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A Picard-type iterative algorithm for general variational inequalities and nonexpansive mappings

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Abstract

In this paper, a normal S-iterative algorithm is studied and analyzed for solving a general class of variational inequalities involving a set of fixed points of nonexpansive mappings and two nonlinear operators. It is shown that the proposed algorithm converges strongly under mild conditions. The rate of convergence of the proposed iterative algorithm is also studied. An equivalence of convergence between the normal S-iterative algorithm and Algorithm 2.6 of Noor (J. Math. Anal. Appl. 331, 810–822, 2007) is established and a comparison between the two is also discussed. As an application, a modified algorithm is employed to solve convex minimization problems. Numerical examples are given to validate the theoretical findings. The results obtained herein improve and complement the corresponding results in Noor (J. Math. Anal. Appl. 331, 810–822, 2007).

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Acknowledgements

Authors are thankful to the reviewers for their useful comments and suggestions to improve quality and presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Adiyaman University, 02040, Adiyaman, Turkey

    Faik Gürsoy & Müzeyyen Ertürk

  2. Department of Mathematics, Government College University, Lahore, Pakistan

    Mujahid Abbas

  3. Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa

    Mujahid Abbas

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  1. Faik Gürsoy
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  2. Müzeyyen Ertürk
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  3. Mujahid Abbas
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Correspondence to Müzeyyen Ertürk.

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Gürsoy, F., Ertürk, M. & Abbas, M. A Picard-type iterative algorithm for general variational inequalities and nonexpansive mappings. Numer Algor 83, 867–883 (2020). https://doi.org/10.1007/s11075-019-00706-w

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