Abstract and Applied Analysis

Volume 2013 (2013), Article ID 682167, 11 pages

http://dx.doi.org/10.1155/2013/682167

Research Article

Convergence Behavior for Newton-Steffensen’s Method under -Condition of Second Derivative

Yonghui Ling,¹ Xiubin Xu,² and Shaohua Yu²

¹Department of Mathematics, Zhejiang University, Hangzhou 310027, China

²Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received 2 September 2013; Accepted 30 September 2013

Academic Editor: Jen-Chih Yao

Copyright © 2013 Yonghui Ling 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.

The present paper is concerned with the semilocal as well as the local convergence problems of Newton-Steffensen’s method to solve nonlinear operator equations in Banach spaces. Under the assumption that the second derivative of the operator satisfies -condition, the convergence criterion and convergence ball for Newton-Steffensen’s method are established.

1. Introduction

Let and be real or complex Banach spaces, let be an open subset, and let be the Fréchet differentiable nonlinear operator. Approximating a solution of a nonlinear equation

is widely studied in both theoretical and applied areas of mathematics.

One of the most famous methods to solve this problem is Newton’s method defined by

where is an initial point. Usually, the study about convergence issue of Newton’s method includes local and semilocal convergence analyses. The local convergence issue is, based on the information around a solution, to seek estimates of the radii of convergence balls, while the semilocal one is, based on the information around an initial point, to give criteria ensuring the convergence. Among the semilocal convergence results on Newton’s method, one of the famous results is Smale’s point estimate theory which gives a convergence criterion of Newton’s method only based on the information at the initial point for analytic functions; see for example, [1–6]. To extend and improve Smale’s theory, Wang and Han proposed in [7, 8] the notion of-condition, which is weaker than Smale’s assumption in [5] for analytic operators.

There are several kinds of cubic generalizations for Newton’s method. The most important family is the Euler-Halley family and its variations which include Chebyshev’s method and Halley’s method as special cases; see for example, [9–16] and references therein. However, the disadvantage of this family is that the evaluation of the second derivative of the operatoris required at every step, the operation cost of which may be very large in fact. To reduce the operation cost but also retain the cubic convergence, Sharma in [17] proposed the following Newton-Steffensen’s method which avoids the computation of the second Fréchet derivative. Let. The method is defined as follows:

where. The author obtained cubic convergence for (3) under the assumption thatis sufficiently smooth in the neighborhood of the solution.

Motivated by the work mentioned above, we extend this method to Banach spaces and present its semilocal and local convergence. The extension is described as follows:

where the divided difference operator is defined by

In Section 2, we introduce some preliminary notions and important majorizing functions with properties. In Sections 3 and 4, we study the semilocal convergence and local convergence results of Newton-Steffensen’s method under-condition, respectively. We obtain the uniqueness ball and the convergence ball.

2. Notations and Preliminary Results

Throughout this paper, we assume that and are two Banach spaces. Let be an open subset and let be a nonlinear operator with the continuous twice Fréchet derivative. Forand, we useandto denote the open ball with radiusand centerand its closure, respectively. Letbe such that exists and.

Letbe some positive constant and. We say thatsatisfies-condition onif the following relation holds:

For simplicity, we write

The lemma below is useful in the next two sections.

Lemma 1. Suppose thatand thatsatisfies-condition (6) on. Then for any,exists and the following inequality holds:

Proof. We can derive the following relation:

For any, it follows from-condition andthat

Then, by Banach lemma, one has thatexists and the following inequality holds:

Letbe some positive constant. The following majorizing functionintroduced by Wang and Han in [18] will be used to obtain a Smale-type semilocal convergence criterion:

Letanddenote the corresponding sequences generated by Newton-Steffensen’s method for the majorizing functionwith the initial point; that is,

The following lemma taken from [19] describes some useful properties about.

Lemma 2. Suppose that

Thenhas two zeros indenoted byand. They satisfy the following relations:

Moreover,is decreasing monotonically in interval, while it is increasing monotonically in interval.

The lemma below describes the convergence property of the sequencesand, which is crucial for the semilocal convergence analysis of Newton-Steffensen’s method (4) under-condition.

Lemma 3. Suppose that (14) holds. Letandbe the sequences generated by (13). Then

Moreover,andconverge increasingly to the same point.

Proof. To show that (16) holds for, we note thatand that. By (15), we have

This implies that. It remains to show thatfor the case. To this end, we define a real function as

It is clear thatand thatis decreasing monotonically in. It follows from (15) that. In view of the fact thatis the unique zero ofin, we obtain. This is equivalent to

Hence (16) holds for.

Now we assume that

From Lemma 2, we have, for each, and. The later one implies that. Define function

Then,, which implies thatis increasing monotonically in. Hence, we have

Sinceis convex in, we getand so.

Furthermore, we claim that

for all, , and. Indeed, it follows from the convexity ofthat

from which we have

where

Noting thatfor all, we obtain

Then (23) follows. By (23), we conclude that

Therefore, (16) holds for all. The inequlities in (16) imply thatandconverge increasingly to some same points, say. Clearlyandis a zero ofin. Noting thatis the unique zero ofin, one has that. The proof is complete.

3. Convergence Criterion

Throughout this subsection, letbe the initial point such that the inverseexists and let, whereis defined by (7). Moreover, we assume thatsatisfies-condition on; that is, the following relation holds:

Then, for any, it follows from Lemma 1 thatexists and the following inequality holds:

Below we list two useful lemmas.

Recall that the divided difference operatoris defined by (5). The following lemma gives the expressions of some desired estimates in the proof of Lemma 5.

Lemma 4. Let. Define

Then the following formulas hold:

(i). (ii)(iii)

Proof. For (i), we notice that

As for (ii), one has

Similarly, we obtain

The proof is complete.

Lemma 5. Suppose that (14) holds. Then the sequencegenerated by (4) with the initial pointis well defined and the following estimates hold for any natural number:

(i),,. (ii). (iii).

Proof. For the casein (i), it is clear that. By Lemma 4 and (29), we have

In view of the monotonicity of, one has that. Hence, we get from Banach lemma thatexists and satisfies

Combining (36) inequality with the definitions ofandgiven in (13), one has

As for the estimate, by Lemma 4, we have

This together with the obtained bounds,and (29) yields that

This implies that statement (i) holds for.

Statement (ii) for the caseis justified by (36). Below, we consider the casefor (iii). First we have the following expression ofdue to Lemma 4:

from which we obtain that

Therefore statement (iii) holds for.

Assume that statements (i)–(iii) are true for. Below, we will show that they also hold for. First, by statement (i), we have

Hence,exists by Lemma 1.

Note that

by the inductive hypotheses of (i) and (iii). Then it follows from (30) and (13) that

Hence by (29), (44), Lemma 4, and the inductive hypothesis of (i), we have

It follows from Banach lemma thatexists and satisfies

Hence, (ii) holds for.

Combining (46) with the inductive hypothesis (iii), one has

which implies that.

On the other hand, by (29), (30), (44), and Lemma 4, we conclude that

which leads to. Thus, (i) holds for.

Next, we will show that (iii) also holds for. In fact, by using Lemma 4, (29), (44), and (48), we obtain

Therefore statement (iii) holds for. Hence (i)–(iii) hold for all. Furthermore, by statement (i), one has, for any,. Thus by Lemma 1 we know thatexists for each;is well defined. The proof is complete.

Recall thatandare defined in (13). Based on the preceding useful lemmas, we are now ready to prove a Smale-type semilocal convergence theorem for Newton-Steffensen’s method (4) under-condition.

Theorem 6. Suppose that (14) holds. Then the sequencegenerated by (4) with the initial pointis well defined and converges to a solutionof (1) with Q-cubic rate, and this solutionis unique in, where. Moreover, the following error bounds

are valid, whereandare defined in Lemma 2.

Proof. The uniqueness ball can be obtained by Theorem  5.2 in [19]. It follows from Lemma 1 thatis well defined. In addition, from Lemmas 3 and 5 (i), we can see thatis convergent to a limit, say. Below, we show thatis a solution of (1). It follows from Lemma 5 (iii) that

Lettingin the preceding relation gives that the limitis a solution of (1). Moreover, we have

Next, we verify that estimate (62) is true. By (29) and Lemma 5, one has

In order to estimate, we first notice that

where. This together with Lemma 5(i), (29), (52), and (53) gives that

Combining the above inequality with (46), we have

Therefore, the error estimate (62) follows. Also, from the previous inequality, we know that the convergence rate oftois -cubic. This completes the proof.

One typical and important class of examples satisfying-condition is the one of analytic functions. Smale [5] studied the convergence and error estimation of Newton’s method (2) under the hypotheses thatis analytic and satisfies

whereis a fixed point inandis defined by

The following lemma taken from [20] shows that an analytic operator satisfies-condition.

Lemma 7. Letandbe defined by (7) and (86), respectively. Suppose thatis analytic and satisfies (85). Thensatisfies-condition

on.

According to this lemma, we can conclude that the semilocal results obtained in Theorem 6 also hold whenis an analytic operator.

Corollary 8. Suppose that (14) holds. Suppose thatis analytic and satisfies

where is defined by

Then the sequencegenerated by (4) with the initial pointis well defined and converges to a solutionof (1) with Q-cubic rate and this solutionis unique in, where. Moreover, the following error bounds

are valid, whereandare defined in Lemma 2.

4. Convergence Ball

Now we begin to study the local convergence properties for Newton-Steffensen’s method (4) under-condition. Recall thatis defined by (7). Throughout this section, suppose thatsuch that,, and the inverseexists. Moreover, we assume thatsatisfies the-condition on; that is, the following relation holds:

Then, for any, it follows from Lemma 1 that

Let

Define functionas follows:

It is clear thatand that. Moreover,increases monotonically in.

Theorem 9. Letbe defined in (65). Then, for any, the sequencegenerated by (4) converges toand satisfies that

where

Proof. For, we write. It is sufficient to show that

In fact, by noticing the monotonicity of, we have

From this we can easily establish (67) by mathematical induction.

We now prove (69). First we get the following expression of:

Similarly, we also have

This together with (63) and (64) yields

On the other hand, we notice that

It follows from (63) that

For the case, by (88) and (73), we get

Combining Lemma 2 with (75) and (76), we obtain

It follows from Banach lemma that

This together with (63), (71) and (76) yields

Hence (69) holds for.

Now assume that the inequalities in (69) hold for up to some. Then by (73), one has

Thus (75) can be further reduced to

Using Banach lemma again, one has

This together with (63), (71), and (73) yields

Thus

This and (83) show that the inequalities in (69) hold forand hence they hold for each. The proof is complete.

Applying Lemma 7 to the above theorem, we get the following corollary:

Corollary 10. Suppose thatis analytic and satisfies

where is defined by

Letandbe defined by (65) and (66), respectively. Then, for any, the sequencegenerated by (4) converges toand satisfies that

where

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

Xiubin Xu’s work was supported in part by the National Natural Science Foundation of China (Grant no. 61170109).

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