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A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem

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Abstract

In this paper, we propose a new extragradient method consisting of the hybrid steepest descent method, a single projection method and an Armijo line searching the technique for approximating a solution of variational inequality problem and finding the fixed point of demicontractive mapping in a real Hilbert space. The essence of this algorithm is that a single projection is required in each iteration and the step size for the next iterate is determined in such a way that there is no need for a prior estimate of the Lipschitz constant of the underlying operator. We state and prove a strong convergence theorem for approximating common solutions of variational inequality and fixed points problem under some mild conditions on the control sequences. By casting the problem into an equivalent problem in a suitable product space, we are able to present a simultaneous algorithm for solving the split equality problem without prior knowledge of the operator norm. Finally, we give some numerical examples to show the efficiency of our algorithm over some other algorithms in the literature.

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Acknowledgements

The authors thank the referees of this paper whose valuable comments and suggestions have improved the presentation of the paper. The first author acknowledges with thanks the bursary and financial support from the 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 (Grant no. BA-2019-067). The second author acknowledges with thanks the International Mathematical Union Breakout Graduate Fellowship (IMU-BGF) (Grant no. IMU-BGF-2019-10) Award for his doctoral study. The fourth 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 NRF and CoE-MaSS.

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Authors and Affiliations

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

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

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  1. L. O. Jolaoso
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  2. A. Taiwo
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  3. T. O. Alakoya
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  4. O. T. Mewomo
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Correspondence to O. T. Mewomo.

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Communicated by Pablo Pedregal.

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Jolaoso, L.O., Taiwo, A., Alakoya, T.O. et al. A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem. Comp. Appl. Math. 39, 38 (2020). https://doi.org/10.1007/s40314-019-1014-2

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