Affiliation:
1. Nizhny Novgorod State Tehnical University named after R. E. Alekseev
Abstract
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.
Publisher
National Research Mordovia State University MRSU
Reference4 articles.
1. I. P. Ryazantseva, “First-order continuous regularization methods for generalized variational inequalities”, Computational Mathematics and Mathematical Physics, 50:4 (2010), 606–619.
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3. Ya. Alber, I. Ryazantseva, Nonlinear ill-posed problems of monotone type, Springer, Dordrecht, 2006 DOI: https://doi.org/10.1007/1-4020-4396-1, 410 p.
4. I. P. Ryazantseva, [Selected topics of the theory of operators of monotone type], Nizhny Novgorod State Tehnical University named after R. E. Alekseev Publ., Nizhny Novgorod, 2008 (In Russ.), 272 p.