The Dynamical Systems Method for solving nonlinear equations with monotone operators

A review of the authors's 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 methods consist of 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.Comment: 50p

Dynamical systems method for solving linear finite-rank operator equations

A version of the Dynamical Systems Method (DSM) for solving ill-conditioned linear algebraic systems is studied in this paper. An {\it a priori} and {\it a posteriori} stopping rules are justified. An iterative scheme is constructed for solving ill-conditioned linear algebraic systems.Comment: 16 pages, 1 table, 1 figur

Some nonlinear inequalities and applications

Sufficient conditions are given for the relation $\lim_{t\to\infty}y(t) = 0$ to hold, where $y(t)$ is a continuous nonnegative function on $[0,1)$ satisfying some nonlinear inequalities. The results are used for a study of large time behavior of the solutions to nonlinear evolution equations. Example of application is given for a solution to some evolution equation with a nonlinear partial differential operator.Comment: 16 page