Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method (Q5928287)

The paper is devoted to a weakening of the known assumptions for the convergence of the third-order Chebyshev method that is intended for the solution of nonlinear equations \(F(x)=0\) in Banach spaces. The method is defined as follows: NEWLINE\[NEWLINE x_{n + 1} = x_{n} - [I + \tfrac 12 F'(x_{n})^{-1} F''(x_{n}) F'(x_{n})^{-1} F(x_{n})] F'(x_{n})^{-1} F(x_{n}). NEWLINE\]NEWLINE Here the authors' previous results [Appl. Math. Comput. 95, No. 1, 51-62 (1998; Zbl 0943.65071)] are developed. The only requirement ensuring semilocal convergence of the method is that the operator \(F\)'s second Fréchet derivative be bounded and a Lipschitz-type condition be verified. The improving convergence results are obtained due to a new technique developed by the authors. They construct some numerical successions and certain recurrence relations which allow to check the convergence of the method. The obtained results are applied to the resolution of two nonlinear integral equations of Fredholm type.