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 $F^{\mathrm{\prime }}(u)$ 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.