Journal of Applied Mathematics

Volume 2011, Article ID 786306, 15 pages

http://dx.doi.org/10.1155/2011/786306

## Semilocal Convergence for a Fifth-Order Newton's Method Using Recurrence Relations in Banach Spaces

^{1}Department of Mathematics, Shanghai University, Shanghai 200444, China^{2}School of Mathematical Science, Huaibei Normal University, Anhui, Huaibei 235000, China^{3}School of Computer Science and Technology, Huaibei Normal University, Anhui, Huaibei 235000, China

Received 13 August 2011; Accepted 3 October 2011

Academic Editor: Ke Chen

Copyright © 2011 Liang Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study a modified Newton's method with fifth-order convergence for nonlinear equations in Banach spaces. We make an attempt to establish the semilocal convergence of this method by using recurrence relations. The recurrence relations for the method are derived, and then an existence-uniqueness theorem is given to establish the R-order of the method to be five and a priori error bounds. Finally, a numerical application is presented to demonstrate our approach.

#### 1. Introduction

Many scientific problems can be expressed in the form of a nonlinear equation where is a nonlinear operator on an open convex subset of a Banach space with values in a Banach space .

The Newton’s method [1, 2], which has quadratically order convergence, is one of the most well-known methods for solving nonlinear equations. Recently, numerous variants of Newton’s method with high-order convergence are developed in the literature [3–7]; these methods improve the local order of convergence of Newton’s method by an additional evaluation of the function. In this paper, we consider the semilocal convergence for the method proposed in [7]. We first extend this method to Banach spaces and write it as where is the identity operator on , , .

In many papers, the convergence of iterative methods for solving nonlinear operator equation in Banach spaces is established from the convergence of majorizing sequences, which is obtained by applying the real function to a polynomial [8–10]. An alternative approach is developed to establish this convergence by using recurrence relations. For example, the recurrence relations are used in establishing the convergence of Newton’s method [2] and some high-order methods [11–17]. In this paper, we consider the semilocal convergence of the method given by (1.2) using the recurrence relations. We construct the system of recurrence relations and prove the convergence of the method, along with an error estimate. Finally, numerical results are presented to demonstrate our approach.

#### 2. Preliminary Results

Let , and the nonlinear operator be continuously second-order Fréchet differentiable, where is an open set and and are Banach spaces. We assume that (), (), (, ()there exists a positive real number such that

Now, we define the following scalar functions which will be often used in the later developments. Let Let , and let be the smallest positive zero of the scalar function for . Then, using MATLAB, we obtain that is decreasing and for all and Some properties of the functions defined above are given in the following lemma.

Lemma 2.1. *Let the real functions , and be given in (2.2). Then *(a)* and are increasing and , for all and , *(b)* is increasing for all and , *(c)*, , and for , , and . *

Assume that the conditions ()–() hold. we now denote , , , , and . Let and , then we can define the following sequences for : From the definition of , , (2.4) and (2.5), we also have

Nextly we will study some properties of the previous scalar sequences defined in (2.4)–(2.10), and later developments will require the following lemma.

Lemma 2.2. *Let the real functions , and be given in (2.2). If
**
then one has *(a)* and for , *(b)*the sequences , , , and are decreasing while is increasing, *(c)* and for . *

*Proof. *By Lemma 2.1 and (2.11), and hold. It follows from the definitions that , , . Moreover, by Lemma 2.1, we have and . This yields , , and (b) holds. Based on these results we obtain and and (c) holds. By induction we can derive that the items (a), (b), and (c) hold.

Lemma 2.3. *Under the assumptions of Lemma 2.2 and defining , then
**
where . Also, for , one has
*

*Proof. *By the definition of and given in (2.9)-(2.10), we obtain , ; by Lemma 2.1 we have
Suppose , . Then, by Lemma 2.2, we have , and . Thus,
Therefore, it holds that , .

By (2.12), we get
This shows that (2.13) holds. The proof is completed.

Lemma 2.4. *Under the assumptions of Lemma 2.2, let and . The sequence satisfies
**
Hence, the sequence converges to 0. Moreover, for any , , it holds that
*

*Proof. *From the definition of sequence given in (2.4) and (2.13), we have
Because and , it follows that as ; hence, the sequence converges to 0.

Since
where , , we can obtain
Furthermore,
Therefore, exists. The proof is completed.

#### 3. Recurrence Relations for the Method

We firstly give an approximation of the operator in the following lemma, which will be used in the next derivation.

Lemma 3.1. *Assume that the nonlinear operator is continuously second-order Fréchet differentiable, where is an open set and and are Banach spaces. Then, one has
**
where and . *

*Proof. *By the Taylor Expansion, we obtain
and we obtain
By the first two steps of method given in (1.2) and (3.4), (3.6), (3.7), we obtain
Substituting (3.4) and (3.8) into (3.3), we obtain (3.1).

We now consider . Since
using Taylor’s formula, we have
Similarly, we obtain
It follows that
Substituting (3.12) into (3.10), we can obtain (3.2). The proof is completed.

We denote and in this paper.

In the following, the recurrence relations are derived for the method given by (1.2) under the assumptions mentioned in the previous section.

For , the existence of implies the existence of . This gives us and this means that , where . By the initial hypotheses, we have Because of the assumption , by the Banach lemma [2] it follows that It is followed that

Consequently is well defined and It is similar to obtain By Lemma 3.1, we can get Therefore, we have From the assumption , it follows that .

By and is increasing in and , we have and it follows by the Banach lemma [2] that exists and Then, from (3.20) and (3.24), we have Because of , we obtain which shows .

In addition, we have Repeating the above derivation, we can obtain the system of recurrence relations given in the next lemma.

Lemma 3.2. *Let the assumptions and the conditions ()– () hold. Then, the following items are true for all : *(i)*there exists and , *(ii)*, *(iii)*, *(iv)*, *(v)*, *(vi)*, where . *

*Proof. *The proof of (I)–(V) follows by using the above-mentioned way and invoking the induction hypothesis. We only consider (VI). By (V) and Lemma 2.4, we obtain
So the lemma is proved.

#### 4. Semilocal Convergence

Lemma 4.1. *Let . If , then .*

*Proof. *Since
we obtain .

Now we give a theorem to establish the semilocal convergence of (1.2), 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 -order of convergence at least five of iteration (1.2).

Theorem 4.2. *Let be a nonlinear two times Fréchet differentiable operator in an open convex subset of a Banach space with values in a Banach space . Assume that and all conditions ()– () hold. Let , , and satisfy and , where , and are defined by (2.2). Let , where , then, starting from , the sequence generated by the method (1.2) converges to a solution of with , belonging to and being the unique solution of in . Moreover, a priori error estimate is given by
**
where and . *

*Proof. *By Lemma 3.2, the sequence is well defined in . Now we prove that is a Cauchy sequence. Since
it follows that is a Cauchy sequence, and hence the sequence is convergent. So there exists an such that .

By letting , in (4.3), we obtain
This shows that .

Now we prove that is a solution of . Since
by letting in (4.5), we obtain since and . Hence, by the continuity of in , we obtain .

We prove the uniqueness of in . Firstly we can obtain , since it follows by Lemma 3.2 that
and then . Let be another zero of in . By Taylor theorem, we have
Since
it follows by the Banach lemma that is invertible and hence .

Finally, by letting in (4.3), we obtain (4.2) and furthermore
This means that the method given by (1.2) is of -order of convergence at least five. This ends the proof.

#### 5. Numerical Example

Let be the space of continuous functions defined on the interval , with the max-norm, and consider the integral equation , where with , . Integral equations of this kind (called Chandrasekhar equations) arise in elasticity or neutron transport problems.

It is easy to obtain the derivatives of as The derivative satisfies and the Lipschitz condition with since the norm is taken as max-norm.

Starting at , we have since and by the Banach lemma that exists and It follow that Consequently, we obtain which satisfy This means that the hypothesis of Theorem 4.2 is satisfied. Hence, the recurrence relations for the method given by (1.2) are demonstrated in Table 1. Besides, the solution belongs to , and it is unique in .

#### 6. Conclusions

A family of recurrence relations is developed for establishing the semilocal convergence of a modified Newton’s method (1.2) used for solving in Banach spaces. Based on these recurrence relations, an existence uniqueness theorem is established to show the -order convergence of the method to be five. Also a priori error bounds are given. A numerical example is worked out to demonstrate our approach and show that our method can be of practical interest.

#### Acknowledgments

The work is supported by the Shanghai Natural Science Foundation (no. 10ZR1410900), the Key Disciplines of Shanghai Municipality (no. S30104), the Natural Science Research Funds of Anhui Provincial for Universities (Grant no. KJ2011A248), and the Open Fund of Shanghai Key Laboratory of Trustworthy Computing.

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