Abstract
In this paper, we consider the split common null point problem in two Banach spaces. Then, using the generalized resolvents of maximal monotone operators and the generalized projections, we prove a strong convergence theorem for finding a solution of the split common null point problem in two Banach spaces. It seems that such a theorem for generalized resolvents is the first of its kind outside Hilbert spaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alber, Y.I.: Metric and generalized projections in Banach spaces: properties and applications. In: Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15–50 (1996)
Alber, Y.I., Reich, S.: An iterative method for solving a class of nonlinear operator in Banach spaces. Panamer. Math. J. 4, 39–54 (1994)
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)
Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Editura Acad. R. S. R., Bucuresti (1976)
Browder, F.E.: Nonlinear maximal monotone operators in Banach spaces. Math. Ann. 175, 89–113 (1968)
Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)
Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994)
Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its application. Inverse Problems 21, 2071–2084 (2005)
Censor, Y., Gibali, A., Reich, R.: Algorithms for the split variational inequality problems. Numer. Algorithms 59, 301–323 (2012)
Censor, Y., Segal, A.: The split common fixed-point problem for directed operators. J. Convex Anal. 16, 587–600 (2009)
Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer, Dordrecht (1990)
Garcia-Falset, J., Muniz-Perez, O., Reich, S.: Domains of accretive operators in Banach spaces. Proceedings of the Royal Soc. Edinburgh 146, 325–336 (2016)
Hojo, M., Takahashi, W.: A strong convegence theorem by shrinking projection method for the split common null point problem in Banach spaces. Numer. Funct. Anal. Optim. 37, 541–553 (2016)
Kamimura, S., Takahashi, W.: Strong convergence of a proximal-type algorithm in a Banach space. SIAM. J. Optim. 13, 938–945 (2002)
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)
Moudafi, A.: The split common fixed point problem for demicontractive mappings. Inverse Problems 26, 6 (2010). 055007
Nakajo, K., Takahashi, W.: Strong convergence theorems for nonexpansive mapping and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003)
Ohsawa, S., Takahashi, W.: Strong convergence theorems for resolvents of maximal monotone operators in Banach spaces. Arch. Math. (Basel) 81, 439–445 (2003)
Reich, S.: Book Review: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Bull. Amer. Math. Soc. 26, 367–370 (1992)
Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Amer. Math. Soc. 149, 75–88 (1970)
Schopfer, F., Schuster, T., Louis, A.K.: An iterative regularization method for the solution of the split feasibility problem in Banach spaces. Inverse Problems 24, 20 (2008). 055008
Shehu, Y., Iyiola, O.S., Enyi, C.D.: An iterative algorithm for solving split feasibility problems and fixed point prblems in Banach spaces. Inverse Problems 72, 835–864 (2016)
Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Programming Ser. A. 87, 189–202 (2000)
Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)
Takahashi, W.: Convex Analysis and Approximation of Fixed Points, (Japanese). Yokohama Publishers, Yokohama (2000)
Takahashi, W.: The split feasibility problem in Banach spaces. J. Nonlinear Convex Anal. 15, 1349–1355 (2014)
Takahashi, W.: The split common null point problem in Banach spaces. Arch. Math. 104, 357–365 (2015)
Takahashi, W.: The split common null point problem in two Banach spaces. J. Nonlinear Convex Anal. 16, 2343–2350 (2015)
Takahashi, W., Xu, H.-K., Yao, J.-C.: Iterative methods for generalized split feasibility problems in Hilbert spaces. Set-Valued Var. Anal. 23, 205–221 (2015)
Wang, F.: A new algorithm for solving the multiple-sets split feasi- bility problem in Banach spaces. Numer. Funct. Anal. Optim. 35, 99–110 (2014)
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 for generalized resolvents in two banach spaces. Numer Algor 75, 1065–1078 (2017). https://doi.org/10.1007/s11075-016-0230-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-016-0230-8
Keywords
- Split common null point problem
- Maximal monotone operator
- Fixed point
- Generalized projection
- Generalized resolvent
- Hybrid method
- Duality mapping