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

Dynamical systems method for solving nonlinear equations with monotone operators

A version of the Dynamical Systems Method (DSM) for solving ill-posed nonlinear equations with monotone operators in a Hilbert space is studied in this paper. An a posteriori stopping rule, based on a discrepancy-type principle is proposed and justified mathematically. The results of two numerical experiments are presented. They show that the proposed version of DSM is numerically efficient. The numerical experiments consist of solving nonlinear integral equations.Comment: 19 pages, 4 figures, 4 table

Nonlinear differential inequality

A nonlinear inequality is formulated in the paper. An estimate of the rate of growth/decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations. It can be applied to a study of global existence of solutions to nonlinear PDE

Existence of solution to an evolution equation and a justification of the DSM for equations with monotone operators

An evolution equation, arising in the study of the Dynamical Systems Method (DSM) for solving equations with monotone operators, is studied in this paper. The evolution equation is a continuous analog of the regularized Newton method for solving ill-posed problems with monotone nonlinear operators $F$. Local and global existence of the unique solution to this evolution equation are proved, apparently for the firs time, under the only assumption that $F'(u)$ exists and is continuous with respect to $u$. The earlier published results required more smoothness of $F$. The Dynamical Systems method (DSM) for solving equations $F(u)=0$ with monotone Fr\'echet differentiable operator $F$ is justified under the above assumption apparently for the first time