Forcing sequences and inexact Newton iterates in Banach space

We use inexact Newton iterates to approximate a solution of a nonlinear equation in a Banach space. Solving a nonlinear equation using Newton iterates at each stage is very expensive in general. That is why we consider inexact Newton methods, where the Newton equations are solved only approximately and in some unspecified manner. In the elegant paper 1], natural assumptions under which the forcing sequence is uniformly less than one were given based on the first Fréchet-derivative of the operator involved. Here, we use assumption on the second Fréchet-derivative. This way, we essentially reproduce all results found earlier. However, our upper error bounds on the distances involved are smaller.