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On trajectory convergence of dissipative flows in Banach spaces

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Abstract

LetX be a convex compact in a real Banach spaceE. An actionU(t) (t≥0) of the semigroupℝ + onX is called dissipative if allU(t) are nonexpanding: ∥U(t)x 1−U(t)x 2∥≤∥x 1−x 2∥. Let the spaceE be strongly normed. We prove that all trajectoriest→U(t)x of the dissipative flowU(t) are converging fort→∞ if there are no two-dimensional Euclidean subspaces in the spaceE. In every two dimensional non-Euclidean spaceE (not necessarily strongly normed) all trajectories of the flow under consideration are converging.

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References

  1. Ljubich, Y.I.: Dissipative action and almost periodic representation of Abelian semigroups, Ukr. Mat. Journ. 40 (1) (1988), 70–74 (in Russian).

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  2. Ljubich, Y.I.: Introduction to the theory of Banach representations of groups, Birkhauser-Verlag, Basel, 1988.

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  1. Lenin Prospect 76, Apt. 12, 310164, Kharkov, USSR

    Yu. I. Ljubich

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  1. Yu. I. Ljubich
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Ljubich, Y.I. On trajectory convergence of dissipative flows in Banach spaces. Integr equ oper theory 13, 138–144 (1990). https://doi.org/10.1007/BF01195297

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  • DOI: https://doi.org/10.1007/BF01195297

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