Asymptotic behaviour of unbounded nonexpansive sequences in Banach spaces
Author:
Behzad Djafari Rouhani
Journal:
Proc. Amer. Math. Soc. 117 (1993), 951956
MSC:
Primary 47H10; Secondary 46B15
MathSciNet review:
1120510
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Abstract: Let be a real Banach space, a nonexpansive sequence in (i.e., for all ), and the closed convex hull of the sequence . We prove that and deduce a simple short proof for the following result, (i) If is reflexive and strictly convex, then converges weakly in to the element of minimum norm in with (ii) If has Fréchet differentiable norm, then converges strongly to . This result contains previous results by Pazy, Kohlberg and Neyman, Plant and Reich, and Reich and is also optimal since the assumptions made on in (i) or (ii) are also necessary for the respective conclusion to hold.
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B. Djafari Rouhani, Ergodic theorems for non expansive sequences in Hilbert spaces and related problems, thesis, Yale University, 1981.
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 [1]
 J. Diestel, Geometry of Banach spacesSelected topics, Lecture Notes in Math., vol. 485, SpringerVerlag, Berlin and New York, 1975. MR 0461094 (57:1079)
 [2]
 B. Djafari Rouhani, Ergodic theorems for non expansive sequences in Hilbert spaces and related problems, thesis, Yale University, 1981.
 [3]
 B. Djafari Rouhani and S. Kakutani, Ergodic theorems for non expansive non linear operators in a Hilbert space, preprint, 1984.
 [4]
 B. Djafari Rouhani, Asymptotic behaviour of quasiautonomous dissipative systems in Hilbert spaces, preprint, ICTP, Trieste, no. IC/88/31, 1988 and J. Math. Anal. Appl. 147 (1990), 465476. MR 1050218 (91h:47069)
 [5]
 , Asymptotic behaviour of almost non expansive sequences in a Hilbert space, preprint, ICTP, Trieste, no. IC/88/188, 1988 and J. Math. Anal. Appl. 151 (1990), 226235. MR 1069458 (92a:47065)
 [6]
 , A note on the convergence of a numerical sequence, internal report, ICTP, Trieste, no. IC/89/204, 1989.
 [7]
 , A non linear ergodic theorem and application to a theorem of A. Pazy, internal report, ICTP, Trieste, no. IC/89/203, 1989.
 [8]
 , A simple proof to an extension of a theorem of A. Pazy in Hilbert space, preprint, ICTP, Trieste, no. IC/90/219, 1990.
 [9]
 , Asymptotic behaviour of unbounded trajectories for some non autonomous systems in a Hilbert space, preprint, ICTP, Trieste, no. IC/90/181, 1990; Nonlinear Anal. (to appear). MR 1186787 (93m:47070)
 [10]
 K. Fan and I. Glicksberg, Some geometric properties of the spheres in a normed linear space, Duke Math. J. 25 (1958), 553568. MR 0098976 (20:5421)
 [11]
 K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry and non expansive mappings, Dekker, New York and Basel, 1984. MR 744194 (86d:58012)
 [12]
 E. Kohlberg and A. Neyman, Asymptotic behaviour of non expansive mappings in uniformly convex Banach spaces, Amer. Math. Monthly 88 (1981), 698700. MR 643273 (83c:47077)
 [13]
 , Asymptotic behaviour of non expansive mappings in normed linear spaces, Israel J. Math. 38 (1981), 269275.
 [14]
 U. Krengel, Ergodic theorems, de Gruyter Studies in Math., vol. 6, de Gruyter, Berlin and New York, 1985. MR 797411 (87i:28001)
 [15]
 A. Pazy, Asymptotic behaviour of contractions in Hilbert space, Israel J. Math. 9 (1971), 235240. MR 0282276 (43:7988)
 [16]
 , Non linear analysis and mechancis, HeriotWatt Symposium, Vol. III (R. J. Knops, ed.), Pitman Research Notes in Math., vol. 30, Longman Sci. Tech., Harlow, 1979, pp. 36134.
 [17]
 A. T. Plant and S. Reich, The asymptotics of non expansive iterations, J. Funct. Anal. 54 (1983), 308319. MR 724526 (85a:47055)
 [18]
 S. Reich, Asymptotic behaviour of contractions in Banach spaces, J. Math. Anal. Appl. 44 (1973), 5770. MR 0328689 (48:7031)
 [19]
 , Asymptotic behaviour of semigroups of non linear contractions in Banach spaces, J. Math. Anal. Appl. 53 (1976), 277290.
 [20]
 , On the asymptotic behaviour of non linear semigroups and the range of accretive operators I, II, Math. Research Center Report 2198, 1981; J. Math. Anal. Appl. 79 (1981), 113126; 87 (1982), 134146.
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DOI:
http://dx.doi.org/10.1090/S00029939199311205108
PII:
S 00029939(1993)11205108
Article copyright:
© Copyright 1993
American Mathematical Society
