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Inertial-Type Algorithm for Solving Split Common Fixed Point Problems in Banach Spaces

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Abstract

In this paper, motivated by the works of Kohsaka and Takahashi (SIAM J Optim 19:824–835, 2008) and Aoyama et al. (J Nonlinear Convex Anal 10:131–147, 2009) on the class of mappings of firmly nonexpansive type, we explore some properties of firmly nonexpansive-like mappings [or mappings of type (P)] in p-uniformly convex and uniformly smooth Banach spaces. We then study the split common fixed point problems for mappings of type (P) and Bregman weak relatively nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. We propose an inertial-type shrinking projection algorithm for solving the two-set split common fixed point problems and prove a strong convergence theorem. Also, we apply our result to the split monotone inclusion problems and illustrate the behaviour of our algorithm with several numerical examples s. The implementation of the algorithm does not require a prior knowledge of the operator norm. Our results complement many recent results in the literature in this direction. To the best of our knowledge, it seems to be the first to use the inertial technique to solve the split common fixed point problems outside Hilbert spaces.

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Acknowledgements

The authors sincerely thank the anonymous reviewer for his careful reading, constructive comments and fruitful suggestions that substantially improved the manuscript. The first author acknowledges with thanks the International Mathematical Union Breakout Graduate Fellowship (IMU-BGF) Award for his doctoral study. The second author acknowledges with thanks the bursary and financial support from Department of Science and Technology and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DST-NRF COE-MaSS) Doctoral Bursary. The third author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS, IMU and NRF

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  1. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

    A. Taiwo, L. O. Jolaoso & O. T. Mewomo

  2. DST-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Johannesburg, South Africa

    L. O. Jolaoso

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  1. A. Taiwo
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Taiwo, A., Jolaoso, L.O. & Mewomo, O.T. Inertial-Type Algorithm for Solving Split Common Fixed Point Problems in Banach Spaces. J Sci Comput 86, 12 (2021). https://doi.org/10.1007/s10915-020-01385-9

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