#### 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).