Renormings of $c_0$ and the minimal displacement problem

Łukasz Piasecki

Renormings of

c_{0}

and the minimal displacement problem

Łukasz Piasecki

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 68 (2014), / Harvested from The Polish Digital Mathematics Library

Access to full text
Full (PDF)

Résumé

The aim of this paper is to show that for every Banach space $(X, ∥ \cdot ∥)$ containing asymptotically isometric copy of the space $c_{0}$ there is a bounded, closed and convex set $C \subset X$ with the Chebyshev radius $r (C) = 1$ such that for every $k \geq 1$ there exists a $k$ -contractive mapping $T : C \to C$ with $∥ x - T x ∥ > 1 - 1 / k$ for any $x \in C$ .

Publié le : 2014-01-01
EUDML-ID : urn:eudml:doc:289825

@article{bwmeta1.element.ojs-doi-10_17951_a_2014_68_2_85,
     author = {\L ukasz Piasecki},
     title = {Renormings of $c\_0$ and the minimal displacement problem},
     journal = {Annales Universitatis Mariae Curie-Sk\l odowska, sectio A -- Mathematica},
     volume = {68},
     year = {2014},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2014_68_2_85}
}

Łukasz Piasecki. Renormings of $c_0$ and the minimal displacement problem. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 68 (2014) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2014_68_2_85/

Bibliographie

Bolibok, K., The minimal displacement problem in the space $l_{1}$ , Cent. Eur. J. Math. 10 (2012), 2211–2214.

Bolibok, K., Constructions of lipschitzian mappings with non zero minimal displacement in spaces $L_{1} (0, 1)$ and $L_{2} (0, 1)$ , Ann. Univ. Mariae Curie-Skłodowska Sec. A 50 (1996), 25–31.

Dowling, P. N., Lennard C. J., Turett, B., Reflexivity and the fixed point property for nonexpansive maps, J. Math. Anal. Appl. 200 (1996), 653–662.

Dowling, P. N., Lennard, C. J., Turett, B., Some fixed point results in $l_{1}$ and $c_{0}$ , Nonlinear Anal. 39 (2000), 929–936.

Dowling, P. N., Lennard C. J., Turett , B., Asymptotically isometric copies of $c_{0}$ in Banach spaces, J. Math. Anal. Appl. 219 (1998), 377–391.

Goebel, K., On the minimal displacement of points under lipschitzian mappings, Pacific J. Math. 45 (1973), 151–163.

Goebel, K., Concise Course on Fixed Point Theorems, Yokohama Publishers, Yokohama, 2002.

Goebel, K., Kirk, W. A., Topics in metric fixed point theory, Cambridge University Press, Cambridge, 1990.

Goebel, K., Marino, G., Muglia, L., Volpe, R., The retraction constant and minimal displacement characteristic of some Banach spaces, Nonlinear Anal. 67 (2007), 735– 744.

James, R. C., Uniformly non-square Banach spaces, Ann. of Math. 80 (1964), 542– 550.

Kirk, W. A., Sims, B. (Eds.), Handbook of Metric Fixed Point Theory, Kluwer Academic Publishers, Dordrecht, 2001.

Lin, P. K., Sternfeld, Y., Convex sets with the Lipschitz fixed point property are compact, Proc. Amer. Math. Soc. 93 (1985), 633–639.

Piasecki, Ł., Retracting a ball onto a sphere in some Banach spaces, Nonlinear Anal. 74 (2011), 396–399.