•  

Iterative regularization method for finding the generalized fixed point of non-stretching operator on the set of Hilbert space

TitleIterative regularization method for finding the generalized fixed point of non-stretching operator on the set of Hilbert space
AuthorsI. P. Ryazantseva1
1Nizhny Novgorod state technical University n. a. R. E. Alekseeva
AnnotationFor the non-stretching operator $A$, the concept of a generalized fixed point on a convex closed set $\Omega$ of the Hilbert space $H$ is introduced. The problem of finding such a point belongs to the class of incorrect ones, so let`s assume that the problem in question is solvable, and the operator $A$ and the set $\Omega$ are perturbed. To find its solution, an operator regularization method with exact data is constructed, in which the projection operator is used on the set $\Omega$. Conditions are established under which the solution of the constructed auxiliary regularized problem converges according to the norm of the space $H$ to the normal generalized fixed point $A$ on $\Omega$. Next, for the problem with the perturbed set $\Omega$ and the perturbed operator $A$, an implicit regularized iterative process is constructed, and conditions are established that ensure strong convergence in $H$ of the constructed iterative approximations to the normal generalized fixed point of the non-stretching operator. Examples of parametric functions that pecivaing the convergence of the constructed approximations to normal generalized fixed point of the operator $A$ on the set $\Omega$.
KeywordsHilbert space, a convex closed set, narastayushey operator, operator regularization method, regularized iterational process, fixed point design operator on a convex closed set, the perturbed data, convergence.
CitationRyazantseva I. P. ''Iterative regularization method for finding the generalized fixed point of non-stretching operator on the set of Hilbert space'' [Electronic resource]. Proceedings of the International Scientific Youth School-Seminar "Mathematical Modeling, Numerical Methods and Software complexes" named after E.V. Voskresensky (Saransk, July 14-18, 2022). Saransk: SVMO Publ, 2022. - pp. 168-177. Available at: https://conf.svmo.ru/files/2022/papers/paper27.pdf. - Date of access: 30.11.2022.