Let (X,d) be a metric space, f, f_n:X\to X, with F_f=F_{f_n}, n\in\mathbb{N}. For the fixed point equation

(1)   \begin{equation*} x=f(x) \end{equation*}

we consider the following iterative algorithm,

(2)   \begin{equation*} x\in X,\ x_0=x,\ x_{n+1}(x)=f_n(x_n(x)),\ n\in\mathbb{N}. \end{equation*}

By definition, the algorithm \eqref{equ2} is convergent if,

    \[x_n(x)\to x^*(x)\in F_f\text{ as }n\to\infty,\ \forall\ x\in X.\]

In this paper we give some conditions on \underline{f_n and f} which imply the convergence of algorithm \eqref{equ2}. In this way we improve some results given in [ Rus, I. A., {\em An abstract point of view on iterative approximation of fixed points: impact on the theory of fixed point equations}, Fixed Point Theory, \textbf{13} (2012), No. 1, 179–192].
In our results, in general we do not suppose that, F_f\not=\emptyset. Some research directions are formulated.

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Rus, Ioan A.