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 Hernández 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 Hernández 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+2p, which is higher than the ones of Chebyshev method considered in Hernández and Salanova (J Comput Appl Math 126:131–143, 2000) and the variant of Chebyshev method considered in Hernández (J Optim Theory Appl 104(3): 501–515, 2000) under the same conditions.

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