Fixed point properties for a degenerate Lorentz–Marcinkiewicz space

We construct an equivalent renorming of ℓ1 , which turns out to produce a degenerate ℓ1 -analog Lorentz–Marcinkiewicz space ℓ;1 , where the weight sequence = (n)n2N = (2; 1; 1; 1;) is a decreasing positive sequence inℓ1nc0 , rather than in c0nℓ1 (the usual Lorentz situation). Then we obtain its isometrically isomorphic predual ℓ0;1 anddual ℓ;1, corresponding degenerate c0 -analog and ℓ1-analog Lorentz–Marcinkiewicz spaces, respectively. We provethat both spaces ℓ;1 and ℓ0;1 enjoy the weak fixed point property (w-fpp) for nonexpansive mappings yet they failto have the fixed point property (fpp) for nonexpansive mappings since they contain an asymptotically isometric copyof ℓ1 and c0 , respectively. In fact, we prove for both spaces that there exist nonempty, closed, bounded, and convexsubsets with invariant fixed point-free affine, nonexpansive mappings on them and so they fail to have fpp for affinenonexpansive mappings. Also, we show that any nonreflexive subspace of l0;1 contains an isomorphic copy of c0 and sofails fpp for strongly asymptotically nonexpansive maps. Finally, we get a Goebel and Kuczumow analogy by provingthat there exists an infinite dimensional subspace of ℓ;1 with fpp for affine nonexpansive mappings.

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[1] Alspach DE. A fixed point free nonexpansive map. Proceedings of the American Mathematical Society 1981; 82 (3): 423-424.
[2] Astashkin SV, Sukochev FA. Banach-Saks property in Marcinkiewicz spaces. Journal of Mathematical Analysis and Applications 2007; 336 (2): 1231-1258.
[3] Beauzamy B. Introduction to Banach Spaces and Their Geometry. Vol. 11. Amsterdam, the Netherlands: Elsevier Science Pub. Co., 1982.
[4] Browder FE. Fixed-point theorems for noncompact mappings in Hilbert space. Proceedings of the National Academy of Sciences of the United States of America 1965; 53 (6): 1272-1276.
[5] Browder FE. Fixed-point theorems for noncompact mappings in Hilbert space. Proceedings of the National Academy of Sciences of the United States of America 1965; 54 (4): 1041-1044.
[6] Carothers NL, Dilworth SJ, Lennard CJ. On a localization of the UKK property and the fixed point property in Lw;1 . Lecture Notes in Pure and Applied Mathematics 1996; 175: 111-124.
[7] Carothers NL, Dilworth SJ, Lennard CJ, Trautman DA. A fixed point property for the Lorentz space Lp;1() . Indiana University Mathematics Journal 1991; 40 (1): 345-352.
[8] Dowling PN, Lennard CJ. Every nonreflexive subspace of L1[0; 1] fails the fixed point property. Proceedings of the American Mathematical Society 1997; 125 (2): 443-446.
[9] Dowling PN, Lennard CJ, Turett B. Renormings of l1 and c0 and Fixed Point Properties. Handbook of Metric Fixed Point Theory. Berlin, Germany: Springer, 2001.
[10] Everest T. Fixed points of nonexpansive maps on closed, bounded, convex sets in l1 . PhD, University of Pittsburgh, Pittsburgh, PA, USA, 2013.
[11] Gamboa de Buen B, Nuñez-Medina F. A generalization of a renorming theorem by Lin and a new nonreflexive space with the fixed point property which is nonisomorphic to l1 . Journal of Mathematical Analysis and Applications 2013; 405 (1): 57-70.
[12] Goebel K, Kirk WA. Topics in Metric Fixed Point Theory. Cambridge, UK: Cambridge University Press, 1990.
[13] Goebel K, Kuczumow T. Irregular convex sets with fixed-point property for nonexpansive mappings. Colloquium Mathematicum 1979; 40 (2): 259-264.
[14] Göhde D. Zum Prinzip der kontraktiven Abbildung. Mathematische Nachrichten 1965; 30 (3-4): 251-258 (in German).
[15] Khamsi MA, Kirk WA. An Introduction to Metric Spaces and Fixed Point Theory. New York, NY, USA: John Wiley & Sons, 2011.
[16] Kirk WA. A fixed point theorem for mappings which do not increase distances. American Mathematical Monthly 1965; 72 (9): 1004-1006.
[17] Lennard CJ, Nezir V. Semi-strongly asymptotically non-expansive mappings and their applications on fixed point theory. Hacettepe Journal of Mathematics and Statistics 2017; 46 (4): 613-620.
[18] Lin PK. Unconditional bases and fixed points of nonexpansive mappings. Pacific Journal of Mathematics 1985; 116 (1): 69-76.
[19] Lin PK. There is an equivalent norm on ℓ1 that has the fixed point property. Nonlinear Analysis: Theory, Methods & Applications 2008; 68 (8): 2303-2308.
[20] Lin PK. Renorming of ℓ1 and the fixed point property. Journal of Mathematical Analysis and Applications 2010; 362 (2): 534-541.
[21] Lindenstrauss J, Tzafriri L. Classical Banach Spaces I. Berlin, Germany: Springer-Verlag, 1977.
[22] Lorentz GG. Some new functional spaces. Annals of Mathematics 1950; 51: 37-55.
[23] Maurey B. Points fixes des contractions de certains faiblement compacts de L1 . Seminaire d’Analyse Fonctionelle, 1980-1981, Centre de Mathématiques, École Polytech., Palaiseau, Exp. No. VIII, 19 pp., 1981 (in French).
[24] Nezir V. Fixed point properties for c0 -like spaces. PhD, University of Pittsburgh, Pittsburgh, PA, USA, 2012.
[25] Nezir V, Mustafa N. c0 can be renormed to have the fixed point property for affine nonexpansive mappings. Filomat 2018; 32 (16): 5645-5663.
[26] Rakov SA. Banach-Saks property of a Banach space. Mathematical notes of the Academy of Sciences of the USSR 1979; 26 (6): 909-916.
[27] Rakov SA. Banach-Saks exponent of certain Banach spaces of sequences. Mathematical Notes of the Academy of Sciences of the USSR 1982; 32 (5): 791-797.
[28] Schauder J. Der Fixpunktsatz in Funktionalraümen. Studia Mathematica 1930; 2 (1): 171-180 (in German).
[29] Soardi PM. Existence of fixed points of nonexpansive mappings in certain Banach lattices. Proceedings of the American Mathematical Society 1979; 73: 25-29.

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

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