Convergence analysis on a class of improved Chebyshev methods for nonlinear equations in Banach spaces

In this paper, we study the semilocal convergence on a class of improved Chebyshev methods for solving nonlinear equations in Banach spaces. Different from the results for Chebyshev method considered in Hernandez and Salanova (J Comput Appl Math 126:131---143, 2000), these methods are free from the second derivative, the R-order of convergence is also improved. We prove a convergence theorem to show the existence-uniqueness of the solution. Under the convergence conditions used in Hernandez and Salanova (J Comput Appl Math 126:131---143, 2000), the R-order for this class of methods is proved to be at least $$3+2p$$ 3 + 2 p , which is higher than the ones of Chebyshev method considered in Hernandez and Salanova (J Comput Appl Math 126:131---143, 2000) and the variant of Chebyshev method considered in Hernandez (J Optim Theory Appl 104(3): 501---515, 2000) under the same conditions.