A convergence result on random products of mappings in metric spaces

Mohamed Amine Khamsi; Issam Louhichi

doi:10.1186/1687-1812-2012-43

A convergence result on random products of mappings in metric spaces

Mohamed Amine Khamsi^*, Issam Louhichi

^*Corresponding author for this work

King Fahd University of Petroleum & Minerals

Research output: Contribution to journal › Article › peer-review

Abstract

Let X be a metric space and {T₁,⋯, 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 x₀ ∈ D; and x_n+1 = T_r(n)(x _n), for all n ≥ 0. In particular, we extend the study of Bauschke [1] from the linear case of Hilbert spaces to metric spaces. Similarly we show that the examples of convergence hold in the absence of compactness. These type of methods have been used in areas like computerized tomography and signal processing.

Original language	English
Article number	43
Journal	Fixed Point Theory and Algorithms for Sciences and Engineering
Volume	2012
DOIs	https://doi.org/10.1186/1687-1812-2012-43
State	Published - 2012

Bibliographical note

Funding Information:
The authors gratefully acknowledge the financial support from King Fahd University of Petroleum & Minerals for supporting research project SB 101021. The authors would also like to thank the referees for excellent suggestions and useful comments.

Keywords

Computerized tomography
Convex feasibility problem
Convex programming
Fejér monotone sequence
Image reconstruction
Image recovery
Innate bounded regularity
Kaczmarz's method
Nonexpansive mapping
Projection algorithm
Projective mapping
Random product
Signal processing
Unrestricted iteration
Unrestricted product

ASJC Scopus subject areas

Geometry and Topology
Applied Mathematics

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10.1186/1687-1812-2012-43

Cite this

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title = "A convergence result on random products of mappings in metric spaces",

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 x0 ∈ D; and xn+1 = Tr(n)(x n), for all n ≥ 0. In particular, we extend the study of Bauschke [1] from the linear case of Hilbert spaces to metric spaces. Similarly we show that the examples of convergence hold in the absence of compactness. These type of methods have been used in areas like computerized tomography and signal processing.",

keywords = "Computerized tomography, Convex feasibility problem, Convex programming, Fej{\'e}r monotone sequence, Image reconstruction, Image recovery, Innate bounded regularity, Kaczmarz's method, Nonexpansive mapping, Projection algorithm, Projective mapping, Random product, Signal processing, Unrestricted iteration, Unrestricted product",

author = "Khamsi, {Mohamed Amine} and Issam Louhichi",

note = "Funding Information: The authors gratefully acknowledge the financial support from King Fahd University of Petroleum & Minerals for supporting research project SB 101021. The authors would also like to thank the referees for excellent suggestions and useful comments. ",

year = "2012",

doi = "10.1186/1687-1812-2012-43",

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journal = "Fixed Point Theory and Algorithms for Sciences and Engineering",

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AU - Khamsi, Mohamed Amine

AU - Louhichi, Issam

N1 - Funding Information: The authors gratefully acknowledge the financial support from King Fahd University of Petroleum & Minerals for supporting research project SB 101021. The authors would also like to thank the referees for excellent suggestions and useful comments.

PY - 2012

Y1 - 2012

N2 - 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 x0 ∈ D; and xn+1 = Tr(n)(x n), for all n ≥ 0. In particular, we extend the study of Bauschke [1] from the linear case of Hilbert spaces to metric spaces. Similarly we show that the examples of convergence hold in the absence of compactness. These type of methods have been used in areas like computerized tomography and signal processing.

AB - 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 x0 ∈ D; and xn+1 = Tr(n)(x n), for all n ≥ 0. In particular, we extend the study of Bauschke [1] from the linear case of Hilbert spaces to metric spaces. Similarly we show that the examples of convergence hold in the absence of compactness. These type of methods have been used in areas like computerized tomography and signal processing.

KW - Computerized tomography

KW - Convex feasibility problem

KW - Convex programming

KW - Fejér monotone sequence

KW - Image reconstruction

KW - Image recovery

KW - Innate bounded regularity

KW - Kaczmarz's method

KW - Nonexpansive mapping

KW - Projection algorithm

KW - Projective mapping

KW - Random product

KW - Signal processing

KW - Unrestricted iteration

KW - Unrestricted product

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SN - 1687-1820

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JO - Fixed Point Theory and Algorithms for Sciences and Engineering

JF - Fixed Point Theory and Algorithms for Sciences and Engineering

M1 - 43

ER -

A convergence result on random products of mappings in metric spaces

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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Cite this