Convergence for a class of improved sixth-order Chebyshev‐Halley type methods

In this paper, we consider the semilocal convergence on a class of improved Chebyshev–Halley type methods for solving $F\left(x\right)=0,$ where F: Ω ⊆ X → Y is a nonlinear operator, X and Y are two Banach spaces, Ω is a non‐empty open convex subset in X. To solve the problems that F‴(x) is unbounded in Ω and it can not satisfy the whole Lipschitz or Hölder continuity, ∥F‴(x)∥ ≤ N is replaced by $\parallel F^{‴}\left(x_0\right)\parallel \le \overline{N},$ for all x ∈ Ω, where $N,\overline{N}\ge 0,$ x 0 is an initial point. Moreover, F‴(x) is assumed to be local Hölder continuous. So the convergence conditions are relaxed. We prove an existence‐uniqueness theorem for the solution, which shows that the R‐order of these methods is at least $5+q,$ where q ∈ (0, 1]. Especially, when F‴(x) is local Lipschitz continuous, the R‐order will become six.