How large is the class of operator equations solvable by a DSM Newton-type method?

It is proved that the class of operator equations F(y) = f solvable by a DSM (dynamical systems method) Newton-type method, u̇ = −[F (u) + a(t)I]−1[Fu(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 ∃!u(t) ∀t ≥ 0; ∃u(∞); F(u(∞)) = f . This justifies the DSM (∗). © 2010 Elsevier Ltd. All rights reserved.