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Journal of Function Spaces

Volume 2014 (2014), Article ID 467980, 10 pages

http://dx.doi.org/10.1155/2014/467980

## A Semilocal Convergence for a Uniparametric Family of Efficient Secant-Like Methods

^{1}Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA^{2}Department of Mathematics and Computation, University of La Rioja, Logroño, 26006 La Rioja, Spain

Received 8 May 2013; Accepted 21 October 2013; Published 6 February 2014

Academic Editor: Alfonso Montes-Rodriguez

Copyright © 2014 I. K. Argyros 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 present a semilocal convergence analysis for a uniparametric family of efficient secant-like methods (including the secant and Kurchatov method as special cases) in a Banach space setting (Ezquerro et al., 2000–2012). Using our idea of recurrent functions and tighter majorizing sequences, we provide convergence results under the same or less computational cost than the ones of Ezquerro et al., (2013, 2010, and 2012) and Hernández et al., (2000, 2005, and 2002) and with the following advantages: weaker sufficient convergence conditions, tighter error estimates on the distances involved, and at least as precise information on the location of the solution. Numerical examples validating our theoretical results are also provided in this study.

#### 1. Introduction

Let and stand, respectively, for the open and closed ball in with center and radius . Denote by the space of bounded linear operators from into .

In this study, we are concerned with the problem of approximating a locally unique solution of nonlinear equation as follows: where is a Fréchet-differentiable operator defined on a nonempty convex subset of a Banach space with values in a Banach space .

Many problems from computational sciences, physics and other disciplines can be taken in the form of (1) using mathematical modelling [1–7]. The solution of these equations can rarely be found in closed form. That is why the solution methods for these equations are iterative. In particular, the practice of numerical analysis for finding such solutions is essentially connected to variants of Newton's method [1, 2, 4–10]. The study about the convergence of iterative procedures is usually focused on two types: semilocal and local convergence analysis. The semilocal convergence is, based on the information around an initial point, to give criteria ensuring the convergence of iterative procedure; while the local one is, based on the information around a solution, to find estimates of the radii of convergence balls. There are a lot of studies on the weakness and/or extension of the hypothesis made on the underlying operators; see, for example, [1–26] and the references therein.

Hernández and Rubio used in [22] the uniparametric family of secant-like methods defined by and the method of recurrent relations to generate a sequence approximating . Here, for each is a divided difference of order one, which is a bounded linear operator such that [1, 4, 6–8, 14, 15]

Secant-like method (2) can be considered as a combination of the secant and Newton’s method. Indeed, if , we obtain the secant method and if , we get Newton’s method provided that is Fréchet-differentiable on , since then and .

It was shown in [20, 21] that the -order of convergence is at least for , the same as that of the secant method. Later in [5], another uniparametric family of secant-like methods defined by was studied. It was shown that there exists , and that the -order of convergence is at least if and , and if , the -order of convergence is quadratic. Note that if , we obtain the secant method, whereas if , we obtain the Kurchatov method [4, 5, 7, 8].

We present a semilocal convergence analysis for secant-like method (2) using our idea of recurrent functions instead of recurrent relations and tighter majorizing sequences. This way, our analysis provided the following advantages over the work in [5] under the same computational cost.Weaker sufficient convergence conditions.Tighter estimates on the distances and for each .At least as precise information on the location of the solution.The results are presented in affine invariant form, whereas the ones in [5] are given in nonaffine invariant forms. The advantages of affine versus nonaffine results have been explained in [1, 4, 6–8, 14, 15].

Our hypotheses for the semilocal convergence of secant-like method (4) are as follows.There exists a divided difference of order one satisfying (3).There exist , such that and .There exist and such that There exists such that We will denote by conditions . In view of , there exist , , such that,, for each .

Clearly, hold in general and , can be arbitrarily large [1, 2, 4]. Note that , , and are not additional to hypotheses. In practise, the computation of requires the computation of , , and . It also follows from that is differentiable [1–3, 8, 9].

The paper is organized as follows. In Section 2, we show that under the same hypotheses as in [23] and using recurrent relations, we obtain at least as precise information on the location of the solution. Section 3 contains the semilocal convergence analysis using weaker hypotheses and recurrent functions. We also show the advantages . The results are also extended to cover the case of equations with nondifferentiable operators. Numerical examples are presented in the concluding Section 4.

#### 2. Semilocal Convergence Using Recurrent Relations

As in [5], let us define sequences and for each by and functions , on by

Next, we present the main result in this section in affine invariant form.

Theorem 1. *Under the hypotheses, further suppose that
**
and for ,
**
where
**
Then, sequence generated by secant-like method (4) is well defined, remains in for each , and converges to a solution of equation . Moreover, the following estimates hold
**
Furthermore, the solution is unique in , where , provided that
*

*Proof. *The proof with the exception of the uniqueness part is given in Theorem in [5] if we use instead of and set , where .

To prove the uniqueness of the solution, let us assume that is a solution of . Let . Then, using and the definition of , we get in turn that
It follows from (15) and the Banach lemma on invertible operators [1, 2, 4, 6–8, 14] that . Using the identity , we deduce that . That completes the proof of the theorem.

*Remark 2. *If , Theorem 1 reduces to Theorem in [5]. Otherwise, that is, if , then our Theorem 1 constitutes an improvement over Theorem , since
where
where given in [5] (for ). Hence, (16) justify our claim for this section which was made in the Introduction of this study.

#### 3. Semilocal Convergence Using Recurrent Functions

We present the semilocal convergence of secant-like methods. First, we need some auxiliary results on majorizing sequences for secant-like method.

Lemma 3. *Let , , , and . Set , , and . Define scalar sequences , , for each by
**
functions on for each by
**
and polynomial on by
**
Denote by the only root of polynomial in . Suppose that
**
Then, sequence is nondecreasing, bounded from above by that is defined by
**
and converges to its unique least upper bound which satisfies
**
Moreover, the following estimates are satisfied for each :
*

*Proof. *We will first show that polynomial has roots in . Indeed, we have and . Using the intermediate value theorem, we deduce that there exists at least one root of in . Moreover . Hence, crosses the positive axis only once. Denote by the only root of in . It follows from (18) and (19) that estimate (25) is certainly satisfied if
Estimate (27) is true by (22) for . Then, we have by (18) that
Suppose that
Estimate (27) will be true for replacing if
or
where is defined by (20). We need a relationship between two consecutive recurrent functions for each . Using (20) and (21), we deduce that
since . Define function on by
Then, we get from (20) and (33) that
Hence, by (32)–(34), (31) is satisfied if
which is true by (22). The induction for (25) is complete. That is, sequence is nondecreasing, bounded from above by that is given by (23), and as such it converges to some which satisfies (24). Estimate (26) follows from (25) by using standard majorization techniques [1, 2, 4, 6–8, 14]. The proof of Lemma 3 is complete.

Lemma 4. *Let , , , , , , and . Set , , and . Define scalar sequences , for each by
**
and functions on by
**Suppose that
**
where is defined in Lemma 3. Then, sequence is nondecreasing, bounded from above by that is defined by
**
and converges to its unique least upper bound which satisfies
**
Moreover, the following estimates are satisfied for each :
*

*Proof. *We will show using induction that
Estimate (43) is true for by (39). Then, we have by (36) that
Suppose that (43) holds for each . Then, using (36), we get that
Estimate (43) will be satisfied if
Using (38), we get the following relationship between two consecutive recurrent functions :
Define function on by
Then, we get from (38) that
Then, (46) is satisfied if
which is true by the choice of and the right hand side inequality in hypothesis (39). The induction for (43) (i.e., (42)) is complete. The rest of the proof as identical to Lemma 3 is omitted. The proof is complete.

*Remark 5. *(a) Let us consider an interesting choice for . Let (secant method). Then, using (21) and (22), we have that

The corresponding condition for the secant method is given by [2, 4, 9, 23] as follows:

Condition (52) can be weaker than (53) (see also the numerical examples at the end of the study). Moreover, the majorizing sequence for the secant method related to (53) is given by

A simple inductive argument shows that if , then for each :

(b) The majorizing sequence used in [5] is essentially given by

Then, again we have

Moreover, our sufficient convergence conditions can be weaker than [5].

(c) Clearly, iteration is tighter than and as we have in (57) than for or as follows:

Next, we present obvious and useful extensions of Lemmas 3 and 4, respectively.

Lemma 6. *Let be fixed. Suppose that
**
Then, sequence generated by (19) is nondecreasing, bounded from above by , and converges to which satisfies . Moreover, the following estimates are satisfied for each :
*

Lemma 7. *Let be fixed. Suppose that
**
Then, sequence generated by (36) is nondecreasing, bounded from above by and converges to which satisfies . Moreover, the following estimates are satisfied for each *

Next, we present the following semilocal convergence result for secant-like method under the conditions.

Theorem 8. *Suppose that the , Lemma 3 (or Lemma 6) conditions and
**
hold. Then, sequence generated by secant-like method is well defined, remains in for each , and converges to a solution of equation . Moreover, the following estimates are satisfied for each :
**
Furthermore, if there exists such that
**
then, the solution is unique in .*

*Proof. *We use mathematical induction to prove that
for each . Let . Then, we obtain that
which implies that . Let also . We get that
hence, . Note that and . That is, . Hence, estimates (67) and (68) hold for and . Suppose that (67) and (68) hold for all . Then, we obtain that
Hence, , .

Using , Lemma 3, and the introduction hypotheses, we get that
It follows from (72) and the Banach lemma on invertible operators [1, 2, 4, 6–8, 14] that exists and
In view of (4), we obtain the following identity:
Using (4), (34), and the induction hypotheses, we get in turn that
It now follows from (4), (18), (74), and (75) that
which completes the induction for (67). Moreover, let . Then, we get that
which implies that . The induction for (68) is complete.

Lemma 3 implies that is a complete sequence. It follows from (67) and (68) that is a complete sequence in a Banach space and as such it converges to some (since is a closed set). By letting in (75), we obtain . Furthermore, estimate (65) follows from (64) by using standard majorization techniques [1–4, 6–8]. To show the uniqueness part, let be such that . We have that
It follows from (78) and the Banach lemma on invertible operators that exists. Then, using the identity , we deduce that . The proof of Theorem 8 is complete.

*Remark 9. *(a) The limit point can be replaced in Theorem 8 by which is given in closed form by (23).

(b) It follows from the proof of Theorem 8 that is also a majorizing sequence for . Hence, Lemma 4 (or Lemma 7), , can replace Lemma 3 (or Lemma 6) , in Theorem 8.

Hence, we arrive at the following.

Theorem 10. *Suppose that the conditions, Lemma 4 (or Lemma 7), and
**
hold. Then, sequence generated by secant-like method is well defined, remains in for each , and converges to a solution of equation . Moreover, the following estimates are satisfied for each :
**
Furthermore, if there exists such that
**
then, the solution is unique in .*

Let us consider the following equation: where is a before and is continuous. The corresponding secant-like method is given by where is an initial guess.

Suppose thatClearly, holds and can be arbitrarily large [1–4, 14].

We will denote by the conditions , , and . Then, we can present the corresponding result along the same lines as in Lemmas 3, 4, 6, and 7 and Theorems 8 and 10. However, we will only present the results corresponding to Lemma 4 and Theorem 10, respectively. The rest combination of results can be given in an analogous way.

Lemma 11. *Let , , , , , , , , and . Set , , and . Define scalar sequences , by
**
and functions on by
**
Suppose that function given by
**
has a minimal zero in and
**
where was defined in Lemma 3. Then, sequence is nondecreasing, bounded from above by that is defined by
**
and converges to its unique least upper bound which satisfies
**
Moreover, the following estimates are satisfied for each :
*

*Proof. *Simply use , , , , instead of , , , , in the proof of Lemma 4.

Theorem 12. *Suppose that the , Lemma 11 conditions,
**
hold, where was defined in Theorem 8 and . Then, sequence generated by the secant-like method (83) in well defined, remains in for each , and converges to a solution of equation . Moreover, the following estimates are satisfied for each :
**
Furthermore, if there exists such that
**
then, the solution is unique in .*

*Proof. *The proof until the uniqueness part follows as in Theorem 8 but using the following identity:
instead of (74). Finally, for the uniqueness part, let be such that . Then, we get from (83) the identity
This identity leads to
Hence, we deduce . But we know that . That is, we conclude that . That completes the proof of the theorem.

#### 4. Numerical Examples

*Example 1. *Let , equipped with the max-norm. Consider the following nonlinear boundary value problem:
It is well known that this problem can be formulated as the integral equation
where is the Green function as follows:
We observe that
Then, problem (101) is in the form (1), where is defined as
The Fréchet derivative of the operator is given by
Then, we have that
Hence, if , then
It follows that exists and
We also have that . Define the divided difference defined by
Choosing such that and . Then, we have for ,
where is such that
Set and . It is easy to verify that since . If and , the operator satisfies conditions of Theorem 8, with

Choosing , , and , we obtain that

Moreover, we obtain that and , but conditions of Theorem 1 are not satisfied since
Notice also that the popular condition (53) is also not satisfied, since . Hence, there is no guarantee under the old conditions that the secant-type method converges to . However, conditions of Lemma 3 are satisfied since
The convergence of the secant-type method is also ensured by Theorem 8.

*Example 2. *Let , and consider the real function
and we are going to apply secant-type method with . We take the starting points , and we consider the domain . In this case, we obtain
Notice that the conditions of Theorem 1 and Lemma 3 are satisfied, but since , Remark 2 ensures that our uniqueness ball is larger. It is clear as .

#### Conflict of Interests

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

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