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Convergence of almost orbits of semigroups

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Abstract

In this paper we establish the following general theorem. Let S be a semitopological reversible semigroup and let Y be a linear subspace of \({\ell }^\infty (S)\) which contains the constants and is left translation invariant. Suppose that Y has a left invariant mean. Let \((E, \Vert \cdot \Vert _E)\) be a uniformly convex Banach space and let C be a nonempty, bounded, closed and convex subset of E. Assume that C has nonempty interior, is locally uniformly rotund and \(\mathcal {T}=\left\{ T\left( s\right) \right\} _{s \in S}\) is an asymptotically nonexpansive semigroup which acts on C. Assume also that \(\mathcal {T}\) has a unique fixed point \(\tilde{x}\) and that, in addition, this point \(\tilde{x}\) lies on the boundary \(\partial C\) of C. Then \(\{u(s) \}_{s \in S}\) converges strongly to \(\tilde{x}\) for each almost orbit \(u \in AO_Y (\mathcal {T})\).

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Acknowledgements

The third author was partially supported by the Israel Science Foundation (Grant Nos. 389/12 and 820/17), the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund. All the authors are grateful to an anonymous referee for his/her useful comments and helpful suggestions.

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Authors and Affiliations

  1. Instytut Matematyki, UMCS, 20-031, Lublin, Poland

    Wiesława Kaczor & Tadeusz Kuczumow

  2. Department of Mathematics, The Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Simeon Reich

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  1. Wiesława Kaczor
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  2. Tadeusz Kuczumow
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  3. Simeon Reich
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Correspondence to Simeon Reich.

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Kaczor, W., Kuczumow, T. & Reich, S. Convergence of almost orbits of semigroups. Anal.Math.Phys. 11, 51 (2021). https://doi.org/10.1007/s13324-021-00495-3

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