Browse > Article
http://dx.doi.org/10.5666/KMJ.2014.54.3.375

Halpern's Iteration for Strongly Relatively Nonexpansive Mappings in Banach Spaces  

Suantai, Suthep (Department of Mathematics, Faculty of Science, Chiang Mai University)
Cholamjiak, Prasit (School of Science, University of Phayao)
Publication Information
Kyungpook Mathematical Journal / v.54, no.3, 2014 , pp. 375-385 More about this Journal
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;
Citations & Related Records
연도 인용수 순위
  • Reference
1 C. E. Chidume and C. O. Chidume, Iterative approximation of fixed points of nonexpansive mappings, J. Math. Anal. Appl., 318(2006), 288-295.   DOI   ScienceOn
2 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.
3 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.   DOI
4 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.   DOI
5 R. E. Bruck and S. Reich, Nonexpansive projections and resolvents of accretive operators in Banach spaces, Houston J. Math., 3(1977), 459-470.
6 Y. Censor and S. Reich, Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization 37(1996), 323-339.   DOI
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.   DOI
9 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.   DOI
10 P. L. Lions, Approximation de points fixes de contractions, C.R. Acad. Sci. Paris Ser. A-B., 284(1977), A1357-A1359.
11 P. E. Mainge, The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces, Comput. Math. Appl., 59(2010), 74-79.   DOI
12 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.   DOI
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.   DOI   ScienceOn
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 S. Reich, Approximating fixed points of nonexpansive mappings, PanAmer. Math. J., 4(1994), 23-28.
17 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.
18 S. Saejung, Halpern's iteration in Banach spaces, Nonlinear Anal., 73(2010), 3431-3439.   DOI
19 N. Shioji and W. Takahashi, Strong convergence of approximated sequences for nonexpansive mapping in Banach spaces, Proc. Amer. Math. Soc., 125(1997), 3641-3645.   DOI   ScienceOn
20 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.
21 W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama (2000).
22 R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math., 58(1992), 486-491.   DOI   ScienceOn
23 H. K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65(2002), 109-113.   DOI
24 H. K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116(2003), 659-678.   DOI   ScienceOn
25 S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim., 13(2002), 938-945.   DOI   ScienceOn