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x_(n+1)=avg(x_n,f(x_n)) converges to fixed point of f

Source: VJIMC 1999 2.2

July 29, 2021
limitsreal analysisfunction

Problem Statement

Let a,b∈Ra,b\in\mathbb R, a≤ba\le b. Assume that f:[a,b]→[a,b]f:[a,b]\to[a,b] satisfies f(x)−f(y)≤∣x−y∣f(x)-f(y)\le|x-y| for every x,y∈[a,b]x,y\in[a,b]. Choose an x1∈[a,b]x_1\in[a,b] and define xn+1=xn+f(xn)2,n=1,2,3,….x_{n+1}=\frac{x_n+f(x_n)}2,\qquad n=1,2,3,\ldots.Show that {xn}n=1∞\{x_n\}^\infty_{n=1} converges to some fixed point of ff.
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