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Weak almost periodicity and the strong ergodic limit theorem for contraction semigroups

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Abstract

We show that if (S(t))t≧0 is a contraction semigroup on a closed convex subset of a uniformly convex Banach space, then every bounded and “asymptotically isometric” almost-orbit of (S(t))t≧0 is weakly almost periodic in the sense of Eberlein. As one consequence, results on the existence of almost periodic solutions to the abstract Cauchy problem are obtained without the need fora priori compactness assumptions. As a further consequence, the known strong ergodic limit theorems for (almost-) orbits of nonlinear contraction semigroups turn out to be special cases of Eberlein’s classical ergodic theorem for weakly almost periodic functions.

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Authors and Affiliations

  1. Fachbereich Mathematik, Universität Essen, 43 Essen 1, FRG

    W. M. Ruess

  2. Mathematical Sciences, University of Arkansas, 72701, Fayetteville, AR, USA

    W. H. Summers

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  1. W. M. Ruess
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  2. W. H. Summers
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Ruess, W.M., Summers, W.H. Weak almost periodicity and the strong ergodic limit theorem for contraction semigroups. Israel J. Math. 64, 139–157 (1988). https://doi.org/10.1007/BF02787219

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  • DOI: https://doi.org/10.1007/BF02787219

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