Proceedings of the American Mathematical Society

A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings

Author(s): Tomonari Suzuki
Journal: Proc. Amer. Math. Soc. 135 (2007), 99-106.
MSC (2000): Primary 47H09; Secondary 47H10
Posted: June 13, 2006
Retrieve article in: PDF DVI PostScript

Abstract: In this paper, we prove a Halpern-type strong convergence theorem for nonexpansive mappings in a Banach space whose norm is uniformly Gâteaux differentiable. Also, we discuss the sufficient and necessary condition about this theorem. This is a partial answer of the problem raised by Reich in 1983.

1.: J. B. Baillon, ``Quelques aspects de la théorie des points fixes dans les espaces de Banach. I, II.'' (in French), Séminaire d'Analyse Fonctionnelle (1978-1979), Exp. No. 7-8, 45 pp., École Polytech., Palaiseau, 1979. MR 0557363 (81d:47036)
2.: S. Banach, ``Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales'', Fund. Math., 3 (1922), 133-181.
3.: F. E. Browder, ``Fixed-point theorems for noncompact mappings in Hilbert space'', Proc. Nat. Acad. Sci. USA, 53 (1965), 1272-1276. MR 0178324 (31:2582)
4.: F. E. Browder, ``Nonexpansive nonlinear operators in a Banach space'', Proc. Nat. Acad. Sci. USA, 54 (1965), 1041-1044. MR 0187120 (32:4574)
5.: F. E. Browder, ``Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces'', Arch. Ration. Mech. Anal., 24 (1967), 82-90. MR 0206765 (34:6582)
6.: R. E. Bruck, ``Nonexpansive retracts of Banach spaces'', Bull. Amer. Math. Soc., 76 (1970), 384-386. MR 0256135 (41:794)
7.: R. E. Bruck, ``Properties of fixed-point sets of nonexpansive mappings in Banach spaces'', Trans. Amer. Math. Soc., 179 (1973), 251-262. MR 0324491 (48:2843)
8.: C. E. Chidume, J. Li and A. Udomene, ``Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings'', Proc. Amer. Math. Soc., 133 (2005), 473-480. MR 2093070 (2005f:47122)
9.: Y. J. Cho, S. M. Kang and H. Y. Zhou, ``Some control conditions on the iterative methods'', to appear in Internat. J. Comput. Numer. Anal. Appl.
10.: D. Göhde, ``Zum Prinzip def kontraktiven Abbildung'', Math. Nachr., 30 (1965), 251-258. MR 0190718 (32:8129)
11.: B. Halpern, ``Fixed points of nonexpanding maps'', Bull. Amer. Math. Soc., 73 (1967), 957-961. MR 0218938 (36:2022)
12.: J. L. Kelley, ``General Topology'', Van Nostrand Reinhold Company (1955). MR 0070144 (16:1136c)
13.: W. A. Kirk, ``A fixed point theorem for mappings which do not increase distances'', Amer. Math. Monthly, 72 (1965), 1004-1006. MR 0189009 (32:6436)
14.: P.-L. Lions, ``Approximation de points fixes de contractions'', C. R. Acad. Sci. Paris Ser. A-B 284 (1977), A1357-A1359. MR 0470770 (57:10515)
15.: S. Reich, ``Strong convergence theorems for resolvents of accretive operators in Banach spaces'', J. Math. Anal. Appl., 75 (1980), 287-292. MR 0576291 (82a:47050)
16.: S. Reich, ``Some problems and results in fixed point theory'', Contemp. Math., 21 (1983), 179-187. MR 0729515 (85e:47082)
17.: N. Shioji and W. Takahashi, ``Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces'', Proc. Amer. Math. Soc., 125 (1997), 3641-3645. MR 1415370 (98e:47088)
18.: T. Suzuki, ``Strong convergence theorem to common fixed points of two nonexpansive mappings in general Banach spaces'', J. Nonlinear Convex Anal., 3 (2002), 381-391. MR 1947106 (2003i:47066)
19.: T. Suzuki, ``Convergence theorems to common fixed points for infinite families of nonexpansive mappings in strictly convex Banach spaces'', Nihonkai Math. J., 14 (2003), 43-54. MR 1986983 (2004d:47089)
20.: T. Suzuki, ``Krasnoselskii and Mann's type sequences and Ishikawa's strong convergence theorem'', in Proceedings of the Third International Conference on Nonlinear Analysis and Convex Analysis (W. Takahashi and T. Tanaka Eds.), pp. 527-539, Yokohama Publishers, 2004. MR 2144072 (2006b:47107)
21.: T. Suzuki, ``Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces'', Fixed Point Theory Appl., 2005 (2005), 103-123. MR 2172156
22.: T. Suzuki, ``Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals'', J. Math. Anal. Appl., 305 (2005), 227-239. MR 2128124 (2005j:47055)
23.: R. Wittmann, ``Approximation of fixed points of nonexpansive mappings'', Arch. Math. (Basel), 58 (1992), 486-491. MR 1156581 (93c:47069)
24.: H. K. Xu, ``Another control condition in an iterative method for nonexpansive mappings'', Bull. Austral. Math. Soc., 65 (2002), 109-113. MR 1889384 (2002m:47076)
25.: H. K. Xu, ``Iterative algorithms for nonlinear operators'', J. London Math. Soc., 66 (2002), 240-256. MR 1911872 (2003e:47114)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47H09, 47H10

Tomonari Suzuki
Affiliation: Department of Mathematics, Kyushu Institute of Technology, Sensuicho, Tobata, Kitakyushu 804-8550, Japan
Email: suzuki-t@mns.kyutech.ac.jp

DOI: 10.1090/S0002-9939-06-08435-8
PII: S 0002-9939(06)08435-8
Keywords: Nonexpansive mapping, fixed point, Halpern-type strong convergence theorem
Received by editor(s): March 1, 2005
Received by editor(s) in revised form: July 18, 2005
Posted: June 13, 2006
Additional Notes: The author was supported in part by Grants-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology.
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS