The Krasnoselskii's Method for Real Differentiable Functions

We study the convergence of the Krasnoselskii sequence x_{n+۱}=\frac{x_n+g(x_n)}{۲} for non-self mappings on closed intervals. We show that if g satisfies g^{'}\ge -۱ along with some other conditions, this sequence converges to a fixed point of g. We extend this fixed-point result to a novel and efficient root-finding method. We present concrete examples at the end. In these examples, we make a comparison between Newton-Raphson and our method. These examples also reveal how our method can be applied efficiently to find the fixed points of a real-valued function.

Keywords:

Krasnoselskii's theorem , Iterative sequence , Newton-Raphson method , Root estimation , Real function