Convergence of the family of the deformed Euler-Halley iterations under the Hölder condition of the second derivative

The convergence problem of the family of the deformed Euler-Halley iterations with parameters for solving nonlinear operator equations in Banach spaces is studied. Under the assumption that the second derivative of the operator satisfies the Hölder condition, a convergence criterion of the order 2+p of the iteration family is established. An application to a nonlinear Hammerstein integral equation of the second kind is provided.