一类发展方程解的存在性及单调算子方程的DSM的证明
Existence of solution to an evolution equation and a justification of the DSM for equations with monotone operators
摘要：
本文研究了在求解单调算子方程的动力系统方法（DSM）研究中出现的一个发展方程。发展方程是求解单调非线性算子不适定问题的正则化牛顿方法$F$的连续模拟。在唯一的假设$F'(u)$存在且关于$u$连续的条件下，证明了该发展方程的局部和整体唯一解的存在性.早期发表的结果要求$F$更平滑。在上述假设下，首次证明了求解具有单调Fréchet可微算子$F$的方程$F(u)=0$的动力系统方法（DSM）的正确性.
英文摘要：
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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