Block Iterative Methods for a Finite Family of Relatively Nonexpansive Mappings in Banach Spaces
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Using the convex combination based on Bregman distances due to Censor and Reich, we define an operator from a given family of relatively nonexpansive mappings in a Banach space. We first prove that the fixed-point set of this operator is identical to the set of all common fixed points of the mappings. Next, using this operator, we construct an iterative sequence to approximate common fixed points of the family. We finally apply our results to a convex feasibility problem in Banach spaces.
KeywordsBanach Space Iterative Method Differential Geometry Nonexpansive Mapping Convex Combination
- 13.Matsushita S, Takahashi W: An iterative algorithm for relatively nonexpansive mappings by a hybrid method and applications. In Nonlinear Analysis and Convex Analysis. Edited by: Takahashi W, Tanaka T. Yokohama Publishers, Yokohama, Japan; 2004:305–313.Google Scholar
- 15.Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Appl. Math.. Volume 178. Edited by: Kartsatos AG. Markel Dekker, New York, NY, USA; 1996:15–50.Google Scholar
- 21.Diestel J: Geometry of Banach Spaces—Selected Topics, Lecture Notes in Mathematics. Volume 485. Springer, Berlin, Germany; 1975.Google Scholar
- 23.Reich S: A weak convergence theorem for the alternating method with Bregman distances. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Appl. Math.. Volume 178. Edited by: Kartsatos AG. Markel Dekker, New York, NY, USA; 1996:313–318.Google Scholar
- 27.Takahashi W: Convex Analysis and Approximation of Fixed Points, Mathematical Analysis Series. Volume 2. Yokohama Publishers, Yokohama, Japan; 2000.Google Scholar
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