Simplification method for nonlinear equations of monotone type in Banach space

I. P. Ryazantseva

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 23 (2021) no. 2, pp. 185-192

Résumé

In a Banach space, we study an operator equation with a monotone operator $T.$ The operator is an operator from a Banach space to its conjugate, and $T=AC,$ where $A$ and $C$ are operators of some classes. The considered problem belongs to the class of ill-posed problems. For this reason, an operator regularization method is proposed to solve it. This method is constructed using not the operator $T$ of the original equation, but a more simple operator $A,$ which is $B$-monotone, $B=C^{-1}.$ The existence of the operator $B$ is assumed. In addition, when constructing the operator regularization method, we use a dual mapping with some gauge function. In this case, the operators of the equation and the right-hand side of the given equation are assumed to be perturbed. The requirements on the geometry of the Banach space and on the agreement conditions for the perturbation levels of the data and of the regularization parameter are established, which provide a strong convergence of the constructed approximations to some solution of the original equation. An example of a problem in Lebesgue space is given for which the proposed method is applicable.

Détail

Keywords: Banach space, conjugate space, strictly convex space, $E$-space, monotone operator, $B$-monotone operator, dual map with gauge function, operator regularization method, perturbed data
Mots-clés : convergence.