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## Studia Mathematica

1991 | 100 | 2 | 95-108
Tytuł artykułu

### A new convexity property that implies a fixed point property for $L_{1}$

Autorzy
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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper we prove a new convexity property for L₁ that resembles uniform convexity. We then develop a general theory that leads from the convexity property through normal structure to a fixed point property, via a theorem of Kirk. Applying this theory to L₁, we get the following type of normal structure: any convex subset of L₁ of positive diameter that is compact for the topology of convergence locally in measure, must have a radius that is smaller than its diameter. Indeed, a stronger result holds. The Chebyshev centre of any norm bounded, convergence locally in measure compact subset of L₁ must be norm compact. Immediately from normal structure, we get a new proof of a fixed point theorem for L₁ due to Lami Dozo and Turpin.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
95-108
Opis fizyczny
Daty
wydano
1991
otrzymano
1990-07-31
poprawiono
1990-11-27
Twórcy
autor
• Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, U.S.A.
Bibliografia
• [Be] M. Besbes, Points fixes des contractions définies sur un convexe $L^0$-fermé de L¹, C. R. Acad. Sci. Paris Sér. I 311 (1990), 243-246.
• [B-M] M. S. Brodskiĭ and D. P. Mil'man, On the center of a convex set, Dokl. Akad. Nauk SSSR (N.S.) 59 (1948), 837-840 (in Russian).
• [Br] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041-1044.
• [C-D-L-T] N. L. Carothers, S. J. Dilworth, C. J. Lennard and D. A. Trautman, A fixed point property for the Lorentz space $L_{p,1}(μ)$, Indiana Univ. Math. J. 40 (1991). 345-352.
• [D-S] D. van Dulst and B. Sims, Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK), in: Banach Space Theory and its Applications, Proc. Bucharest 1981, Lecture Notes in Math. 991, Springer, 1983, 35-43.
• [D-V] D. van Dulst and V. de Valk, (KK)-properties, normal structure and fixed points of nonexpansive mappings in Orlicz spaces, Canad. J. Math. 38 (1986), 728-750.
• [H] R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), 743-749.
• [I-P] V. I. Istrăţescu and J. R. Partington, On nearly uniformly convex and k-uniformly convex spaces, Math. Proc. Cambridge Philos. Soc. 95 (1984), 325-327.
• [Kh] M. A. Khamsi, Note on a fixed point theorem in Banach lattices, preprint, 1990.
• [K-T] M. A. Khamsi and Ph. Turpin, Fixed points of nonexpansive mappings in Banach lattices, Proc. Amer. Math. Soc. 105 (1989), 102-110.
• [Ki₁] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006.
• [Ki₂] W. A. Kirk, An abstract fixed point theorem for nonexpansive mappings, Proc. Amer. Math. Soc. 82 (1981), 640-642.
• [K-F] A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Dover Publ., 1975.
• [L-T] E. Lami Dozo and Ph. Turpin, Nonexpansive maps in generalized Orlicz spaces, Studia Math. 86 (1987), 155-188.
• [L-M] A. T. Lau and P. F. Mah, Normal structure in dual Banach spaces associated with a locally compact group, Trans. Amer. Math. Soc. 310 (1988), 341-353.
• [L₁] C. J. Lennard, Operators and geometry of Banach spaces, Ph.D. dissertation, 1988.
• [L₂] C. J. Lennard, C₁ is uniformly Kadec-Klee, Proc. Amer. Math. Soc. 109 (1990), 71-77.
• [Pa] J. R. Partington, On nearly uniformly convex Banach spaces, Math. Proc. Cambridge Philos. Soc. 93 (1983), 127-129.
• [Pe] J. P. Penot, Fixed point theorems without convexity, in: Analyse Non Convexe (Pau 1977), Bull. Soc. France Mem. 60 (1979), 129-152.
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