scholarly journals Separable Lindenstrauss spaces whose duals lack the weak$^*$ fixed point property for nonexpansive mappings

2017 ◽  
Vol 238 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Emanuele Casini ◽  
Enrico Miglierina ◽  
Łukasz Piasecki
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
Weak Fixed Point Property

scholarly journals The generalized Day norm. Part II. Applications

2017 ◽  
Vol 71 (2) ◽  
pp. 51
Author(s):  
Monika Budzyńska ◽  
Aleksandra Grzesik ◽  
Mariola Kot
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
Empty Interior ◽  
Weak Fixed Point Property ◽  
Complete Set ◽  
Alpha Beta

In this paper we prove that for each \(1< p, \tilde{p} < \infty\), the Banach space \((l^{\tilde{p}}, \left\|\cdot\right\|_{\tilde{p}})\) can be equivalently renormed in such a way that  the Banach space \((l^{\tilde{p}},\left\|\cdot\right\|_{L,\alpha,\beta,p,\tilde{p}})\) is LUR and has a diametrically complete set with empty interior. This result extends the Maluta theorem about existence of such a set in \(l^2\) with the Day norm. We also show that the Banach space \((l^{\tilde{p}},\left\|\cdot\right\|_{L,\alpha,\beta,p,\tilde{p}})\) has the weak fixed point property for nonexpansive mappings.

scholarly journals THE FIXED POINT PROPERTY IN DIRECT SUMS AND MODULUS

2013 ◽  
Vol 89 (1) ◽  
pp. 79-91 ◽  
Author(s):  
ANDRZEJ WIŚNICKI
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
Direct Sums ◽  
Definitive Answer ◽  
Direct Sum ◽  
Asymptotically Nonexpansive ◽  
Weak Fixed Point Property ◽  
Monotone Norm

AbstractWe show that the direct sum $\mathop{({X}_{1} \oplus \cdots \oplus {X}_{r} )}\nolimits_{\psi } $ with a strictly monotone norm has the weak fixed point property for nonexpansive mappings whenever $M({X}_{i} )\gt 1$ for each $i= 1, \ldots , r$. In particular, $\mathop{({X}_{1} \oplus \cdots \oplus {X}_{r} )}\nolimits_{\psi } $ enjoys the fixed point property if Banach spaces ${X}_{i} $ are uniformly nonsquare. This combined with the earlier results gives a definitive answer for $r= 2$: a direct sum ${X}_{1} {\mathop{\oplus }\nolimits}_{\psi } {X}_{2} $ of uniformly nonsquare spaces with any monotone norm has the fixed point property. Our results are extended to asymptotically nonexpansive mappings in the intermediate sense.

scholarly journals Towards the fixed point property for superreflexive spaces

2001 ◽  
Vol 64 (3) ◽  
pp. 435-444 ◽  
Author(s):  
Andrzej Wiśnicki
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Unit Sphere ◽  
Convex Set ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
Geometrical Condition

A Banach space X is said to have property (Sm) if every metrically convex set A ⊂ X which lies on the unit sphere and has diameter not greater than one can be (weakly) separated from zero by a functional. We show that this geometrical condition is closely connected with the fixed point property for nonexpansive mappings in superreflexive spaces.

scholarly journals Remarks on the stability of the fixed point property for nonexpansive mappings

2012 ◽  
Vol 1 (4) ◽  
pp. 417-430 ◽  
Author(s):  
Krzysztof Bolibok ◽  
Kazimierz Goebel ◽  
W. A. Kirk
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
The Stability

The fixed point property for some uniformly nonoctahedral Banach spaces

1999 ◽  
Vol 59 (3) ◽  
pp. 361-367 ◽  
Author(s):  
A. Jiménez-Melado
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property

Roughly speaking, we show that a Banach space X has the fixed point property for nonexpansive mappings whenever X has the WORTH property and the unit sphere of X does not contain a triangle with sides of length larger than 2.

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