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Zubov's condition revisited

Published online by Cambridge University Press:  20 January 2009

Ronald A. Knight
Ronald A. Knight
Affiliation:
Northeast Missouri State University
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Zubov states an elegant necessary and sufficient limit set condition for positive orbital stability of compact invariant sets in his book “Metody A. M. Lyapunova i ih Primenenie” [11]. Stated in terms of our terminology of L– for the negative limit set, Zubov's proposition is as follows: A necessary and sufficient condition for positive stability of a compact invariant set M isL–(X\M)∩M=Ø. Unfortunately, Zubov's condition L–(X\M)∩M=Ø has subsequently been shown to be necessary but not sufficient (see [9]). Bass and Ura devote considerable effort in [2] and [9[ to correcting Zubov's proposition and Desbrow obtains additional results principally concerning unstable sets in [6] and [7]. Ura gives his classical corrected prolongational version of Zubov's assertion on locally compact phase spaces in [9] and extends it to any closed invariant set with compact boundary on such spaces in [10].

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 26 , Issue 2 , June 1983 , pp. 253 - 257
Copyright
Copyright © Edinburgh Mathematical Society 1983

References

REFERENCES

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