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Published in:
2016 | OriginalPaper | Chapter
The Retraction-Displacement Condition in the Theory of Fixed Point Equation with a Convergent Iterative Algorithm
Authors : V. Berinde, A. Petruşel, I. A. Rus, M. A. Şerban
Published in: Mathematical Analysis, Approximation Theory and Their Applications
Publisher: Springer International Publishing
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Abstract
Let (X, d) be a complete metric space and f: X → X be an operator with a nonempty fixed point set, i.e., \(F_{f}:=\{ x \in X: x = f(x)\}\neq \emptyset\). We consider an iterative algorithm with the following properties:
(1)
for each x ∈ X there exists a convergent sequence (x
n
(x)) such that \(x_{n}(x) \rightarrow x^{{\ast}}(x) \in F_{f}\) as \(n \rightarrow \infty\);
(2)
if x ∈ F
f
, then x
n
(x) = x, for all \(n \in \mathbb{N}\).
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 \(f^{\infty }\), 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 \(\psi: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}\) which is continuous at 0 and satisfies ψ(0) = 0, such that 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.
$$\displaystyle{d(x,r(x)) \leq \psi (d(x,f(x)),\mbox{ for all }x \in X.}$$