Renormings of \(c_0\) and the minimal displacement problem

Łukasz Piasecki

Abstract


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 \(\| x - Tx \| > 1 − 1/k\) for any \(x \in C\).

Full Text:

PDF

References


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.




DOI: http://dx.doi.org/10.17951/a.2014.68.2.85
Date of publication: 2015-05-23 16:29:45
Date of submission: 2015-05-09 13:53:37


Statistics


Total abstract view - 537
Downloads (from 2020-06-17) - PDF - 364

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2015 Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica