Semilocal convergence of a family of third-order Chebyshev-type methods under a mild differentiability condition

The aim of this paper is to establish the semilocal convergence of a family of third-order Chebyshev-type methods used for solving nonlinear operator equations in Banach spaces under the assumption that the second Fréchet derivative of the operator satisfies a mild ω-continuity condition. This is done by using recurrence relations in place of usual majorizing sequences. An existence–uniqueness theorem is given that establishes the R-order and existence–uniqueness ball for the method. Two numerical examples are worked out and comparisons being made with a known result.

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