On the structure of fixed-point sets of asymptotically regular semigroups

Wiśnicki, Andrzej

doi:10.1016/j.jmaa.2012.03.036

Cited by 6 publications

(4 citation statements)

References 28 publications

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“…Asymptotically regular mappings were studied in many papers, in different contexts, for instance [3,8,13,16,[19][20][21][22][23]41,48].…”

Section: Example 23 the Mappingmentioning

confidence: 99%

Remarks on asymptotic regularity and fixed points

Górnicki

2019

J. Fixed Point Theory Appl.

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Asymptotic regularity allows to provide simple proofs of Banach's theorem and Kannan's theorem. Using asymptotic regularity and Kannan's type conditions we generalize these results, in particular, the Banach contraction principle (see Theorem 2.6 and Corollary 2.10). Further, we discuss the analogous results for monotone mappings on preordered metric spaces, where a preordered binary relation is weaker than a partial order. Next, we will prove a random version of the presented deterministic fixed-point theorems.

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“…Asymptotically regular mappings were studied in many papers, in different contexts, for instance [3,8,13,16,[19][20][21][22][23]41,48].…”

Section: Example 23 the Mappingmentioning

confidence: 99%

Remarks on asymptotic regularity and fixed points

Górnicki

2019

J. Fixed Point Theory Appl.

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“…Then there exists a mapping T : S → M for which T x ∈ T * x for x ∈ S and for which d(T x, T y) ≤ D(T * x, T * y) for each x, y ∈ S. Theorem 3.9 (see, e.g., [4,Prop. 1.10], [26,Lemma 2.2]). Let (X, d) be a complete bounded metric space and let f : X → X be a k-Lipschitzian mapping.…”

Section: Thenmentioning

confidence: 99%

“…(see, e.g.,[4, Prop. 1.10],[26, Lemma 2.2]). Let (X, d) be a complete bounded metric space and let f : X → X be a k-Lipschitzian mapping.…”

mentioning

confidence: 99%

Uniformly Lipschitzian group actions on hyperconvex spaces

Wiśnicki¹,

Wośko²

2016

Proc. Amer. Math. Soc.

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Abstract. Suppose that {T a : a ∈ G} is a group of uniformly LLipschitzian mappings with bounded orbits {T a x : a ∈ G} acting on a hyperconvex metric space M . We show that if L < √ 2, then the set of common fixed points Fix G is a nonempty Hölder continuous retract of M. As a consequence, it follows that all surjective isometries acting on a bounded hyperconvex space have a common fixed point. A fixed point theorem for L-Lipschitzian involutions and some generalizations to the case of λ-hyperconvex spaces are also given.

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“…It is clear that α-Hölder nonexpansive maps (0 < α < 1) are 1-nonexpansive in the above sense. In [37, Theorem 2.3], he proved under certain conditions that d(T, K) ≤ h. As another interesting work, A. Wiśnicki [50] had shown under some geometric conditions (cf. [50,Theorem 4.2]) that if K ∈ B c (X) is a weakly compact set, then the fixed point set of asymptotically regular semigroups acting on K are Hölder-Lipschitz retracts.…”

Section: Introductionmentioning

confidence: 99%

Hölder-contractive mappings, nonlinear extension problem and fixed point free results

Barroso¹

2022

Preprint

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For a bounded closed convex set K, in this note, we study the FPP for α-Hölder nonexpansive maps, i.e. mappings T : K → K for which T x − T y ≤ x − y α for all x, y ∈ K, α ∈ (0, 1). First, we note that only finite-dimensional spaces have the Hölder-FPP. Moreover, the unit ball B X of any infinite-dimensional space fails the FPP for Hölder maps with d(T, B X ) > 0, where d(T, K) denotes the minimal displacement of T . We further show that reflexivity and weak sequential continuity are sufficient conditions to capture fixed points of Hölder-Lipschitz maps with bounded orbits. Next we focus on the existence of fixed point free α-Hölder maps T : K → K with d(T, K) ≤ ϕ(α) where either ϕ(α) = 0 or ϕ(α) → 0 as α → 1. Interesting results are obtained for the spaces c, c 0 and ℓ 1 , and also for L p -spaces with p ∈ {1, ∞}. We also study the problem in spaces containing copies of c 0 and ℓ 1 . Some questions are left open.

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On the structure of fixed-point sets of asymptotically regular semigroups

Cited by 6 publications

References 28 publications

Remarks on asymptotic regularity and fixed points

Remarks on asymptotic regularity and fixed points

Uniformly Lipschitzian group actions on hyperconvex spaces

Hölder-contractive mappings, nonlinear extension problem and fixed point free results

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