Asymptotic behaviour of unbounded nonexpansive sequences in Banach spaces

Author:
Behzad Djafari Rouhani

Journal:
Proc. Amer. Math. Soc. **117** (1993), 951-956

MSC:
Primary 47H10; Secondary 46B15

MathSciNet review:
1120510

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Abstract | References | Similar Articles | Additional Information

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

**[1]**Joseph Diestel,*Geometry of Banach spaces—selected topics*, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR**0461094****[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]**Behzad Djafari Rouhani,*Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces*, J. Math. Anal. Appl.**147**(1990), no. 2, 465–476. MR**1050218**, 10.1016/0022-247X(90)90361-I**[5]**Behzad Djafari Rouhani,*Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space*, J. Math. Anal. Appl.**151**(1990), no. 1, 226–235. MR**1069458**, 10.1016/0022-247X(90)90253-C**[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]**Behzad Djafari Rouhani,*Asymptotic behaviour of unbounded trajectories for some nonautonomous systems in a Hilbert space*, Nonlinear Anal.**19**(1992), no. 8, 741–751. MR**1186787**, 10.1016/0362-546X(92)90218-4**[10]**Ky Fan and Irving Glicksberg,*Some geometric properties of the spheres in a normed linear space*, Duke Math. J.**25**(1958), 553–568. MR**0098976****[11]**Kazimierz Goebel and Simeon Reich,*Uniform convexity, hyperbolic geometry, and nonexpansive mappings*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker, Inc., New York, 1984. MR**744194****[12]**Elon Kohlberg and Abraham Neyman,*Asymptotic behavior of nonexpansive mappings in uniformly convex Banach spaces*, Amer. Math. Monthly**88**(1981), no. 9, 698–700. MR**643273**, 10.2307/2320677**[13]**-,*Asymptotic behaviour of non expansive mappings in normed linear spaces*, Israel J. Math.**38**(1981), 269-275.**[14]**Ulrich Krengel,*Ergodic theorems*, de Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR**797411****[15]**A. Pazy,*Asymptotic behavior of contractions in Hilbert space*, Israel J. Math.**9**(1971), 235–240. MR**0282276****[16]**-,*Non linear analysis and mechancis*, Heriot-Watt Symposium, Vol. III (R. J. Knops, ed.), Pitman Research Notes in Math., vol. 30, Longman Sci. Tech., Harlow, 1979, pp. 36-134.**[17]**Andrew T. Plant and Simeon Reich,*The asymptotics of nonexpansive iterations*, J. Funct. Anal.**54**(1983), no. 3, 308–319. MR**724526**, 10.1016/0022-1236(83)90003-4**[18]**Simeon Reich,*Asymptotic behavior of contractions in Banach spaces*, J. Math. Anal. Appl.**44**(1973), 57–70. MR**0328689****[19]**-,*Asymptotic behaviour of semi-groups of non linear contractions in Banach spaces*, J. Math. Anal. Appl.**53**(1976), 277-290.**[20]**-,*On the asymptotic behaviour of non linear semi-groups and the range of accretive operators*I, II, Math. Research Center Report 2198, 1981; J. Math. Anal. Appl.**79**(1981), 113-126;**87**(1982), 134-146.

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DOI:
https://doi.org/10.1090/S0002-9939-1993-1120510-8

Article copyright:
© Copyright 1993
American Mathematical Society