Convergence of the variants of the Chebyshev–Halley iteration family under the Hölder condition of the first derivative

The present paper is concerned with the convergence problem of the variants of the Chebyshev–Halley iteration family with parameters for solving nonlinear operator equations in Banach spaces. Under the assumption that the first derivative of the operator satisfies the Hölder condition of order pp, a convergence criterion of order 1+p1+p for the iteration family is established. An application to a nonlinear Hammerstein integral equation of the second kind is provided.