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Hyperconvexity and nonexpansive multifunctions


Author: Robert Sine
Journal: Trans. Amer. Math. Soc. 315 (1989), 755-767
MSC: Primary 54C65; Secondary 47H09, 47H10, 54G05
DOI: https://doi.org/10.1090/S0002-9947-1989-0954603-6
MathSciNet review: 954603
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Abstract: It is shown that a ball intersection valued nonexpansive multifunction on a hyperconvex space admits a nonexpansive point valued selection. This implies fixed point theorems for such multifunctions and to certain point valued nonexpansive maps. The result is used to study best approximation and to show the space of all nonexpansive maps of a bounded hyperconvex space is hyperconvex.


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  • [1] N. Aronszajn and P. Panitchpakdi, Extensions of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405-439. MR 0084762 (18:917c)
  • [2] J.-B. Baillon, Nonexpansive mapping and hyperconvex spaces, Fixed Point Theory and Its Applications, (R. F. Brown, ed.), Contemp. Math., vol. 72, Amer. Math. Soc., Providence, R.I., 1988, pp. 11-19. MR 956475 (89k:54068)
  • [3] B. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Acad., Bucuresti, 1976.
  • [4] Kim Border, Fixed point theorems with applications to economics and game theory, Cambridge Univ. Press, 1985. MR 790845 (86j:90002)
  • [5] A. Cellina and J.-P. Aubin, Differential inclusions, Springer-Verlag, 1984. MR 755330 (85j:49010)
  • [6] R. B. Holmes, A course on optimization and best approximation, Lecture Notes in Math., vol. 257, 1972. MR 0420367 (54:8381)
  • [7] E. Jawhari, D. Misane, and M. Pouzet, Retracts: graphs and ordered sets from the metric point of view, Contemp. Math., vol. 56, Amer. Math. Soc., Providence, R.I., 1986. MR 856237 (88i:54022)
  • [8] W. A. Kirk, Fixed point theory for nonexpansive mappings, II, Contemp. Math., vol. 18, Amer. Math. Soc., Providence, R.I., 1983, pp. 121-140. MR 728596 (85a:47062)
  • [9] H. E. Lacey, The isometric theory of classical Banach spaces, Springer-Verlag, 1974. MR 0493279 (58:12308)
  • [10] M. Lin and R. Sine, On the fixed point set of nonexpansive order preserving maps (to appear). MR 1033433 (91a:47074)
  • [11] W. O. Ray and R. C. Sine, Nonexpansive mappings with precompact orbits, Fixed Point Theory, Lecture Notes in Math., vol. 886, Springer, 1981. MR 643018 (84h:47058)
  • [12] R. C. Sine, On nonlinear contractions in sup norm spaces, Nonlinear Analysis 3 (1979), 885-890. MR 548959 (80i:47082)
  • [13] -, Hyperconvexity and approximate fixed points (to appear). MR 999336 (90g:54041)
  • [14] P. Soardi, Existence of fixed points of nonexpansive mappings in certain Banach lattices, Proc. Amer. Math. Soc. 73 (1979), 25-29. MR 512051 (80c:47051)
  • [1] M. A. Khamsi, On the fixed point property in metric spaces (preprint). MR 1304040 (95m:47106b)
  • [2] W. A. Kirk, Nonexpansive mappings in product spaces, set-valued mappings and $ k$-uniform rotundity, Proc. Sympos. Pure Math., vol. 45, Part 2, Amer. Math. Soc., Providence, R.I., 1986, pp. 51-64. MR 843594 (87i:47068)
  • [3] T. C. Lim, A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space, Bull. Amer. Math. Soc. 80 (1974), 1123-1126. MR 0394333 (52:15136)
  • [4] S. Reich, Approximate selections, best approximations, fixed points and invariant sets, J. Math. Anal. Appl. 62 (1978), 104-113. MR 0514991 (58:24180)
  • [5] S. Reich, Integral equations, hyperconvex spaces, and the Hilbert ball, Nonlinear Analysis and Applications, Dekker, New York, 1987, pp. 517-525. MR 912333 (90d:47066)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0954603-6
Keywords: Multifunction, selection, fixed point, retract, best approximation, nonexpansive, hyperconvex
Article copyright: © Copyright 1989 American Mathematical Society

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