2010/12/13 by А. Г. Рамм, Ramm, A. G.
Computer Science · Mathematics · #4705 #4706 #47J35 #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1012.2762
openalex publication_date 2010/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Dynamical Systems Method (DSM) is justified for solving operator equations F(u)=f, where F is a nonlinear operator in a Hilbert space H. It is assumed that F is a global homeomorphism of H onto H, that F∈ C1loc, that is, it has a continuous with respect to u Fréchet derivative F'(u), that the operator [F'(u)]-1 exists for all u∈ H and is bounded, ||[F'(u)]-1||≤ m(u), where m(u)>0 is a constant, depending on u, and not necessarily uniformly bounded with respect to u. It is proved under these assumptions that the continuous analog of the Newton's method u=-[F'(u)]-1(F(u)-f), u(0)=u0, (*) converges strongly to the solution of the equation F(u)=f for any f∈ H and any u0∈ H. The global (and even local) existence of the solution to the Cauchy problem (*) was not established earlier without assuming that F'(u) is Lipschitz-continuous. The case when F is not a global homeomorphism but a monotone operator in H is also considered.