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ABSTRACT. We provide a definition of an attractor to a multivalued iterated function system (IFS) modelled on previous ones existing in the literature (e.g. [Hale, J. K.: Asymptotic Behavior of Dissipative Systems. Math. Surveys Monographs 25, Amer. Math. Soc, Providence, RI, 1988]). Such an attractor expressing asymptotic behaviour of a system does not need to be invariant. Then, as a remedy there serves the uniform Hausdorff upper semicontinuity It was recently shown that condensing multifunctions possess a maximal invariant set which is
compact. The theorem ensuring the existence of attractors considered here also exploits compactness-like hypothesis slightly stronger than condensity, namely contractivity with respect to measure of noncompactness. Hence contractivity in measure and uniform Hausdorff upper semicontinuity together do guarantee existence of a compact attractor which is maximal invariant and unique. We also supply examples (e.g. unbounded attractor) and state further questions.

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She believed in nothing; only her skepticism kept her from being an atheist. -- Jean-Paul Sartre

by ATinNM on Fri Sep 28th, 2012 at 02:03:30 PM EST
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