Asymptotic behaviour of firmly nonexpansive sequences
Author:
Behzad Djafari Rouhani
Journal:
Proc. Amer. Math. Soc. 123 (1995), 771777
MSC:
Primary 47H09; Secondary 46B15
MathSciNet review:
1234631
Fulltext 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), 246250]. 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
(57 #13590)
 [2]
Felix
E. Browder, Convergence theorems for sequences of nonlinear
operators in Banach spaces, Math. Z. 100 (1967),
201–225. MR 0215141
(35 #5984)
 [3]
Ronald
E. Bruck Jr., Nonexpansive projections on subsets of Banach
spaces, Pacific J. Math. 47 (1973), 341–355. MR 0341223
(49 #5973)
 [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
(57 #10507)
 [5]
Behzad
Djafari Rouhani, Asymptotic behaviour of quasiautonomous
dissipative systems in Hilbert spaces, J. Math. Anal. Appl.
147 (1990), no. 2, 465–476. MR 1050218
(91h:47069), http://dx.doi.org/10.1016/0022247X(90)90361I
 [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
(92a:47065), http://dx.doi.org/10.1016/0022247X(90)90253C
 [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
(93e:47073), http://dx.doi.org/10.1090/S00029939199311205108
 [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 (20 #5421)
 [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
(92c:47070)
 [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
(86d:58012)
 [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
(54 #1030), http://dx.doi.org/10.1090/S0002993919760412909X
 [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 (83g:47056), http://dx.doi.org/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
(85a:47055), http://dx.doi.org/10.1016/00221236(83)900034
 [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
(82c:47066), http://dx.doi.org/10.1016/0022247X(81)900135
 [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
(88i:47030), http://dx.doi.org/10.1090/S00029939198709025367
 [1]
 J. B. Baillon, R. E. Bruck, and S. Reich, On the asymptotic behaviour of nonexpansive mappings and semigroups in Banach spaces, Houston J. Math. 4 (1978), 19. MR 0473932 (57:13590)
 [2]
 F. E. Browder, Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Z. 100 (1967), 201225. MR 0215141 (35:5984)
 [3]
 R. E. Bruck, Nonexpansive projections on subsets of Banach spaces, Pacific J. Math. 47 (1973), 341355. MR 0341223 (49:5973)
 [4]
 R. E. Bruck and S. Reich, Nonexpansive projections and resolvents of accretive operators in Banach spaces, Houston J. Math. 3 (1977), 459470. MR 0470761 (57:10507)
 [5]
 B. Djafari Rouhani, Asymptotic behaviour of quasiautonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl. 147 (1990), 465476. MR 1050218 (91h:47069)
 [6]
 , Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl. 151 (1990), 226235. MR 1069458 (92a:47065)
 [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]
 , Asymptotic behaviour of unbounded nonexpansive sequences in Banach spaces, Proc. Amer. Math. Soc. 117 (1993), 951956. MR 1120510 (93e:47073)
 [9]
 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)
 [10]
 K. Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge Stud. in Adv. Math., vol. 28, Cambridge, New York, Melbourne, and Sydney, 1990. MR 1074005 (92c:47070)
 [11]
 K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry and nonexpansive mappings, Dekker, New York and Basel, 1984. MR 744194 (86d:58012)
 [12]
 S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976), 6571. MR 0412909 (54:1030)
 [13]
 E. Kohlberg and A. Neyman, Asymptotic behaviour of nonexpansive mappings in normed linear spaces, Israel J. Math. 38 (1981), 269275. MR 617673 (83g:47056)
 [14]
 A. T. Plant and S. Reich, The asymptotics of nonexpansive iterations, J. Funct. Anal. 54 (1983), 308319. MR 724526 (85a:47055)
 [15]
 S. Reich, On the asymptotic behaviour of nonlinear semigroups and the range of accretive operators, I, II, Mathematics Research Center Report 2198 (1981); J. Math. Anal. Appl. 79 (1981), 113126; 87 (1982), 134146. MR 603380 (82c:47066)
 [16]
 S. Reich and I. Shafrir, The asymptotic behaviour of firmly nonexpansive mappings, Proc. Amer. Math. Soc. 101 (1987), 246250. MR 902536 (88i:47030)
Similar Articles
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/S00029939199512346314
PII:
S 00029939(1995)12346314
Article copyright:
© Copyright 1995
American Mathematical Society
