A new convergence theorem for the inexact Newton methods based on assumptions involving the second Fréchet derivative

We provide sufficient conditions for the convergence of inexact Newton methods to a solution of a nonlinear equation in a Banach space. Earlier results have used conditions on the first Fréchet-derivative. Our results differ from earlier results in that we use Lipschitz conditions on the second Fréchet-derivative of the operator involved. Finally we consider some particular situations to which our general results apply.