Convergence results for fixed point iterative algorithms in metric spaces

Rus, Ioan A.

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Issue no: Vol 35/2019 no. 2 Tags: Metric space, Stability, Convergence, Demicompact operator, iterative algorithm, asymptotic regularity, quasinonexpansive operator, well posedness of fixed point problem, Fixed point, open problem

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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.