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Dynamical Systems Method for ill-posed equations with monotone operators

A. G. Ramm
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Abstract

Consider an operator equation (*) $B(u)-f=0$ in a real Hilbert space. Let us call this equation ill-posed if the operator $B'(u)$ 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 $B$.

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A. G. Ramm. 2004-09-19. Dynamical Systems Method for ill-posed equations with monotone operators. https://doi.org/10.1016/j.cnsns.2003.07.002

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