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

Consider an operator equation () $B(u)+\text{ep}u=0$ in a real Hilbert space, where $\text{ep}>0$ 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,
2. 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$andequation(\ast \ast )$B(u)=0$ is solvable, the DSM yields$ solution to (**).