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Applicable Analysis
An International Journal
Volume 101, 2022 - Issue 1
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Articles

Convergence properties of a restricted Newton-type method for generalized equations with metrically regular mappings

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Pages 14-34 | Received 27 Sep 2017, Accepted 30 Sep 2017, Published online: 20 Oct 2017
 

Abstract

Let X and Y be Banach spaces. Let f:XY be a Fréchet differentiable function and F:X2Y be a set-valued mapping with closed graph. In this paper, a restricted Newton-type method has been introduced for solving the generalized equations of the form 0f(x)+F(x). Under some suitable conditions, we will establish the convergence criteria of the restricted Newton-type method, which ensures the existence and the convergence of any sequence generated by this method. More precisely, when the Fréchet derivative of f is continuous and Lipschitz continuous as well as f+F is metrically regular, we analyze the semilocal and local convergence of the restricted Newton-type method. In addition, an example is given to show the reason for considering the metrically regular property instead of Lipschitz-like property of set-valued mapping in this paper.

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Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research work of first author is fully supported by CAS-President International Fellowship Initiative(PIFI), Chinese Academy of Sciences, China and research work of the second author is partially supported by the National Natural Science Foundation of China [grant number 11331012; 11321061; 11461161005].

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