The Krasnoselskii’s Method for Real Differentiable Functions

We study the convergence of the Krasnoselskii sequence xn+1 = xn+g(xn)/2 for non-self mappings on closed intervals. We show that if g satisfies g′ ≥ −1 along with some other conditions, this sequence converges to a fixed point of g. We extend this fixedpoint 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.