Abstract
In this paper, we consider the split common null point problem in Banach spaces. Then using the metric resolvents of maximal monotone operators and the metric projections, we prove a strong convergence theorem for finding a solution of the split common null point problem in Banach spaces. The result of this paper seems to be the first one to study it outside Hilbert spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alsulami S.M., Takahashi W.: The split common null point problem for maximal monotone mappings in Hilbert spaces and applications. J. Nonlinear Convex Anal. 15, 793–808 (2014)
Aoyama K., Kohsaka F., Takahashi W.: Three generalizations of firmly nonexpansive mappings: Their relations and continuous properties. J. Nonlinear Convex Anal. 10, 131–147 (2009)
Browder F.E.: Nonlinear maximal monotone operators in Banach spaces, Math. Ann. 175, 89–113 (1968)
C. Byrne, Y. Censor, A. Gibali, and S. Reich, The split common null point problem, J. Nonlinear Convex Anal. 13 (2012), 759–775.
Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms 8 (1994), 221–239.
Censor Y., Segal A.: The split common fixed-point problem for directed operators. J. Convex Anal. 16, 587–600 (2009)
I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht, 1990.
F. Kosaka and W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM. J.Optim. 19 (2008), 824–835.
F. Kosaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. (Basel) 91 (2008), 166–177.
Masad E., Reich S.: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8, 367–371 (2007)
A. Moudafi, The split common fixed point problem for demicontractive mappings, Inverse Problems 26 (2010), 055007, 6 pp.
K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mapping and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003), 372–379.
S. Ohsawa and W. Takahashi, Strong convergence theorems for resolvents of maximal mono- tone operators in Banach spaces, Arch. Math. (Basel) 81 (2003), 439–445.
S. Reich, Book Review: Geometry of Banach Spaces, Duality Mappings and Nonlinear Prob- lems, Bull. Amer. Math. Soc. 26 (1992), 367–370.
M. V. Solodov and B. F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Programming Ser. A. 87 (2000), 189–202.
W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.
W. Takahashi, Convex Analysis and Approximation of Fixed Points, Yokohama Publishers, Yoko- hama, 2000 (Japanese).
W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, 2009.
Takahashi W.: The split feasibility problem in Banach spaces. J. Nonlinear Convex Anal. 15, 1349–1355 (2014)
W. Takahashi, The split feasibility problem and the shrinking projection method in Banach spaces, J. Nonlinear Convex Anal., to appear.
W. Takahashi, H.-K. Xu, and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, Set-Valued Var. Anal., to appear.
Xu H.K.: A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Problems 22, 2021–2034 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Takahashi, W. The split common null point problem in Banach spaces. Arch. Math. 104, 357–365 (2015). https://doi.org/10.1007/s00013-015-0738-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-015-0738-5