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Slowly oscillating solutions of Cauchy problems with countable spectrum

Published online by Cambridge University Press:  11 July 2007

W. Arendt  and
C. J. K. Batty
W. Arendt
Affiliation:
Abteilung Mathematik V, Universität Ulm, 89069 Ulm, Germany (arendt@mathematik.uni-ulm.de)
C. J. K. Batty
Affiliation:
St. John's College, Oxford OX1 3JP, UK (charles.batty@sjc.ox.ac.uk)
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Abstract

Let u be a bounded slowly oscillating mild solution of an inhomogeneous Cauchy problem, u̇ (t) = Au (t) + f (t), on R or R+, where A is a closed operator such that σap (A) ∩iR is countable, and the Carleman or Laplace transform of f has a continuous extension to an open subset of the imaginary axis with countable complement. It is shown that u is (asymptotically) almost periodic if u is totally ergodic (or if X does not contain c0 in the case of a problem on R). Similar results hold for second-order Cauchy problems and Volterra equations.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 130 , Issue 3 , June 2000 , pp. 471 - 484
Copyright
Copyright © Royal Society of Edinburgh 2000

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