Application of iterative processes of R-order at least three to operators with unbounded second derivative

In this paper, we apply a family of Newton-like methods, which contains the best known iterative processes, to operator equations where the usual convergence conditions are relaxed. We weaken these conditions by assuming ∥F″(x0)∥ ⩽ α and ∥F″(x) − F″(y)∥ ⩽ ω(∥x − y∥), with ω a non-decreasing continuous real function. Our results include the ones obtained when the convergence of the family is studied under Lipschitz continuous or Hölder continuous conditions for the second derivative of the operator involved. To finish, we apply the study to boundary value problems.