Open Access Research

A convergence result on random products of mappings in metric trees

Saleh A Al-Mezel1* and Mohamed A Khamsi2

Author Affiliations

1 Department of Mathematics, King Abdulaziz University P.O. Box 80203, Jeddah 21589, Saudi Arabia

2 Department of Mathematical Sciences, The University of Texas at El Paso El Paso, TX 79968, USA

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Fixed Point Theory and Applications 2012, 2012:57 doi:10.1186/1687-1812-2012-57

Published: 13 April 2012

Abstract

Let X be a metric space and {T1, ..., TN} 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 (xn) defined by

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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.

Keywords:
computerized tomography; convex feasibility problem; convex programming; metric tree; nonexpansive mapping; projection algorithm; projective mapping; random product; signal processing; unrestricted iteration; unrestricted product