Abstract and Applied Analysis

Volume 2013, Article ID 923898, 8 pages

http://dx.doi.org/10.1155/2013/923898

## Newton-Kantorovich and Smale Uniform Type Convergence Theorem for a Deformed Newton Method in Banach Spaces

^{1}Department of Mathematics, Taizhou University, Linhai, Zhejiang 317000, China^{2}Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technicka 8, 616 00 Brno, Czech Republic^{3}Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, China

Received 12 September 2013; Accepted 18 November 2013

Academic Editor: Miroslava Růžičková

Copyright © 2013 Rongfei Lin 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

Newton-Kantorovich and Smale uniform type of convergence theorem of a deformed Newton method having the third-order convergence is established in a Banach space for solving nonlinear equations. The error estimate is determined to demonstrate the efficiency of our approach. The obtained results are illustrated with three examples.

#### 1. Introduction

In this paper, we study the problem of approximating a unique solution of a nonlinear operator equation where is a Fréchet-differentiable operator defined on an open convex of a Banach space with values in a Banach space .

There are many iterative methods (see [1–3]), which have been used for finding a solution of (1). For example, the well-known iterative method for solving (1) is Newton's method defined by

Under the appropriate assumptions, Newton's method is the second-order convergence. Kantorovich (see [4]) presented the famous convergence result regarding a solution of (1). Many Newton-Kantorovich type of convergence theorems were given in papers [5–11]. Frontini and Sormani (see [12]) presented a new deformed Newton method with

The deformed Newton method can be written as follows: where is a real or a complex function. In papers [13–17], the local convergence theorem has been established and the deformed method in a real or a complex space was discussed.

In the paper, we generalize the deformed Newton method [18] in a Banach space. The deformed Newton method [18] is shown as follows: where is defined on an open convex subset of a Banach space with values in a Banach space , has Fréchet derivatives in , and exists.

We establish Newton-Kantorovich and Smale uniform type convergence theorem (see [18]) for the deformed Newton method with the third-order in a Banach space with new sufficient conditions for the existence of a well-defined sequence which converges to a unique solution of (1).

#### 2. Main Results

Denote , , , and suppose are the positive and nondecreasing continuous functions, , , for , .

Assume that sequences , are generated by the following formulae [18]:

Firstly, we give some lemmas.

Lemma 1. *If , then the function has two positive real roots , ().*

*Proof. *Because , , and , we know that is the convex function for . Hence, is a unique positive root of . So, the necessary and sufficient condition that has two positive roots for is that the minimum of satisfies the condition , which holds for . This completes the proof of Lemma 1.

Lemma 2. *Suppose the sequences , are generated by (6). Then, for , the sequences , are increasing and converge to the minimum positive root of , and
*

*Proof. *Denote

On , we know , , , and is increasing. Denoting , then

Therefore, , are increasing on . Thus, for , , . Moreover,
hence we can easily prove Lemma 2 by the induction.

Suppose and are the Banach spaces, is an open convex subset, has the second-order Fréchet derivative, exists for , and the following conditions hold:
where and .

Lemma 3. *Suppose satisfies (11) and . Then exists, and
*

*Proof. *Firstly, by the conditions (11), we know that

Secondly, we know for . Hence

By Banach Theorem, we know exists, and

This completes the proof of Lemma 3.

Lemma 4. *Suppose and are Banach spaces, is an open convex of the Banach space , has the second-order Fréchet derivative, and the sequences , are generated by (5). Then, for any natural number , the following formula holds:
*

*Proof. *By (5), we have

Hence

This completes the proof of Lemma 4.

Theorem 5. *Suppose and are Banach spaces, is an open convex subset, satisfies condition (11), , and
**
Then the sequence generated by (5) is well defined, , and converges to the unique solution in and
*

*Proof. *By induction, we can prove that the following formulae hold:

In fact, by Lemma 2 we know for any natural number . It is easy to prove that for the above formulae hold. Suppose the above formulae also hold for . Then

By Lemma 3, we get
By Lemmas 3 and 4 and the fact that , are positive and increasing on , we have

Hence we get

By Lemma 3, we get

Moreover, we have

Hence, the sequence generated by (5) is well defined, , and converges to the solution of (1).

Now we prove the uniqueness. Suppose is also a solution of (1) on . We know that for . Then

By Banach Theorem, we know the inverse of exists and
hence we get . This completes the proof of the uniqueness of the solution of (1).

For , we know that

When , we get

This completes the proof of Theorem 5.

Suppose that , . Then , , and .

Corollary 6. *Suppose and are the Banach spaces, is an open convex subset of the Banach space , has the second-order Fréchet derivative, exists for , and the following conditions hold:
**Then the sequence generated by (5) is well defined, , and converges to the unique solution on of (1), where are two positive roots of .*

Suppose , , , and and for , . Hence, for , we get

Corollary 7 (see [10]). *Suppose and are Banach spaces, is an open convex subset of the Banach space , has the third-order Fréchet derivative, exists for , and the following conditions hold:
**Then the sequence generated by (5) is well defined, , and converges to the unique solution of (1) on , where
**
are two positive roots of the equation .*

#### 3. Numerical Examples

In this section, we apply the convergence theorem and show three numerical examples.

*Example 1. *Consider the equation

We choose the initial point , ; then

Hence, by Corollary 6, the sequence generated by (5) is well defined, and converges to the solution of (36).

Now, we will analyze errors by Corollary 6 (see Table 1). In this case, we take ; then .

*Example 2. *Consider the system of equation [18] , where

Then, we have

We choose and . We take the max-norm in and the norm for . Define the norm of a bilinear operator on by
where and

Then we get the following results:

This means that the hypotheses of Corollary 6 are satisfied.

Now, we will analyze errors by Corollary 6 (see Table 2). In this case, we take ; then .

*Example 3. *Consider the following integral equations:
and the space with the norm

This equation arises in the theory of radiative transfer and neutron transport and in the kinetic theory of gases. Define the operator on by

Then, for , we obtain
This means that the hypotheses of Corollary 7 are satisfied.

Now, we will analyze errors by Corollary 7 (see Table 3). In this case, we take ; then .

#### Acknowledgments

This work is supported by the National Basic Research 973 Program of China (no. 2011JB105001), National Natural Science Foundation of China (Grant no. 11371320), the Foundation of Science and Technology Department (Grant no. 2013C31084) of Zhejiang Province, and the Foundation of the Education Department (nos. 20120040 and Y201329420) of Zhejiang Province of China and also by the Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology, Czech Republic.

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