- Open Access
Newton-Kantorovich convergence theorem of a new modified Halley’s method family in a Banach space
© Lin et al.; licensee Springer. 2013
- Received: 19 July 2013
- Accepted: 4 September 2013
- Published: 19 November 2013
A Newton-Kantorovich convergence theorem of a new modified Halley’s method family is established in a Banach space to solve nonlinear operator equations. We also present the main results to reveal the competence of our method. Finally, two numerical examples arising in the theory of the radiative transfer, neutron transport and in the kinetic theory of gasses are provided to show the application of our theorem.
- Banach Space
- Iterative Method
- Radiative Transfer
- Positive Root
- Positive Real Root
where F is defined on an open convex subset Ω of a Banach space X with values in a Banach space Y.
For , , the method becomes Chebyshev’s iterative method (see ). For , , the method becomes inverse-free Jarratt iterative method (see [14, 15]). In this paper, we establish a Kantorovich-type third-order convergence theorem for this kind of method by using majorizing function to improve the result .
where are two positive real roots of the function . Then, for , the sequence generated by (3) is well defined, and converges to the unique solution of equation (1) in .
Then for , the sequence generated by (3) is well defined, and converges to the unique solution of equation (1) in .
To prove Theorems 1 and 2, we first give some lemmas.
Lemma 1 If , the polynomial has two positive real roots , (let ), and a negative real root ().
Proof From definition of the function , there follows that , , hence has a negative root. Denote it . We get that has the unique positive root , and for , . So, the necessary and sufficient condition that has two positive roots for is that the minimum of satisfies , that is also . This completes the proof of Lemma 1. □
Lemma 2 (see )
Then for , , are increasing and converge to .
This completes the proof of Lemma 3. □
This completes the proof of Lemma 4. □
So, the sequence generated by (3) is well defined, and converges to the solution of equation (1) on . Now, we prove the uniqueness. If is also the solution of equation (1) in , then, by the proof of Lemma 1, we know that , .
we have . This completes the proof of uniqueness. Thus, the proof of Theorem 1 is complete. □
Using the same proof method as in Theorem 1, we get assertion of Theorem 2. □
In this section, we apply the convergence ball result and show two numerical examples.
Moreover, by Theorem 2, we get that the sequence () generated by (3) is well defined and convergent.
this means that the hypotheses of Theorem 2 hold.
This work is supported by the National Basic Research 973 Program of China (No. 2011JB105001), the National Natural Science Foundation of China (Grant No. 11371320), the Foundation of Science and Technology Department (Grant No. 2013C31084) of Zhejiang Province and the Foundation of the Education Department (No. 20120040, Y201329420) of Zhejiang Province of China and by the Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology, Czech Republic.
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