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Regularity of Newton’s iteration for general parametric variational system

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Abstract

In this work, we apply the contraction mapping principle for set-valued mappings to study the Lipschitz-like property of the solution mapping to general parametric variational system obtained via canonical perturbation of a generalized equation. Then, under the assumptions of partial metric regularity and partial Lipschitz-like property, we obtained quadratic convergence of the iterative sequence when started close enough to the solution and provided estimates for the convergence parameters. Upon reconsidering all associated infinite sequences of Newton’s iterates as the image of a mapping defined on a sequence space, we established efficient conditions ensuring such a mapping to possess partial Lipschitz-like properties and also provided estimates of the moduli.

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Acknowledgements

The authors would like to express their gratitude to Michel Théra for his helpful comments and suggestions. This work was supported by National Natural Science Foundation of China (11801500, 11771384), Yunnan Provincial Science and Technology Department Research Fund (2017FD070), Yunnan Provincial Department of Education Research Fund (2019J0040) and the Fund for Fostering Talents in Kunming University of Science and Technology (KKSY 201807022).

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Authors and Affiliations

  1. School of Mathematics, Yunnan Normal University, Kunming, 650500, People’s Republic of China

    Wei Ouyang

  2. School of Science, Kunming University of Science and Technology, Kunming, 650500, People’s Republic of China

    Binbin Zhang

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  1. Wei Ouyang
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  2. Binbin Zhang
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Correspondence to Wei Ouyang.

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W. Ouyang research was supported by the National Natural Science Foundation of the People’s Republic of China (Grant 11801500) and the Yunnan Provincial Science and Technology Research Program (Grant 2017FD070). B. Zhang research was supported by the National Natural Science Foundation of the People’s Republic of China (Grant 11771384), the Yunnan Provincial Department of Education Research Fund (Grant 2019J0040), and the Fund for Fostering Talents in Kunming University of Science and Technology (no. KKSY 201807022).

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Ouyang, W., Zhang, B. Regularity of Newton’s iteration for general parametric variational system. J. Fixed Point Theory Appl. 21, 92 (2019). https://doi.org/10.1007/s11784-019-0733-8

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  • DOI: https://doi.org/10.1007/s11784-019-0733-8

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