Convergence of almost orbits of semigroups


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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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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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).

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  • Almost orbit
  • Asymptotically nonexpansive mapping
  • Fixed point
  • Locally uniformly rotund set
  • Semigroup of mappings
  • Uniform convexity

Mathematics Subject Classification

  • 41A65
  • 47H10
  • 47H20