Convergence for variants of Chebyshev–Halley methods using restricted convergence domains

Abstract

We present a local convergence analysis for some variants of Chebyshev–Halley methods of approximating a locally unique solution of a nonlinear equation in a Banach space setting. We only use hypotheses reaching up to the second Fréchet derivative of the operator involved in contrast to earlier studies using Lipschitz hypotheses on the second Fréchet derivative and other more restrictive conditions. This way the applicability of these methods is expanded. We also show how to improve the semilocal convergence in the earlier studies under the same conditions using our new idea of restricted convergence domains leading to: weaker sufficient convergence criteria, tighter error bounds on the distances involved and an at least as precise information on the location of the solution. Numerical examples where earlier results cannot be applied but our results can, are also provided.