Dynamical systems method for solving nonlinear equations with non-smooth monotone operators

Abstract

Consider an operator equation (*) $B(u)+\ep u=0$ in a real Hilbert space, where $\ep>0$ is a small constant. The DSM (dynamical systems method) for solving equation (*) consists of a construction of 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, 3) the limit solves the equation $B(u)=0$. Existence of the unique solution is proved by the DSM for equation (*) with monotone hemicontinuous operators $B$ defined on all of$ If $\ep=0$ and equation (**) $B(u)=0$ is solvable, the DSM yields$ solution to (**).