Сontinuous regularization analog of Newton method for m-accretive equations

I. P. Ryazantseva

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 19 (2017) no. 1, pp. 77-87

Résumé

Paper studies nonlinear equations with approximate data (i.e. the operator and the right-hand side of an operator equation) with the Frechet-differentiable m-accretive operator in Banach space. For such equation a regularized continuous analog of Newton's method is constructed; sufficient conditions for method’s strong convergence to a certain uniquely determined solution of the given equation are obtained. Previously we prove auxiliary assertions of continuity of values that are determined in terms of regularized solutions and their derivatives. The approximations of the operator are assumed to be differentiable. One-valued solvability of the differential equation defining the investigated regularization method is proved. In the proof of the continuous method convergence known convergence of operator regularization method for accretive equations is used. Requirements on the geometry of Banach space and its conjugate are performed for a wide class of Banach spaces. For approximate right-hand side definition of the equation the cases of the unperturbed and of the perturbed operator are studied separately. Examples are built for parametric functions that are used in the equation that defines the method studied. The example is specified of an operator arising in the theory of a scalar density function, for which the conditions of convergence of the method are performed.

Détail

Keywords: Banach space, m-accretive operator, duality mapping, Newton’s method, continuous method, perturbed data, regularization
Mots-clés : convergence.