A note on the semilocal convergence of Chebyshev's method (Q2847601)

An iterative algorithm is presented to solve nonlinear equations. This method generalizes the Chebyshev's method to equations in Banach spaces. The Chebyshev method is also well-known as Euler's method, Euler-Chebyshev method or Schröder's method, respectively. A family of iterative processes was studied by \textit{J. M. Gutiérez} and \textit{M. A. Hernández} [Bull. Aust. Math. Soc. 55, No.~1, 113--130 (1997; Zbl 0893.47043)]. They proved the Chebyshev method has (at least) quadratic convergence order.NEWLINENEWLINE In the paper under review a convergence theorem is provided that guarantees cubic convergence. The Kantorovich-like theorem for the Chebyshev method follows \textit{Sh. Zheng}'s and \textit{D. Robbie}'s idea for Halley's method [J. Aust. Math. Soc., Ser. B 37, No. 1, 16--25 (1995; Zbl 0842.65035)].