The retraction-displacement condition in the theory of fixed point equation with a convergent iterative algorithm

Abstract

Let (X, d) be a complete metric space and f: X → X be an operator with a nonempty fixed point set, i.e., Ff:={x∈X:x=f(x)}≠∅. We consider an iterative algorithm with the following properties: (1) for each x ∈ X there exists a convergent sequence (xn(x)) such that xn(x)→x∗(x)∈Ff as n→∞; (2) if x ∈ Ff, then xn (x) = x, for all n∈N. In this way, we get a retraction mapping r: X → Ff, given by r(x) = x∗(x). Notice that, in the case of Picard iteration, this retraction is the operator f∞, see I.A. Rus (Picard operators and applications, Sci. Math. Jpn. 58(1):191-219, 2003).By definition, the operator f satisfies the retraction-displacement condition if there is an increasing function ψ:R+→R+ which is continuous at 0 and satisfies ψ(0) = 0, such that (Formula presented). In this paper, we study the fixed point equation x = f(x) in terms of a retraction-displacement condition. Some examples, corresponding to Picard, Krasnoselskii, Mann and Halpern iterative algorithms, are given. Some new research directions and open questions are also presented.