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.