Local convergence of Newton’s method for solving generalized equations with monotone operator

ABSTRACT

In this paper, we study Newton’s method for solving the generalized equation $F(x)+T(x)\ni 0$ in Hilbert spaces, where F is a Fréchet differentiable function and T is set-valued and maximal monotone. We show that this method is locally quadratically convergent to a solution. Using the idea of a majorant condition on the nonlinear function, which is associated with the generalized equation, the convergence of the method, the optimal convergence radius, and results of the convergence rate are established. The advantage of working with a majorant condition rests in the fact that it allows unifying of several convergence results pertaining to Newton’s method.

KEYWORDS:

• Generalized equation
• Newton’s method
• majorant condition
• Banach lemma

AMS SUBJECT CLASSIFICATIONS:

• Primary 65K05
• 65K15
• 49M15
• 49M37

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This works was supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior [grant number 10.13039/501100002322].