A note on the Halley method in Banach spaces
Applied Mathematics and Computation
Third-order iterative methods for operators with bounded second derivative
ICCAM '96 Proceedings of the seventh international congress on Computational and applied mathematics
Reduced recurrence relations for the Chebyshev method
Journal of Optimization Theory and Applications
Second-derivative-free variant of the Chebyshev method for nonlinear equations
Journal of Optimization Theory and Applications
Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method
Journal of Computational and Applied Mathematics
Geometric constructions of iterative functions to solve nonlinear equations
Journal of Computational and Applied Mathematics
| Hi-index | 0.00 |
In this paper, we study the semilocal convergence and R-order for a class of modified Chebyshev-Halley methods for solving non-linear equations in Banach spaces. To solve the problem that the third-order derivative of an operator is neither Lipschitz continuous nor Hölder continuous, the condition of Lipschitz continuity of third-order Fréchet derivative considered in Wang et al. (Numer Algor 56:497---516, 2011) is replaced by its general continuity condition, and the latter is weaker than the former. Furthermore, the R-order of these methods is also improved under the same condition. By using the recurrence relations, a convergence theorem is proved to show the existence-uniqueness of the solution and give a priori error bounds. We also analyze the R-order of these methods with the third-order Fréchet derivative of an operator under different continuity conditions. Especially, when the third-order Fréchet derivative is Lipschitz continuous, the R-order of the methods is at least six, which is higher than the one of the method considered in Wang et al. (Numer Algor 56:497---516, 2011) under the same condition.