scholarly journals A note on properties that imply the fixed point property

2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
S. Dhompongsa ◽  
A. Kaewkhao
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Property ◽  
Point Property ◽  
Geometric Properties ◽  
Weak Fixed Point Property

We give relationships between some Banach-space geometric properties that guarantee the weak fixed point property. The results extend some known results of Dalby and Xu.

scholarly journals A class of spaces with weak normal structure

1994 ◽  
Vol 49 (3) ◽  
pp. 523-528 ◽  
Author(s):  
Brailey Sims
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Normal Structure ◽  
Fixed Point Property ◽  
Point Property ◽  
Weak Fixed Point Property

It has recently been shown that a Banach space enjoys the weak fixed point property if it is ε0-inquadrate for some ε0 < 2 and has WORTH; that is, if then, ║xn — x║ — ║xn + x║ → 0, for all x. We establish the stronger conclusion of weak normal structure under the substantially weaker assumption that the space has WORTH and is ‘ε0-inquadrate in every direction’ for some ε0 < 2.

scholarly journals On \(\ell_1\)-preduals distant by 1

2018 ◽  
Vol 72 (2) ◽  
pp. 41
Author(s):  
Łukasz Piasecki
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Wide Class ◽  
Fixed Point Theory ◽  
General Setting ◽  
Point Theory ◽  
Fixed Point Property ◽  
Sufficient Condition ◽  
Point Property ◽  
Weak Fixed Point Property

For every predual \(X\) of \(\ell_1\) such that the standard basis in \(\ell_1\) is weak\(^*\) convergent, we give explicit models of all Banach spaces \(Y\) for which the Banach-Mazur distance \(d(X,Y)=1\). As a by-product of our considerations, we obtain some new results in metric fixed point theory. First, we show that the space \(\ell_1\), with a predual \(X\) as above, has the stable weak\(^*\) fixed point property if and only if it has almost stable weak\(^*\) fixed point property, i.e. the dual \(Y^*\) of every Banach space \(Y\) has the weak\(^*\) fixed point property (briefly, \(\sigma(Y^*,Y)\)-FPP) whenever \(d(X,Y)=1\). Then, we construct a predual \(X\) of \(\ell_1\) for which \(\ell_1\) lacks the stable \(\sigma(\ell_1,X)\)-FPP but it has almost stable \(\sigma(\ell_1,X)\)-FPP, which in turn is a strictly stronger property than the \(\sigma(\ell_1,X)\)-FPP. Finally, in the general setting of preduals of \(\ell_1\), we give a sufficient condition for almost stable weak\(^*\) fixed point property in \(\ell_1\) and we prove that for a wide class of spaces this condition is also necessary.

scholarly journals The effect of the dual on a Banach space and the weak fixed point propery

2003 ◽  
Vol 67 (2) ◽  
pp. 177-185 ◽  
Author(s):  
Tim Dalby
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Property ◽  
Point Property ◽  
Original Space ◽  
Weak Fixed Point Property

A number of Banach space properties have been shown to imply the weak fixed point property. If the dual of a Banach space were to possess some of these properties then the original space can been shown to satisfy related conditions.

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 \(1&lt; p, \tilde{p} &lt; \infty\), 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 inc0with an Equivalent Norm

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Berta Gamboa de Buen ◽  
Fernando Núñez-Medina
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Dual Space ◽  
Equivalent Norm ◽  
Dimensional Subspace ◽  
Fixed Point Property ◽  
Point Property ◽  
Infinite Dimensional ◽  
Weak Fixed Point Property ◽  
Infinite Dimensional Subspace

We study the fixed point property (FPP) in the Banach spacec0with the equivalent norm‖⋅‖D. The spacec0with this norm has the weak fixed point property. We prove that every infinite-dimensional subspace of(c0,‖⋅‖D)contains a complemented asymptotically isometric copy ofc0, and thus does not have the FPP, but there exist nonempty closed convex and bounded subsets of(c0,‖⋅‖D)which are notω-compact and do not contain asymptotically isometricc0—summing basis sequences. Then we define a family of sequences which are asymptotically isometric to different bases equivalent to the summing basis in the space(c0,‖⋅‖D),and we give some of its properties. We also prove that the dual space of(c0,‖⋅‖D)over the reals is the Bynum spacel1∞and that every infinite-dimensional subspace ofl1∞does not have the fixed point property.

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 James quasi reflexive space has the fixed point property

1989 ◽  
Vol 39 (1) ◽  
pp. 25-30 ◽  
Author(s):  
M.A. Khamsi
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Unconditional Basis ◽  
Fixed Point Property ◽  
Point Property ◽  
Reflexive Space ◽  
Lin's Method ◽  
Lin’S Method

We prove that the classical sequence James space has the fixed point property. This gives an example of Banach space with a non-unconditional basis where the Maurey-Lin's method applies.

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.

scholarly journals On the modulus ofu-convexity of Ji Gao

2003 ◽  
Vol 2003 (1) ◽  
pp. 49-54 ◽  
Author(s):  
Eva María Mazcuñán-Navarro
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Normal Structure ◽  
Fixed Point Property ◽  
Sufficient Condition ◽  
Point Property

We consider the modulus ofu-convexity of a Banach space introduced by Ji Gao (1996) and we improve a sufficient condition for the fixed-point property (FPP) given by this author. We also give a sufficient condition for normal structure in terms of the modulus ofu-convexity.

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