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(x)=0,F(x)=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 Ho¨lder continuity, ‖F ′′′(x)‖ ≤ N   is replaced by ∥F′′′(x0)∥≤N¯, for all x ∈ Ω  , where N,N¯≥0,x0 is an initial point. Moreover, F ′′′(x  ) is assumed to be local Ho¨lder 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,5+q, where q ∈ (0, 1]. Especially, when F ′′′(x) is local Lipschitz continuous, the R-order will become six.