On the DSM Newton-type method

A wide class of the operator equations F(u)=h in a Hilbert space is studied. Convergence of a Dynamical Systems Method (DSM), based on the continuous analog of the Newton method, is proved without any smoothness assumptions on the F??(u). It is assumed that F??(u) depends on u continuously. Existence and uniqueness of the solution to evolution equation $\dot{u}(t)=-F'(u(t))]^{-1}(F(u(t))-h)$ , u(0)=u ₀, is proved without assuming that F??(u) satisfies the Lipschitz condition. The method of the proof is new. This method is based on a novel version of the abstract inverse function theorem.