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In this article, by using the concept of W-mapping introduced by Atsushiba and Takahashi and K-mapping introduced by Kangtunyakarn and Suantai, we define W(T,N)-iteration and K(T,N)-iteration for finding a fixed point of continuous mappings on an arbitrary interval. Then, a necessary and sufficient condition for the strong convergence of the proposed iterative methods for continuous mappings on an arbitrary interval is given. We also compare the rate of convergence of those iterations. It is proved that the W(T,N)-iteration and K(T,N)-iteration are equivalent and the K(T,N)-iteration converges faster than the W(T,N)-iteration. Moreover, we also present numerical examples for comparing the rate of convergence between W(T,N)-iteration and K(T,N)-iteration.

Strong convergence theorems and rate of convergence of multi-step iterative methods for continuous mappings on an arbitrary interval