The Retraction-Displacement Condition in the Theory of Fixed Point Equation with a Convergent Iterative Algorithm

Let (X, d) be a complete metric space and f: X → X be an operator with a nonempty fixed point set, i.e., F f : = { x ∈ X : x = f ( x ) } ≠ ∅ $$F_{f}:=\{ x \in X: x = f(x)\}\neq \emptyset$$ . We consider an iterative algorithm with the following properties: (1) for each x ∈ X there exists a convergent sequence (x n (x)) such that x n ( x ) → x ∗ ( x ) ∈ F f $$x_{n}(x) \rightarrow x^{{\ast}}(x) \in F_{f}$$ as n → ∞ $$n \rightarrow \infty$$ ; (2) if x ∈ F f , then x n (x) = x, for all n ∈ ℕ $$n \in \mathbb{N}$$ . In this way, we get a retraction mapping r: X → F f , given by r(x) = x ∗(x). Notice that, in the case of Picard iteration, this retraction is the operator f ∞ $$f^{\infty }$$ , 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 + $$\psi: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$$ which is continuous at 0 and satisfies ψ(0) = 0, such that d ( x , r ( x ) ) ≤ ψ ( d ( x , f ( x ) ) , for all x ∈ X . $$\displaystyle{d(x,r(x)) \leq \psi (d(x,f(x)),\mbox{ for all }x \in X.}$$ 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.