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

By N. S. Hoang and A. G. Ramm


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 first 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. Key words. Dynamical systems method (DSM), nonlinear operator equations, monotone operators. subject classifications. Primary 47J05; 47J06; 47J35 1. Introduction The Dynamical Systems Method (DSM) for solving an operator equation F (u) = f in a Hilbert space consists of finding a nonlinear map Φ(u,t) such that the Cauchy problem ˙u = Φ(t,u), u(0) = u0; ˙u: = du dt

Year: 2010
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