JOURNAL BROWSE
Search
Advanced SearchSearch Tips
Halpern's Iteration for Strongly Relatively Nonexpansive Mappings in Banach Spaces
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
  • Journal title : Kyungpook mathematical journal
  • Volume 54, Issue 3,  2014, pp.375-385
  • Publisher : Department of Mathematics, Kyungpook National University
  • DOI : 10.5666/KMJ.2014.54.3.375
 Title & Authors
Halpern's Iteration for Strongly Relatively Nonexpansive Mappings in Banach Spaces
Suantai, Suthep; Cholamjiak, Prasit;
  PDF(new window)
 Abstract
We investigate strong convergence of Halpern's iteration for a countable family of strongly relatively nonexpansive mappings in the framework of uniformly convex and uniformly smooth Banach spaces. Our results extend those announced by many authors.
 Keywords
Strongly relatively nonexpansive mapping;Banach space;generalized projection;Halpern's iteration;strong convergence theorem;
 Language
English
 Cited by
 References
1.
Y. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, In: Kartsatos, A. G. (ed.) Theory and Applications of Nonlinear Operator of Accretive and Monotone Type, Marcel Dekker, New York (1996), 15-50.

2.
K. Aoyama, F. Kohsaka and W. Takahashi, Strongly relatively nonexpansive sequences in Banach spaces and applications, J. Fixed Point Theory Appl., 5(2009), 201-225. crossref(new window)

3.
K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Non-linear Anal., 67(2007), 2350-2360. crossref(new window)

4.
R. E. Bruck and S. Reich, Nonexpansive projections and resolvents of accretive operators in Banach spaces, Houston J. Math., 3(1977), 459-470.

5.
Y. Censor and S. Reich, Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization 37(1996), 323-339. crossref(new window)

6.
C. E. Chidume and C. O. Chidume, Iterative approximation of fixed points of nonexpansive mappings, J. Math. Anal. Appl., 318(2006), 288-295. crossref(new window)

7.
Y. J. Cho, S. M. Kang and H. Zhou, Some control conditions on iterative methods, Comm. Appl. Nonlinear Anal., 12(2005), 27-34.

8.
B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73(1967), 957-961. crossref(new window)

9.
S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim., 13(2002), 938-945. crossref(new window)

10.
F. Kohsaka and W. Takahashi, Strong convergence of an iterative sequence for maximal monotone operators in a Banach space, Abstr. Appl. Anal., 2004(2004), 239-249. crossref(new window)

11.
P. L. Lions, Approximation de points fixes de contractions, C.R. Acad. Sci. Paris Ser. A-B., 284(1977), A1357-A1359.

12.
P. E. Mainge, The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces, Comput. Math. Appl., 59(2010), 74-79. crossref(new window)

13.
S. Matsushita and W. Takahashi, Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 2004(2004), 37-47.

14.
S. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theor., 134(2005), 257-266. crossref(new window)

15.
W. Nilsrakoo and S. Saejung, Strong convergence to common fixed points of countable relatively quasi-nonexpansive mappings, Fixed Point Theory Appl., 2008(2008), Article ID 312454.

16.
W. Nilsrakoo and S. Saejung, Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spaces, Appl. Math. Comput., 217(2011), 6577-6586. crossref(new window)

17.
S. Reich, Approximating fixed points of nonexpansive mappings, PanAmer. Math. J., 4(1994), 23-28.

18.
S. Reich, A weak convergence theorem for the alternating method with Bregman distance, In: Kartsatos, A. G. (ed.) Theory and Applications of Nonlinear Operator of Accretive and Monotone Type, Marcel Dekker, New York (1996), 313-318.

19.
S. Saejung, Halpern's iteration in Banach spaces, Nonlinear Anal., 73(2010), 3431-3439. crossref(new window)

20.
N. Shioji and W. Takahashi, Strong convergence of approximated sequences for nonexpansive mapping in Banach spaces, Proc. Amer. Math. Soc., 125(1997), 3641-3645. crossref(new window)

21.
T. Suzuki, A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 135(2007), 99-106.

22.
W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama (2000).

23.
R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math., 58(1992), 486-491. crossref(new window)

24.
H. K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65(2002), 109-113. crossref(new window)

25.
H. K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116(2003), 659-678. crossref(new window)