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, ( ∗ ) u ̇ = − [ F ′ ( u ) + a ( t ) I ] − 1 [ F u ( t ) + a ( t ) u − f ] , u ( 0 ) = u 0 , is large. Here F : X → X is a continuously Fréchet differentiable operator in a Banach space X , a ( t ) : [ 0 , ∞ ) → C is a function, lim t → ∞ | 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 ( ∗ ) .