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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 methodfor solving ill-posed problems with monotone nonlinear operators $F$. Local andglobal existence of the unique solution to this evolution equation are proved,apparently for the firs time, under the only assumption that $F'(u)$ exists andis continuous with respect to $u$. The earlier published results required moresmoothness of $F$. The Dynamical Systems method (DSM) for solving equations$F(u)=0$ with monotone Fr\'echet differentiable operator $F$ is justified underthe above assumption apparently for the first time.

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