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DOI: 10.23952/cot.2026.39
Received May 29, 2025; Accepted September 5, 2025; Published online May 2, 2026
Abstract. In order to prove the existence of periodic solutions of dissipative systems in Banach spaces in the absence of uniqueness, a general notion of a Hausdorff measure of noncompactness is introduced in quasi-uniform spaces. A corresponding class of “condensifying” maps is defined which is shown to be equivalent to maps which have a “uniform” (pre)compact attractor. The results are used to prove that each dissipative and condensing on bounded subsets map has a compact attractor. In particular, the corresponding system has a periodic solution. Most results apply to single- and multivalued maps, even to monotone maps of power sets or not necessarily monotone sequences of sets, and also the connectedness of certain attractors is discussed.
How to Cite this Article:
M. Väth, Periodic solutions of dissipative systems and the Hausdorff measure of noncompactness in quasi-uniform spaces, Commun. Optim. Theory 2026 (2026) 39.