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.