Exploring fixed point property for an l1-like set and l1 like subspace in c0 with an equivalent norm and a c0 like space Lorentz-Marchinkiewicz space lw,∞0

2016 ◽  
Author(s):  
Veysel Nezir
Keyword(s):  
Fixed Point ◽  
Equivalent Norm ◽  
Fixed Point Property ◽  
Point Property

scholarly journals c0 can be renormed to have the fixed point property for affine nonexpansive mappings

Filomat ◽  
2018 ◽  
Vol 32 (16) ◽  
pp. 5645-5663 ◽  
Author(s):  
Veysel Nezir ◽  
Nizami Mustafa
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Reflexive Banach Space ◽  
Nonexpansive Mappings ◽  
Equivalent Norm ◽  
Positive Answer ◽  
Fixed Point Property ◽  
Null Sequence ◽  
Point Property ◽  
Usual Norm

P.K. Lin gave the first example of a non-reflexive Banach space (X,||?||) with the fixed point property (FPP) for nonexpansive mappings and showed this fact for (l1,||?||1) with the equivalent norm ||?|| given by ||x|| = sup k?N 8k/1+8k ?1,n=k |xn|, for all x = (xn)n?N ? l1. We wonder (c0, ||?||1) analogue of P.K. Lin?s work and we give positive answer if functions are affine nonexpansive. In our work, for x = (?k)k ? c0, we define |||x||| := lim p?? sup ?k?N ?k (?1,j=k |?j|p/2j)1/p where ?k ?k 3, k is strictly increasing with ?k > 2, ?k ? N, then we prove that (c0,|||?|||) has the fixed point property for affine |||?|||-nonexpansive self-mappings. Next, we generalize this result and show that if ?(?) is an equivalent norm to the usual norm on c0 such that lim sup n ?(1/n ?n,m=1 xm + x) = lim sup n ?(1/n ?n,m=1 xm) + ?(x) for every weakly null sequence (xn)n and for all x ? c0, then for every ? > 0, c0 with the norm ||?||? = ?(?)+?|||?||| has the FPP for affine ||?||?-nonexpansive self-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 Brush spaces and the fixed point property

2011 ◽  
Vol 158 (8) ◽  
pp. 1085-1089 ◽  
Author(s):  
M.M. Marsh ◽  
J.R. Prajs
Keyword(s):  
Fixed Point ◽  
Fixed Point Property ◽  
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 A product space with the fixed point property

2012 ◽  
Vol 2012 (1) ◽  
Author(s):  
Helga Fetter Nathansky ◽  
Enrique Llorens-Fuster
Keyword(s):  
Fixed Point ◽  
Product Space ◽  
Fixed Point Property ◽  
Point Property
Sign in / Sign up

Export Citation Format

Share Document