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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 \(x \in C\).
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2014
online
2015-05-23
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Bibliografia
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- 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.
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Bibliografia
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bwmeta1.element.ojs-doi-10_17951_a_2014_68_2_85