An iterative method for estimation the roots of real-valued functions

In this paper we study the recursive sequence $x_{n+1}=\frac{x_n+f(x_n)}{2}$ for each continuous real-valued function $f$ on an interval $[a,b]$, where $x_0$ is an arbitrary point in $[a,b]$. First, we present some results for real-valued continuous function $f$ on $[a,b]$ which have a unique fixed point $c\in (a,b)$ and show that the sequence $\{x_n\}$ converges to $c$ provided that $f$ satisfies some conditions. By assuming that $c$ is a root of $f$ instead of being its fixed point, we extend these results. We define two other sequences by $x^{+}_0=x^{-}_0=x_0\in [a,b]$ and $x^{+}_{n+1}=x^{+}_n+\frac{f(x^{+}_n)}{2}$ and $x^{-}_{n+1}= x^{-}_n-\frac{f(x^{-}_n)}{2}$ for each $n\ge 0$. We show that for each real-valued continuous function $f$ on $[a,b]$ with $f(a)>0>f(b)$ which has a unique root $c\in (a,b)$, the sequence $\{x^{+}_n\}$ converges to $c$ provided that $f^{'}\ge -2$ on $(a,b)$. Accordingly we show that for each real-valued continuous function $f$ on $[a,b]$ with $f(a)<0<f(b)$ which has a unique root $c\in (a,b)$, the sequence $\{x^{-}_n\}$ converges to $c$ provided that $f^{'}\le 2$ on $(a,b)$. By an example we also show that there exists some continuous real-valued function $f:[a,b]\to [a,b]$ such that the sequence $\{x_{n}\}$ does not converge for some $x_0\in [a,b]$.