Convergence of a parametric continuation method

Abstract

The aim of this paper is to establish the semilocal convergence of a parameter based continuation method combining the Chebyshev's and the Super-Halley's methods for solving nonlinear equations in Banach spaces. The parameter α $in$ [0,1] be such that for α = 0 it reduces to the Chebyshev's method and for α = 1 to the Super-Halley's method. This convergence is established using recurrence relations under the assumption that the second order Fréchet derivative satisfies the ω-continuity condition. This condition is milder than the Lipschitz and the Hölder continuity conditions used for this purpose. A numerical example is given to show that the second order Fréchet derivative satisfies the ω-continuity condition even when it fails to satisfy the Lipschitz and the Hölder continuity conditions. A number of recurrence relations are derived based on two parameters. The existence and uniqueness regions along with a closed form of the error bounds in terms of a real parameter α $in$ [0, 1] for the solution x* is given. Two numerical examples are worked out to demonstrate the efficacy of the method. It is observed that our method gives better existence and uniqueness regions of the solution for both the examples when compared with the results obtained in [4] for both the Chebyshev's method (α = 0) and the Super-Halley's method (α = 1).