Viscosity approximation method for split common null point problems between Banach spaces and Hilbert spaces
DOI:
https://doi.org/10.26713/jims.v9i1.380Keywords:
Iterative method, Viscosity approximation method, Fixed point problems, Split common null point problems, A zero point, Nonexpansive operator, (Metric) resolvent operatorAbstract
We study an iterative scheme to approximation the split common null point problems for set-valued maximal monotone operators which combine viscosity method and some fixed point technically proving method between Banach spaces and Hilbert space, without using the metric projection. We prove that strong convergence theorem. Also, we show that our result can be solves the split minimization problems.Downloads
References
Y. Censor, and T. Elfving: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms, vol. 8, pp. 221-239, (1994).
SM. Alsulami and W. Takahashi: The split common null point problem for maximal monotone mappings in Hilbert spaces and applications. J. Nonlinear Convex Anal., vol.15, pp.793-808, (2014).
W. Takahashi, H-K. Xu and J-C. Yao: Iterative methods for generalized split feasibility problems in Hilbert spaces. Set-Valued Var. Anal. 23, pp. 205-221,(2015).
C. Byrne, Y. Censor, A. Gibali, and S. Reich: The split common null point problem. J. Nonlinear Convex Anal., vol. 13, pp. 759-775, (2012).
W. Takahashi: The split feasibility problem in Banach spaces. J. Nonlinear Convex Anal., vol. 15, pp. 1349-1355, (2014).
W. Takahashi: The split common null point problem in Banach spaces. Arch. Math. (Basel), vol. 104 (4), pp. 357-365, (2015).
W. Takahashi and J-C Yao:Strong convergence theorems by hybrid methods for the split common null point problem in Banach spaces, Fixed Point Theory and Applications, vol. 2015, article 87, (2015).
A. Moudafi: Viscosity approximation methods for fixed-point problems, J. Math. Anal. Appl., vol.241, pp. 46-55, (2000).
H.K. Xu: Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. vol. 298, pp. 279-291, (2004).
J.P. Gossez and E. Lami Dazo: Some geometric properties related to the fixed point theory for nonexpan- sive mappings, Pac. J. Math., vol. 40, pp. 565-573, (1972).
S-S. Chang, Y.J. ChoZhou, and H.Z. ZHou: Ilterative methods for nonlinear opertor equation in banach spaces, Nova Science Publishers Inc., Huntington, New York, (2002).
C. Chidume: Geometric Properties of Banach Spaces and Nonlinear Iterations,Lecture Notes in Mathe- matics, vol. 1965, Springer-Verlag London Limited, (2009).
R.T. Rockafellar: On the maximality of sums of nonliear monotone opertors,Trans. Amer. Math.Soc., vol. 149, pp. 77-85, (1970).
FE. Browder: Nonlinear maximal monotone operators in Banach spaces. Math. Ann. vol. 175, pp. 89-113, (1968).
K. Ayoma, F. Kohsaka and W.Takahashi: Strong convergence theorems for a family of mappings of type (P) and applications. Proceeding of the Asian Conference on Nonlinear Analysis and Optimization (Matsue, Japan, 2008), Japan, Yokohama Publishers Inc., pp.1-17, (2009).
S. Reich: Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., vol. 67, pp. 274-276, (1979).
S. Kitahara and W. Takahashi: Image recovery by convex combinations of sunny nonexpansive retractions,
Topological Methods in Nonlinear Anal., vol. 2, no. 2, pp. 333-342, (1993).
SS. Chang: Some problems and results in the study of nonlinear analysis. Nonlinear Anal., vol. 33, pp. 4197-4208, (1997).
H.K. Xu: Inequalities in Banach spaces with applications, Nonlinear Analysis, vol. 16 (12), pp. 1127-1138, (1991).
Cho, YJ, Zhou, HY, Guo, G: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl., vol. 47, pp. 707-717, (2004).
V. Barbu: Nonlinear Semigroups and Differential Equations in Banach Space. Noordhoff, Groningen, (1976).
R.T. Rockafellar: Characterization of subdifferencetials of convex function. Pacific J.Math., vol. 17, pp. 497-510, (1966).
R.T. Rockafellar: On maximal monotonicity of subdifferencetial mappings. Pacific J.Math., vol. 33, pp. 209-216, (1970).
F. E. Browder: Fixed-point theorems for noncompact mappings in Hilbert space, Proceedings of the National Academy of Sciences of the United States of America, vol. 53, pp. 1272-1276, (1965).
T. Suzuki: Strong convergence of Krasnoselskii and Mann's type sequence for one-parameter nonexpansive semigroup without Bochner integrals, J. Math. Anal. Appl., vol. 305, pp. 227-239, (2005).
H.K. Xu: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. , vol. 66, pp. 240-256, (2002).
W. Takahashi: Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, (2009).
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
