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

2003
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Journal of Physics A: Mathematical and General
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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 $C^2_{loc}(H)$ 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, 3) the limit is the minimum norm solution to the equation $F(u)=0$. Example of

doi:10.1088/0305-4470/36/16/102
fatcat:7swcygny4ndijfnb7is6ojtco4