Convergence results for fixed point iterative algorithms in metric spaces.

Let (X, d) be a metric space, f, fn : X → X, with Ff = ..., n ∈ N. For the fixed point equation (1) x = f(x) we consider the following iterative algorithm, (2) x ∈ X; x0 = x; xn+1(x) = fn(xn(x)); n ∈ N. By definition, the algorithm (2) is convergent if, xn(x) → x*(x) ∈ Ff as n → ∞, ∀ x ∈ X. In this paper we give some conditions on fn and f which imply the convergence of algorithm (2). In this way we improve some results given in [Rus, I. A., An abstract point of view on iterative approximation of fixed points. impact on the theory of fixed point equations, Fixed Point Theory, 13 (2012), No. 1, 179-192]. In our results, in general we do not suppose that, Ff ≠ ∅. Some research directions are formulated. [ABSTRACT FROM AUTHOR]