Refinement of convergence conditions of the Chebyshev method

The iterative Chebyshev method of an approximate solution of equations of the form $F(x)=0$ in Banach spaces is studied, assuming that $F''$ satisfies the Lipschitz condition. Accurate (attainable) estimates of the domains of existence and uniqueness of solution, non-refinable conditions of existence and convergence of the Chebyshev method as well as asymptotic estimates of the rate of convergence have been obtained.