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, <CitationRef CitationID="CR5">2000</CitationRef>), 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, <CitationRef CitationID="CR5">2000</CitationRef>), the R-order for this class of methods is proved to be at least <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$3+2p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mi>p</mi> </mrow> </math> </EquationSource> </InlineEquation>, which is higher than the ones of Chebyshev method considered in Hernández and Salanova (J Comput Appl Math 126:131–143, <CitationRef CitationID="CR5">2000</CitationRef>) and the variant of Chebyshev method considered in Hernández (J Optim Theory Appl 104(3): 501–515, <CitationRef CitationID="CR16">2000</CitationRef>) under the same conditions. Copyright Springer Science+Business Media New York 2015