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)=fF(y)=f solvable by a DSM (dynamical systems method) Newton-type method, equation(∗∗)u̇=−[F′(u)+a(t)I]−1[Fu(t)+a(t)u−f],u(0)=u0, is large. Here F:X→XF:X→X is a continuously Fréchet differentiable operator in a Banach space XX, a(t):[0,∞)→Ca(t):[0,∞)→C is a function, limt→∞|a(t)|=0limt→∞|a(t)|=0, and there exists a y∈Xy∈X such that F(y)=fF(y)=f. Under weak assumptions on FF and aa it is proved that ∃!u(t)∀t≥0;∃u(∞);F(u(∞))=f. This justifies the DSM (∗)(∗).