Note on Vector Translation with Respect to Thompson's Metric. (3rd August 2017)

• Record Type: Journal Article
• Authors: Herzog, Gerd; Kunstmann, Peer C.
• Is Part Of: Numerical functional analysis and optimization. Volume 38:Number 8(2017)
• Page Start: 1008
• Page End: 1013
• Publication Date: 2017-08-03
• Subjects: Boundary value problems -- contractions -- fixed points -- Thompson's part metric; 46B40 -- 34B15; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2017.1318916
• Languages: English
• ISSNs: 0163-0563

Abstract

ABSTRACT: It is known that vector translations are contractive with respect to Thompson's part metric. Here, we give a simple proof, based on a representation of Thompson's metric through positive functionals. Moreover, we use contractivity of translations to prove a fixed point result for mappings that are Lipschitz continuous with respect to Thompson's metric with Lipschitz constant r >1. The case r = 1 for order preserving or order reversing mappings has been recently studied by Lawson and Lim. We apply our result to a nonlinear boundary value problem.