Dynamical Systems Method for ill-posed equations with monotone operators

Abstract

Consider an operator equation (*) $B(u)-f=0$ in a real Hilbert space. Let us call this equation ill-posed if the operator $B^{\prime }(u)$ is not boundedly invertible, and well-posed otherwise. 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 is the minimal-norm solution to the equation $B(u)=f$. A global convergence theorem is proved for DSM for equation (*) with monotone $C_{loc}^2$ operators $B$.

References in corpus (2)

• Dynamical systems method for solving operator equations
• Global convergence for ill-posed equations with monotone operators: the dynamical systems method

Cited by in corpus (3)

• Dynamical Systems Gradient method for solving nonlinear equations with monotone operators
• Dynamical systems method for solving nonlinear equations with monotone operators
• Discrete L'Hospital's rule