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A Strong Convergence Theorem for Solving Pseudo-monotone Variational Inequalities Using Projection Methods

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Abstract

Several iterative methods have been proposed in the literature for solving the variational inequalities in Hilbert or Banach spaces, where the underlying operator A is monotone and Lipschitz continuous. However, there are very few methods known for solving the variational inequalities, when the Lipschitz continuity of A is dispensed with. In this article, we introduce a projection-type algorithm for finding a common solution of the variational inequalities and fixed point problem in a reflexive Banach space, where A is pseudo-monotone and not necessarily Lipschitz continuous. Also, we present an application of our result to approximating solution of pseudo-monotone equilibrium problem in a reflexive Banach space. Finally, we present some numerical examples to illustrate the performance of our method as well as comparing it with related method in the literature.

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Acknowledgements

The authors sincerely thank the Editor in Chief and the anonymous reviewers for their careful reading, constructive comments and fruitful suggestions that substantially improved the manuscript. The first 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) (BA2019-039) Doctoral Bursary. The second author acknowledges with thanks the International Mathematical Union Breakout Graduate Fellowship (IMU-BGF-20191101) 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 CoE-MaSS, NRF and IMU.

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Authors and Affiliations

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

    Lateef Olakunle Jolaoso, Adeolu Taiwo, Timilehin Opeyemi Alakoya & Oluwatosin Temitope Mewomo

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  1. Lateef Olakunle Jolaoso
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  2. Adeolu Taiwo
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  3. Timilehin Opeyemi Alakoya
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  4. Oluwatosin Temitope Mewomo
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Correspondence to Oluwatosin Temitope Mewomo.

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Jolaoso, L.O., Taiwo, A., Alakoya, T.O. et al. A Strong Convergence Theorem for Solving Pseudo-monotone Variational Inequalities Using Projection Methods. J Optim Theory Appl 185, 744–766 (2020). https://doi.org/10.1007/s10957-020-01672-3

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