Abstract
In this paper, we propose an iterative technique with residual vectors for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of a split inclusion problem (SIP) with a way of selecting the stepsizes without prior knowledge of the operator norm in the framework of p-uniformly convex and uniformly smooth Banach spaces. Then strong convergence of the proposed algorithm to a common element of the above two sets is proved. As applications, we apply our result to find the set of common fixed points of a family of mappings which is also a solution of the SIP. We also give a numerical example and demonstrate the efficiency of the proposed algorithm. The results presented in this paper improve and generalize many recent important results in the literature.
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Acknowledgements
P. Cholamjiak was supported by the Thailand Research Fund and University of Phayao under Grant RSA6180084. S. Suantai was partially supported by Chiang Mai University.
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Communicated by Héctor Ramírez.
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Cholamjiak, P., Suantai, S. & Sunthrayuth, P. An iterative method with residual vectors for solving the fixed point and the split inclusion problems in Banach spaces. Comp. Appl. Math. 38, 12 (2019). https://doi.org/10.1007/s40314-019-0766-z
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DOI: https://doi.org/10.1007/s40314-019-0766-z