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General semi-implicit approximations with errors for common fixed points of nonexpansive-type operators and applications to Stampacchia variational inequality

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Abstract

It is meaningful and valuable to find common fixed points of different nonexpansive-type operators, which are associated with variational inequalities, integral equations, image process and other optimization problems in real life. The purpose of this paper is to suggest and consider a class of general semi-implicit iterative methods involving semi-implicit rule and inaccurate computing errors, which extend the iterative algorithm introduced by Ali et al. in 2020. Using Liu’s lemma, we analyze convergence and stability of the new iterative approximations for common fixed points of three different nonexpansive-type operators. Furthermore, we provide convergence rates of the new iterations and some numerical examples to illustrate the efficiency and stability of the new iterative schemes. As an application of our main results presented in this paper, we use the proposed iterative schemes to solve the known Stampacchia variational inequality.

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Abbreviations

JF:

Iterative scheme introduced by Ali et al. (2020).

JFESD:

General semi-implicit approximation with errors for three different nonexpansive-type operators.

JFSD:

Semi-implicit iteration for three different nonexpansive-type operators.

JFES:

Semi-implicit scheme with errors for a nonexpansive-type operator.

JFS:

Semi-implicit scheme for a nonexpansive-type operator.

PMMI:

Picard–Mann semi-implicit iteration with mixed errors for a nonexpansive-type operator (Li and Lan 2019).

PMI:

Picard–Mann semi-implicit iterative process for a nonexpansive-type operator (Li and Lan 2019).

MANN:

Mann iteration introduced by Mann (1953).

ISHIKAWA:

Ishikawa iterative process due to Ishikawa (1974).

NOOR:

Noor three-step iterative approximation scheme introduced by Noor (2007).

SAKURAI:

Novel fixed point algorithm formulated by Sakurai and Iiduka (2014).

Iter.:

The numbers of iteration.

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Acknowledgements

We are grateful to the anonymous referees and editors for their valuable comments and helpful suggestions to improve the quality of this paper.

Funding

This work was partially supported by Central Government Funds of Guiding Local Scientific and Technological Development for Sichuan Province (Grant No. 2021ZYD0017), the Sichuan Science and Technology Program (2019YJ0541) and the Innovation Fund of Postgraduate, Sichuan University of Science & Engineering (y2021097).

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Author notes

  1. Hai-yang Xu and Fan Zhang contributed equally to this work.

Authors and Affiliations

  1. College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong, 643000, Sichuan, People’s Republic of China

    Hai-yang Xu, Heng-you Lan & Fan Zhang

  2. South Sichuan Center for Applied Mathematics, Zigong, 643000, Sichuan, People’s Republic of China

    Heng-you Lan

Authors
  1. Hai-yang Xu
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  2. Heng-you Lan
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  3. Fan Zhang
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Contributions

H-YX carried out the proof of the theorems and gave some numerical simulations to show the existence results. H-YL and FZ conceived of the research, and participated in its design and coordination. All authors have read and approved the final manuscript.

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Correspondence to Heng-you Lan.

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Xu, Hy., Lan, Hy. & Zhang, F. General semi-implicit approximations with errors for common fixed points of nonexpansive-type operators and applications to Stampacchia variational inequality. Comp. Appl. Math. 41, 190 (2022). https://doi.org/10.1007/s40314-022-01890-7

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