  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.
