The dynamical systems method for solving nonlinear equations with monotone operators

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Show simple item record Hoang, N. S. Ramm, Alexander G. 2011-05-25T14:48:24Z 2011-05-25T14:48:24Z 2011-05-25
dc.description.abstract A review of the authors’ results is given. Several methods are discussed for solving nonlinear equations F(u) = f, where F is a monotone operator in a Hilbert space, and noisy data are given in place of the exact data. A discrepancy principle for solving the equation is formulated and justified. Various versions of the Dynamical Systems Method (DSM)for solving the equation are formulated. These versions of the DSM include a regularized Newton-type method, a gradient-type method, and a simple iteration method. A priori and a posteriori choices of stopping rules for these methods are proposed and justified. Convergence of the solutions, obtained by these methods, to the minimal norm solution to the equation F(u) = f is proved. Iterative schemes with a posteriori choices of stopping rule corresponding to the proposed DSM are formulated. Convergence of these iterative schemes to a solution to equation F(u) = f is justified. New nonlinear differential inequalities are derived and applied to a study of large-time behavior of solutions to evolution equations. Discrete versions of these inequalities are established. en_US
dc.relation.uri en_US
dc.rights Electronic version of an article published as Asian-European Journal of Mathematics, 3(1),2010,57-105, doi:10.1142/S1793557110000027 © copyright World Scientific Publishing Company, en
dc.subject Ill-posed problems en_US
dc.subject Nonlinear operator equations en_US
dc.subject Monotone operators en_US
dc.subject Non-linear inequalities en_US
dc.subject Dynamical systems method en_US
dc.title The dynamical systems method for solving nonlinear equations with monotone operators en_US
dc.type Article (author version) en_US 2010 en_US
dc.citation.doi doi:10.1142/S1793557110000027 en_US
dc.citation.epage 105 en_US
dc.citation.issue 1 en_US
dc.citation.jtitle Asian European Journal of Mathematics en_US
dc.citation.spage 57 en_US
dc.citation.volume 3 en_US
dc.contributor.authoreid ramm en_US

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