Middle Volga Mathematical Society Journal

DOI 10.15507/2079-6900.23.202102.185–192

Original article

ISSN 2587-7496 (Print)

ISSN 2079-6900 (Online)

MSC2020 65J15

Simplification method for nonlinear equations of monotone type in Banach space

I. P. Ryazantseva

Nizhny Novgorod State Tehnical University named after R. E. Alekseev (Nizhny Novgorod, Russian Federation)

Abstract. 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.

Key Words: Banach space, conjugate space, strictly convex space, $E$-space, monotone operator, $B$-monotone operator, dual map with gauge function, operator regularization method, perturbed data, convergence

For citation: I. P. Ryazantseva. Simplification method for nonlinear equations of monotone type in Banach space. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 23:2(2021), 185–192. DOI: https://doi.org/10.15507/2079-6900.23.202102.185–192

Submitted: 13.03.2021; Revised: 24.04.2021; Accepted: 04.05.2021