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A self-adaptive projection method with an inertial technique for split feasibility problems in Banach spaces with applications to image restoration problems

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Abstract

In this work, we study the split feasibility problem (SFP) in the framework of p-uniformly convex and uniformly smooth Banach spaces. We propose an iterative scheme with inertial terms for seeking the solution of SFP and then prove a strong convergence theorem for the sequences generated by our iterative scheme under implemented conditions on the step size which do not require the computation of the norm of the bounded linear operator. We finally provide some numerical examples which involve image restoration problems and demonstrate the efficiency of the proposed algorithm. The obtained result of this paper complements many recent results in this direction and seems to be the first one to investigate the SFP outside Hilbert spaces involving the inertial technique.

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Acknowledgements

We would like to thank anonymous referees for their useful comments that have improved the quality of the paper. The research was carried out when the first author was an Alexander von Humboldt Postdoctoral Fellow at the Institute of Mathematics, University of Wurzburg, Germany. He is grateful to the Alexander von Humboldt Foundation, Bonn, for the fellowship and the Institute of Mathematics, Julius Maximilian University of Wurzburg, Germany for the hospitality and facilities. The research of the second author was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) Grant 101.01-2017.315. The third author was supported by the Thailand Research Fund and University of Phayao under Grant RSA6180084.

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Authors and Affiliations

  1. Department of Mathematics, University of Nigeria, Nsukka, Nigeria

    Yekini Shehu

  2. Division of Computational Mathematics and Engineering Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Phan Tu Vuong

  3. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Phan Tu Vuong

  4. School of Science, University of Phayao, Phayao, 56000, Thailand

    Prasit Cholamjiak

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  1. Yekini Shehu
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Correspondence to Phan Tu Vuong.

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Shehu, Y., Vuong, P.T. & Cholamjiak, P. A self-adaptive projection method with an inertial technique for split feasibility problems in Banach spaces with applications to image restoration problems. J. Fixed Point Theory Appl. 21, 50 (2019). https://doi.org/10.1007/s11784-019-0684-0

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