# An iterative method with residual vectors for solving the fixed point and the split inclusion problems in Banach spaces

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## 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.

## Keywords

Resolvent operator Relatively nonexpansive mapping Strong convergence Iterative methods Banach spaces## Mathematics Subject Classification

47H09 47H10 47J25 47J05## Notes

### 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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