Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1991 | 100 | 2 | 95-108
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
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.
Opis fizyczny
  • Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, U.S.A.
  • [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.
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.