Convergence analysis of a variant of Newton-type method for generalized equations

ABSTRACT

Let X and Y be Banach spaces. Consider the following generalized equation problem: (1) $0\in f\left(x\right)+F\left(x\right),$(1) where f is Fréchet differentiable on an open subset Ω of X and F is set-valued mapping with closed graph acting between Banach spaces. In the present paper, we introduce a variant of Newton-type method for solving generalized equation (1). Semi-local and local convergence analysis are provided under the weaker condition that of utilized by Jean-Alexis and Pietrus [A variant of Newton's method for generalized equations, Rev. Colombiana Mat. 39 (2005), pp. 97–112]. In particular, this result extends the corresponding ones Jean-Alexis and Pietrus (2005). Finally, we present a numerical example to validate the convergence result of this method.

KEYWORDS:

• Set-valued mappings
• Lipschitz-like mappings
• generalized equations
• variant of Newton-type method
• semi-local convergence

AMS SUBJECT CLASSIFICATION 2010:

• 49J53
• 47H04
• 65K10

Acknowledgements

The author thanks to the referees and editor for their valuable comments and constructive suggestions which improved the presentation of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the author.