Global convergence for ill-posed equations with monotone operators: the dynamical systems method

Consider an operator equation F(u) = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator F'(u) is not boundedly invertible, and well-posed otherwise. If F is monotone C2loc(H) operator, then we construct a Cauchy problem, which has the following properties: (1) it has a global solution for an arbitrary initial data, (2) this solution tends to a limit as time tends to infinity and (3) the limit is the minimum norm solution to the equation F(u) = 0. An example of applications to linear ill-posed operator equation is given. (letter to the editor)