Convergence for a family of modified Chebyshev methods under weak condition

Abstract

In this paper, we focus on a family of modified Chebyshev methods and study the semilocal convergence for these methods. Different from the results in reference (Hernandez and Salanova, J. Comput. Appl. Math. 126:131---143, 2000), the Holder continuity of the second derivative is replaced by its generalized continuity condition, and the latter is weaker than the former. Using the recurrence relations, we establish the semilocal convergence of these methods and prove a convergence theorem to show the existence-uniqueness of the solution. The R-order of these methods is also analyzed. Especially, when the second derivative of the operator is Holder continuous, the R-order of these methods is at least 3 + 2p, which is higher than the one of Chebyshev method considered in reference (Hernandez and Salanova, J. Comput. Appl. Math. 126:131---143, 2000) under the same condition. Finally, we give some numerical results to show our approach.