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Proceedings of the London Mathematical Society 2000 81(1):72-92; doi:10.1112/S0024611500012429
© 2000 by London Mathematical Society
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© London Mathematical Society

Asymptotic Behavior of Solutions of Differential Equations with Variable Delays

John R. Graef, Chuanxi Qian and Bo Zhang

Department of Mathematics and Statistics, Mississippi State University Mississippi State, MS 39762, USA; graef{at}math.msstate.edu
Department of Mathematics and Statistics, Mississippi State University Mississippi State, MS 39762, USA; qian{at}math.msstate.edu
Department of Mathematics and Computer Science, Fayetteville State University Fayetteville, NC 28301, USA; bzhang{at}sbel.uncfsu.edu

Received 19 February 1999. Revision received 26 July 1999.

The authors consider the system of forced differential equations with variable delays

Formula
where Bj(t) is a continuous n x n matrix on R+, F isin C(R+, Rn) and {tau} isin C(R+, R+). Using Razumikhin-type techniques and Liapunov's direct method, they establish conditions to ensure the ultimate boundedness and the global attractivity of solutions of (*), and when F(t) = 0, the asymptotic stability of the zero solution. Under those same conditions, they also show that Formula is a necessary and sufficient condition for all of the above properties to hold. 1991 Mathematics Subject Classification: 34K15, 34C10.

Key Words: forced equations • systems • variable delays • boundedness • global attractivity • asymptotic stability

Present Address: Department of Mathematics, University of Tennessee at Chattanooga, 615 McCallie Avenue Chattanooga, TN 37403, USA john-graef{at}utc.edu


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