- Research
- Open access
- Published:
Convergence of Halley’s method for operators with the bounded second Fréchet-derivative in Banach spaces
Journal of Inequalities and Applications volume 2013, Article number: 260 (2013)
Abstract
In this paper, we present a semi-local convergence analysis of Halley’s method for approximating a locally unique solution of a nonlinear equation in a Banach space setting, where we assume that the second Fréchet-derivative is bounded. Numerical examples are used to show that the new convergence criteria can provide better information than those provided by the convergence criteria developed earlier.
MSC:65G99, 65J15, 65H10, 47H17, 49M15.
1 Introduction
In this paper, we are concerned with the problem of approximating a locally unique solution of the nonlinear equation
where F is twice Fréchet-differentiable operator defined on a nonempty open and convex subset of a Banach space X with values in a Banach space Y.
Many problems from computational sciences and other disciplines can be brought in a form similar to equation (1.1) using mathematical modeling [1–3]. The solutions of these equations can rarely be found in a closed form. That is why most solution methods for these equations are iterative. The study about convergence matter of iterative procedures is usually based on two types: semi-local and local convergence analysis. The semi-local convergence matter is, based on the information around an initial point, to give conditions ensuring the convergence of the iterative procedure; while the local one is, based on the information around a solution, to find estimates of the radii of convergence balls.
In this paper, we provide a semi-local convergence analysis for Halley’s method defined by [4–6]
for each , where
The convergence of Halley’s method has a long history and has been studied by many authors (cf. [1, 2, 4–7] and the references therein). The most popular conditions for the semi-local convergence of Halley’s method are given as follows.
(C1) There exists such that , the space of bounded linear operator from Y into X;
(C2) for any ;
(C3) for each x in D and ;
(C4) for each and .
The corresponding sufficient convergence condition [1, 7] is given by
There are simple examples to show that (C4) is not satisfied. As an example, let , and define on D by
Then we have
Therefore, there is no constant K satisfying (C4). Other examples where (C4) is not satisfied can be found in [2].
Using the recurrent relations, Equerro and Hernández [7] expanded the applicability of Halley’s method by dropping condition (C4) and replacing (1.3) by
In the present study, we show how to expand even further the applicability of Halley’s method using (C1), (C2), (C3) and the center-Lipschitz condition:
(C5) for each and .
We have that
holds in general and can be arbitrarily large [2, 4]. The semi-local convergence analysis of Halley’s method requires obtaining upper bounds on the norms . In the literature, (C3) is used to obtain
for each . However, if we use (C5) less expensive and tighter if (see (1.5)), the estimate
for each is obtained. This modification can lead to a tighter convergence analysis and weaker sufficient convergence conditions or error bounds or the location of the solution for Halley’s method (see numerical examples). The new approach has already led to such advantages in the case of Newton-type methods [1, 2, 4–6].
We use a different approach than recurrent relations in our semi-local convergence analysis. The paper is organized as follows. Section 2 contains the semi-local convergence of Halley’s method, whereas the numerical examples are presented in the concluding section.
2 Semi-local convergence
We present the semi-local convergence analysis of Halley’s method. We shall use an additional condition.
Define
(C6) Suppose that
and there exists
which is the minimal positive zero of a function g on given by
Then we have
Define
Then we have
We can rewrite (2.4) as
Let us introduce another condition needed in our main theorem.
(C7) Suppose that
We refer to (C1), (C2), (C3), (C5), (C6) and (C7) as the (C) conditions. Let , stand, respectively, for the open and closed balls in X with center x and radius . Then we can show the following semi-local convergence result for Halley’s method.
Theorem 2.1 Let be continuously twice Fréchet differentiable, where X, Y are Banach spaces and D is open and convex. Suppose the (C) conditions and . Then the Halley sequence generated by (1.2) is well defined, remains in for all and converges to a unique solution of the equation . Moreover, the following error estimate holds: for each ,
Furthermore, if there exists such that
and
then is the only solution of equation in .
Proof We have, using (C1)-(C3), that
It follows from (2.9) and the Banach lemma on invertible operators [2, 3] that exists and
Then, by (C2), (1.2) and the above estimate, we get
We have, by (C5),
Hence exists and
In view of Halley’s iteration, we can write
or
or
Hence we get
Moreover, we have
Then exists and
So, is well defined and, using (1.2), we get
Therefore, we have
Hence we have .
The above shows the following items are true for :
-
(a)
exists and
-
(b)
-
(c)
-
(d)
exists and
-
(e)
is well defined and
-
(f)
.
Here, the sequence is defined by
for each . The rest will be shown by induction.
Assume that (a)-(f) are true for all natural integers , where is a fixed integer. Then exists since and
Hence exists and
Next, we estimate . It follows from Halley’s method that
or
Hence we get
and
Thus exists and
Therefore, is well defined. Moreover, we obtain
Furthermore, we have
Hence we deduce that , which completes the induction for (a)-(f).
Let m be a natural integer. Then we have
It follows that is a complete sequence in a Banach space X and as such it converges to some (since is a closed set). By letting in (2.10), we obtain . We also have
To show the uniqueness part, let be a solution of the equation in . Let . Using (C5), we have in turn that
It follows from the Banach lemma on invertible operators [2–4] that . Using the identity
we deduce . This completes the proof. □
3 Numerical examples
In this section, we give some examples to show the application of our theorem.
Example 3.1 Let us define a scalar function on with initial point . Then we have that
So, , and . Moreover, we have, for any ,
and
That is, we can choose and in conditions (C3) and (C5), respectively. Hence, we obtain , and . Furthermore, it is easy to get by using iterative methods such as the secant method. Then we have , and . So, conditions (C6) and (C7) are satisfied. It is clear that
Now, all the conditions in Theorem 2.1 are true and Theorem 2.1 applies. We can compare our results to the ones in [7]. Using (1.4), we get
So, the conditions of [7] are also satisfied. The uniqueness ball is , where
Then we get . That is, we provide better information on the location of the solution. Moreover, by (2.7) and (2.8), we can set , which extends the uniqueness ball from to .
Example 3.2 In this example, we provide an application of our results to a special nonlinear Hammerstein integral equation of the second kind. Consider the integral equation
for any , where G is the Green kernel on defined by
Let and D be a suitable open convex subset of , which is given below. Define a mapping by
for any . The first and second derivatives of F are given by
for any and
for any , respectively.
We use the max-norm. Let for all . Then, for any , we have
for any , which means
It follows from the Banach theorem that exists and
On the other hand, it follows from (3.6) that . Then we get .
Note that is not bounded in X or its subset . Take into account that a solution of equation (1.1) with F given by (3.6) must satisfy
i.e., and , where and are the positive roots of the real equation . Consequently, if we look for a solution such that , we can consider , with , as a nonempty open convex subset of X. For example, choose .
Using (3.7) and (3.8), we have, for any ,
and
Then we get
and
Now, we can choose constants in Theorem 2.1 as follows:
We also have
So, all the conditions in Theorem 2.1 are satisfied and Theorem 2.1 applies. Consequently, the sequence generated by Halley’s method (1.2) with initial point converges to the unique solution of equation (1.1) on . Moreover, by (2.7) and (2.8), we can set , which extends the uniqueness ball from to .
References
Argyros IK: The convergence of Halley-Chebyshev type method under Newton-Kantorovich hypotheses. Appl. Math. Lett. 1993, 6: 71–74.
Argyros IK Studies in Computational Mathematics 15. In Computational Theory of Iterative Methods. Elsevier, New York; 2007. Chui, CK, Wuytack L (eds.)
Deuflhard P: Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms. Springer, Berlin; 2004.
Argyros IK, Cho YJ, Hilout S: On the semilocal convergence of the Halley method using recurrent functions. J. Appl. Math. Comput. 2011, 37: 221–246. 10.1007/s12190-010-0431-6
Argyros IK, Ren HM: Ball convergence theorems for Halley’s method in Banach spaces. J. Appl. Math. Comput. 2012, 38: 453–465. 10.1007/s12190-011-0490-3
Argyros, IK, Ren, HM: On the Halley method in Banach space. Appl. Math. (2012, to appear)
Ezquerro JA, Hernández MA: New Kantorovich-type conditions for Halley’s method. Appl. Numer. Anal. Comput. Math. 2005, 2: 70–77. 10.1002/anac.200410024
Acknowledgements
First, the authors would like to thank the anonymous referee for valuable suggestions that have been implemented in the final version of this paper. The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170) and the third author was supported by the National Natural Science Foundation of China (Grant No. 10871178).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Argyros, I.K., Cho, Y.J. & Ren, H. Convergence of Halley’s method for operators with the bounded second Fréchet-derivative in Banach spaces. J Inequal Appl 2013, 260 (2013). https://doi.org/10.1186/1029-242X-2013-260
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2013-260