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arXiv · 0909.0517

Existence of solution to an evolution equation and a justification of the DSM for equations with monotone operators

N. S. Hoang·A. G. Ramm
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Abstract

An evolution equation, arising in the study of the Dynamical Systems Method (DSM) for solving equations with monotone operators, is studied in this paper. The evolution equation is a continuous analog of the regularized Newton method for solving ill-posed problems with monotone nonlinear operators $F$. Local and global existence of the unique solution to this evolution equation are proved, apparently for the firs time, under the only assumption that $F'(u)$ exists and is continuous with respect to $u$. The earlier published results required more smoothness of $F$. The Dynamical Systems method (DSM) for solving equations $F(u)=0$ with monotone Fréchet differentiable operator $F$ is justified under the above assumption apparently for the first time.

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N. S. Hoang, A. G. Ramm. 2009-09-02. Existence of solution to an evolution equation and a justification of the DSM for equations with monotone operators. https://arxiv.org/abs/0909.0517

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