Abstract and Applied Analysis

Volume 2013 (2013), Article ID 232618, 2 pages

http://dx.doi.org/10.1155/2013/232618

Letter to the Editor

Comment on “A Note on Kang-Rafiq-Kwun Iteration Method for Solving Nonlinear Equations”

Shin Min Kang,¹ Arif Rafiq,² and Young Chel Kwun³

¹Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

²Department of Mathematics, Lahore Leads University, Lahore 54810, Pakistan

³Department of Mathematics, Dong-A University, Busan 614-714, Republic of Korea

Received 11 July 2013; Accepted 25 August 2013

Academic Editor: Allan Peterson

Copyright © 2013 Shin Min Kang 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 point out in this paper that the claims made by Li in “A Note on Kang-Rafiq-Kwun Iteration Method for Solving Nonlinear Equations” are not true.

1. Introduction and Explanation

Consider the following nonlinear equation:

which can be written in the form of the following functional equation:

which is the famous fixed point method.

By using the following iterative relation

and the variational iteration method, He [1] established the following iteration method:

He [1] noted that the value of the auxiliary function should not be zero or small value during all iteration steps, .

In [2], Li claimed that the iteration method introduced by Kang et al. [3],

is not new and the formulation was first derived by He [1].

We comment as follows.

(1) Their claim is wrong, because no such derivation or explanation was presented in [1].(2) For , it can be easily seen that the iteration method (5) reduces to (4). Hence, (4) is the special case of (5).(3) On page 2 of [2], it is interesting to note that for , the claimed new formulation,

 is not new and reduced to

 which is the special case of relation (5).(4)Remark  1 (see [2]). Actually Li pointed out the special case of an already derived method due to Kang et al. [3].(5)Remark  2 (see [2]). The claim is wrong because in [3] the equation is different from the equation (7) in [2].(6)Remark  3 (see [2]). This argument is misleading because in [3] which is not equal to appeared in equation (8) of [2].(7)Remark  4 (see [2]). This statement is meaningless because the approach of Kang et al. [3] is different.

2. Conclusions

All the claims made by Li in [2] are incorrect. However, in order to obtain the variants and generalizations of the Newton-Raphson method, the approach and the performance of the variational iteration formulation can be seen in [4].

Acknowledgment

The authors are greatly thankful to Professor Naseer Shahzad for his valuable suggestions in order to improve this letter.

References

J.-H. He, “Variational iteration method—some recent results and new interpretations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 3–17, 2007.

H. Li, “A note on Kang-Rafiq-Kwun iteration method for solving nonlinear equations,” Abstact and Applied Analysis. In Press

S. M. Kang, A. Rafiq, and Y. C. Kwun, “A new second-order iteration method for solving nonlinear equations,” Abstract and Applied Analysis, vol. 2013, Article ID 487062, 4 pages, 2013.

A. Rafiq, “A note on “new classes of iterative methods for nonlinear equations” and ‘some iterative methods free from second derivatives for nonlinear equations’,” Applied Mathematics and Computation, vol. 193, no. 2, pp. 572–576, 2007.