Abstract
In this paper, we establish the Newton-Kantorovich convergence theorem of a fourth-order super-Halley method under weaker conditions in Banach space, which is used to solve the nonlinear equations. Finally, some examples are provided to show the application of our theorem.
Highlights
1 Introduction For a number of problems arising in scientific and engineering areas one often needs to find the solution of nonlinear equations in Banach spaces
Where F is a third-order Fréchet-differentiable operator defined on a convex subset of a Banach space X with values in a Banach space Y
3 Newton-Kantorovich convergence theorem we give a theorem to establish the semilocal convergence of the method ( ) in weaker conditions, the existence and uniqueness of the solution and the domain in which it is located, along with a priori error bounds, which lead to the R-order of convergence of at least four of the iterations ( )
Summary
We apply majorizing functions to prove the semilocal convergence of the method ( ) to solve nonlinear equations in Banach spaces and establish its convergence theorems in [ ]. Assume that all conditions (C )-(C ) hold and x ∈ , h = Kβη ≤ / , B(x , t∗) ⊂ , the sequence {xn} generated by the method ( ) is well defined, xn ∈ B(x , t∗) and converges to the unique solution x∗ ∈ B(x , t∗∗) of F(x), and xn – x∗ ≤ t∗ – tn, where t∗∗ = + We know the conditions of Theorem cannot be satisfied by some general nonlinear operator equations.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
