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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the homotopy index for infinite-dimensional semiflows


Author: Krzysztof P. Rybakowski
Journal: Trans. Amer. Math. Soc. 269 (1982), 351-382
MSC: Primary 58F25; Secondary 34C35, 47H09, 54H20
MathSciNet review: 637695
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Abstract: In this paper we consider semiflows whose solution operator is eventually a conditional $ \alpha $-contraction. Such semiflows include solutions of retarded and neutral functional differential equations, of parabolic and certain other classes of partial differential equations. We prove existence of (nonsmooth) isolating blocks and index pairs for such semiflows, via the construction of special Lyapunov functionals. We show that index pairs enjoy all the properties needed to define the notion of a homotopy index, thus generalizing earlier results of Conley [2]. Finally, using a result of Mañé [9], we prove that, under additional smoothness assumptions on the semiflow, the homotopy index is essentially a finite-dimensional concept. This gives a formal justification of the applicability of Ważewski's Principle to infinite-dimensional problems. Several examples illustrate the theory.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0637695-7
PII: S 0002-9947(1982)0637695-7
Article copyright: © Copyright 1982 American Mathematical Society