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How large is the class of operator equations solvable by a DSM Newtontype method?
"... It is proved that the class of operator equations F (y) = f solvable by a DSM (Dynamical Systems Method) Newtontype method ˙u = −[F ′ (u) + a(t)I] −1 [F u(t) + a(t)u − f], u(0) = u0, (∗) is large. Here F: X → X is a continuously Fréchet differentiable operator in a Banach space X, a(t) : [0, ∞) ..."
Abstract

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It is proved that the class of operator equations F (y) = f solvable by a DSM (Dynamical Systems Method) Newtontype method ˙u = −[F ′ (u) + a(t)I] −1 [F u(t) + a(t)u − f], u(0) = u0, (∗) is large. Here F: X → X is a continuously Fréchet differentiable operator in a Banach space X, a(t) : [0, ∞) → C is a function, limt→ ∞ a(t)  = 0, and there exists a y ∈ X such that F (y) = f. Under weak assumptions on F and a it is proved that This justifies the DSM (*). ∃!u(t) ∀t ≥ 0; ∃u(∞); F (u(∞)) = f.
DSM of Newton type for solving operator equations F(u) = f with minimal smoothness assumptions on F
 JOURN. COMP. SCI AND MATH., 3, N1/2, (2010), 355
, 2010
"... This paper is a review of the authors’ results on the DSM (Dynamical Systems Method) for solving operator equation (*) F (u) = f. It is assumed that (*) is solvable. The novel feature of the results is the minimal assumption on the smoothness of F. It is assumed that F is continuously Fréchet diffe ..."
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This paper is a review of the authors’ results on the DSM (Dynamical Systems Method) for solving operator equation (*) F (u) = f. It is assumed that (*) is solvable. The novel feature of the results is the minimal assumption on the smoothness of F. It is assumed that F is continuously Fréchet differentiable, but no smoothness assumptions on F ′ (u) are imposed. The DSM for solving equation (*) is developed. Under weak assumptions global existence of the solution u(t) is proved, the existence of u(∞) is established, and the relation F(u(∞)) = f is obtained. The DSM is developed for a stable solution of equation (*) when noisy data fδ are given, ‖‖f − fδ‖‖ ≤ δ.