TY  - JOUR
T1  - A nice copy of a degenerate Lorentz-Marcinkiewicz space that implies the failure of the fixed point property
AU  - Nezir, Veysel
AU  - Mustafa, Nizami
PY  - 2021
DA  - April
JF  - Turkish Journal of Science
JO  - TJOS
PB  - Ahmet Ocak AKDEMİR
WT  - DergiPark
SN  - 2587-0971
SP  - 14
EP  - 23
VL  - 6
IS  - 1
LA  - en
AB  - Introducing the notion of asymptotically isometric copies inside Banach spaces, Dowling, Lennard and Turett made easier to detect failure of the fixed point property for nonexpansive mappings. Their tool was very usefull for indicating the failure. Since then, researchers have investigated alternative tools. Recently, Nezir introduced the notion of asymptotically isometric copies of $\ell^{1\boxplus 0}$. He noticed that a renorming of $\ell^1$ turns out to be a degenerate Lorentz-Marcinkiewicz space and using its structure he introduced his notion which implies the failure of the fixed point property for nonexpansive mappings. In this study, we introduce another notion which is derived from the structure of another degenerate Lorentz-Marcinkiewicz space and we show that detecting our new tool in Banach spaces will indicate the failure of the fixed point property for nonexpansive mappings.
KW  - fixed point property
KW  - nonexpansive mappings
KW  - Lorentz-Marcinkiewicz space
KW  - degenerate Lorentz-Marcinkiewicz space
KW  - asymptotically isometric copy of $\ell^1$
CR  - Alvaro JM,  Cembranos P, Mendoza J.  Renormings of $c_0$ and the fixed point property.  J. Math. Anal. Appl. 454(2), 2017, 1106-1113.
	
Diestel J.  Sequences and series in Banach spaces.  Springer Science & Business Media, 2012.
CR  - Dowling PN, Lennard CJ, Turett B.   Reflexivity and the fixed-point property for nonexpansive maps. J. Math. Anal. Appl. 200(3), 1996, 653-662.
CR  - Dowling PN, Lennard CJ. Every nonreflexive subspace of $L_1[0, 1]$ fails the fixed point property. Proc. Amer. Math. Soc. 125,  1997,  443-446.
CR  - Dowling PN, Johnson WB, Lennard CJ, Turett B.   The optimality of James's distortion theorems.   Proc. Amer. Math. Soc.  125, 1997,  167-174.
CR  - Dowling PN, Lennard CJ, Turett B. Renormings of $\ell_1$  and $c_{0}$ and fixed point properties. In:  Handbook of Metric Fixed Point Theory, Springer, Netherlands, 2001, pp. 269-297.
CR  - Lin  PK. There is an equivalent norm on $\ell_1$ that has the fixed point property.   Nonlinear Anal.    68, 2008, 2303--2308.
CR  - Lindenstrauss J, Tzafriri L.  Classical Banach spaces I: sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 92, Springer-Verlag. 1977.
CR  - Lorentz GG.  Some new functional spaces. Ann. Math. 1950. 37-55.
CR  - Nezir V. Fixed point properties for a degenerate Lorentz-Marcinkiewicz space. Turkish Journal of Mathematics. 43(4), 2019, 1919-1939.
 
Nezir V. Asymptotically isometric copies of $\ell^{1\boxplus 0}$.   Hacet. J. Math. Stat.    49(3), 2020, 984-997.
UR  - https://dergipark.org.tr/tr/pub/tjos/issue//880573
L1  - https://dergipark.org.tr/tr/download/article-file/1578405
ER  -