A convergence result on random products of mappings in metric trees

• Published in: Fixed point theory and algorithms for sciences and engineering Vol. 2012; no. 1; pp. 1 - 10
• Main Authors: Al-Mezel, Saleh Abdullah, Khamsi, Mohamed Amine
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 13.04.2012; Springer Nature B.V
• Subjects: Analysis; Applications of Mathematics; Classification; Convergence; Differential Geometry; Mapping; Mathematical analysis; Mathematical and Computational Biology; Mathematical models; Mathematics; Mathematics and Statistics; Metric space; Tomography; Topology; Trees; random product; signal processing; metric tree; unrestricted iteration; convex feasibility problem; convex programming; nonexpansive mapping; projective mapping; computerized tomography; unrestricted product; projection algorithm
• ISSN: 1687-1812; 1687-1820; 1687-1812; 2730-5422
• DOI: 10.1186/1687-1812-2012-57

Let X be a metric space and { T 1 , ..., T N } be a finite family of mappings defined on D ⊂ X . Let r : ℕ → {1,..., N } be a map that assumes every value infinitely often. The purpose of this article is to establish the convergence of the sequence ( x n ) defined by x 0 ∈ D ; and x n + 1 = T r ( n ) ( x n ) , for all n ≥ 0 . In particular we prove Amemiya and Ando's theorem in metric trees without compactness assumption. This is the first attempt done in metric spaces. These type of methods have been used in areas like computerized tomography and signal processing. Mathematics Subject Classification 2000 : Primary: 06F30; 46B20; 47E10.