Asymptotic behaviour of firmly nonexpansive sequences

Author:
Behzad Djafari Rouhani

Journal:
Proc. Amer. Math. Soc. **123** (1995), 771-777

MSC:
Primary 47H09; Secondary 46B15

MathSciNet review:
1234631

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce the notion of firmly nonexpansive sequences in a Banach space and present several results concerning their asymptotic behaviour extending previous results and giving an affirmative answer to an open question raised by S. Reich and I. Shafrir [Proc. Amer. Math. Soc. **101** (1987), 246-250]. Applications to averaged mappings are also given.

**[1]**J. B. Baillon, R. E. Bruck, and S. Reich,*On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces*, Houston J. Math.**4**(1978), no. 1, 1–9. MR**0473932****[2]**Felix E. Browder,*Convergence theorems for sequences of nonlinear operators in Banach spaces*, Math. Z.**100**(1967), 201–225. MR**0215141****[3]**Ronald E. Bruck Jr.,*Nonexpansive projections on subsets of Banach spaces*, Pacific J. Math.**47**(1973), 341–355. MR**0341223****[4]**Ronald E. Bruck and Simeon Reich,*Nonexpansive projections and resolvents of accretive operators in Banach spaces*, Houston J. Math.**3**(1977), no. 4, 459–470. MR**0470761****[5]**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**[6]**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**[7]**-,*A simple proof to an extension of a theorem of A. Pazy in Hilbert space*, preprint, ICTP, Trieste, No. IC/90/219, 1990.**[8]**Behzad Djafari Rouhani,*Asymptotic behaviour of unbounded nonexpansive sequences in Banach spaces*, Proc. Amer. Math. Soc.**117**(1993), no. 4, 951–956. MR**1120510**, 10.1090/S0002-9939-1993-1120510-8**[9]**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****[10]**Kazimierz Goebel and W. A. Kirk,*Topics in metric fixed point theory*, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. MR**1074005****[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]**Shiro Ishikawa,*Fixed points and iteration of a nonexpansive mapping in a Banach space*, Proc. Amer. Math. Soc.**59**(1976), no. 1, 65–71. MR**0412909**, 10.1090/S0002-9939-1976-0412909-X**[13]**Elon Kohlberg and Abraham Neyman,*Asymptotic behavior of nonexpansive mappings in normed linear spaces*, Israel J. Math.**38**(1981), no. 4, 269–275. MR**617673**, 10.1007/BF02762772**[14]**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**[15]**Simeon Reich,*On the asymptotic behavior of nonlinear semigroups and the range of accretive operators*, J. Math. Anal. Appl.**79**(1981), no. 1, 113–126. MR**603380**, 10.1016/0022-247X(81)90013-5**[16]**Simeon Reich and Itai Shafrir,*The asymptotic behavior of firmly nonexpansive mappings*, Proc. Amer. Math. Soc.**101**(1987), no. 2, 246–250. MR**902536**, 10.1090/S0002-9939-1987-0902536-7

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47H09,
46B15

Retrieve articles in all journals with MSC: 47H09, 46B15

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1995-1234631-4

Article copyright:
© Copyright 1995
American Mathematical Society