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1
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Hyperconvexity of ℝ-trees

100%
Kirk W. A.
Fundamenta Mathematicae
|
1998
|
tom 156
|
nr 1
67-72
EN
It is shown that for a metric space (M,d) the following are equivalent: (i) M is a complete ℝ-tree; (ii) M is hyperconvex and has unique metric segments.
2
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Weak uniform normal structure and iterative fixed points of nonexpansive mappings

88%
Domínguez Benavides T., López Acedo G., Xu H.-K.
Colloquium Mathematicum
|
1995
|
tom 68
|
nr 1
17-23
EN
This paper is concerned with weak uniform normal structure and iterative fixed points of nonexpansive mappings. Precisely, in Section 1, we show that the geometrical coefficient β(X) for a Banach space X recently introduced by Jimenez-Melado [8] is exactly the weakly convergent sequence coefficient WCS(X) introduced by Bynum [1] in 1980. We then show in Section 2 that all kinds of James' quasi-reflexive spaces have weak uniform normal structure. Finally, in Section 3, we show that in a space X with weak uniform normal structure, every nonexpansive self-mapping defined on a weakly sequentially compact convex subset of X admits an iterative fixed point.
3
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Strong convergence of implicit iteration processes for nonexpansive semigroups in Banach spaces

75%
Kozlowski W. M.
Commentationes Mathematicae
|
2014
|
tom 54
|
nr 2
EN
Let \(C\) be a convex compact subset of a uniformly convex Banach space. Let \(\{T_t\}_{t \geq0}\) be a strongly-continuous nonexpansive semigroup on \(C\). Consider the iterative process defined by the sequence of equations $$x_{k+1} =c_k T_{t_{k+1}}(x_{k+1})+(1-c_k)x_k.$$ We prove that, under certain conditions on \(\{c_k\}\) and \(\{t_k\}\), the sequence \(\{x_k\}_{n=1}^\infty\) converges strongly to a common fixed point of the semigroup \(\{T_t\}_{t \geq0}\). There are known results on convergence of such iterative processes for nonexpansive semigroups in Hilbert spaces and Banach spaces with the Opial property, and also weak convergence results in Banach spaces that are simultaneously uniformly convex and uniformly smooth. In this paper, we do not assume the Opial property or uniform smoothness of the norm.
4
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On the Existence of Common Fixed Points for Semigroups of Nonlinear Mappings in Modular Function Spaces

63%
Kozlowski W. M.
Commentationes Mathematicae
|
2011
|
tom 51
|
nr 1
EN
Let \(C\) be a \(\rho\)-bounded, \(\rho\)-closed, convex subset of a modular function space \(L_\rho\). We investigate the existence of common fixed points for semigroups of nonlinear mappings \(T_t\colon C\to C\), i.e. a family such that \(T_0(x) = x\), \(T_{s+t} = T_s (T_t (x))\), where each \(T_t\) is either \(\rho\)-contraction or \(\rho\)-nonexpansive. We also briefly discuss existence of such semigroups and touch upon applications to differential equations.
5
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A new convexity property that implies a fixed point property for $L_{1}$

63%
Lennard C.
Studia Mathematica
|
1991
|
tom 100
|
nr 2
95-108
EN
In this paper we prove a new convexity property for L₁ that resembles uniform convexity. We then develop a general theory that leads from the convexity property through normal structure to a fixed point property, via a theorem of Kirk. Applying this theory to L₁, we get the following type of normal structure: any convex subset of L₁ of positive diameter that is compact for the topology of convergence locally in measure, must have a radius that is smaller than its diameter. Indeed, a stronger result holds. The Chebyshev centre of any norm bounded, convergence locally in measure compact subset of L₁ must be norm compact. Immediately from normal structure, we get a new proof of a fixed point theorem for L₁ due to Lami Dozo and Turpin.
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