]Yorke, J. A., “Asymptotic stability for one-dimensional differential-delay equations”, J. Differential Eq. 7 (1970) 189–202.CrossRefGoogle Scholar