# A Lipschitz constant formula for vector addition in cones with applications to Stein-like equations

A Lipschitz constant formula for vector addition in cones with applications to Stein-like equations For a cone C equipped with the Thompson metric and $${a\in C}$$ , we show that the translation map $${x\mapsto x+a}$$ is a strict contraction on any lower (or initial) set $${C\cap x-C}$$ of the cone and derive an explicit formula for the Lipschitz constant. We apply our results to Stein equations, Riccati equations, and Ferrante–Levy equations on normal cones of Banach spaces to establish the existence, uniqueness and continuous dependence on parameters of positive solutions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# A Lipschitz constant formula for vector addition in cones with applications to Stein-like equations

Positivity, Volume 16 (1) – Feb 24, 2011
15 pages

Publisher
Springer Journals
Copyright © 2011 by Springer Basel AG
Subject
Mathematics; Potential Theory; Operator Theory; Fourier Analysis; Econometrics; Calculus of Variations and Optimal Control; Optimization
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-011-0112-1
Publisher site
See Article on Publisher Site

### Abstract

For a cone C equipped with the Thompson metric and $${a\in C}$$ , we show that the translation map $${x\mapsto x+a}$$ is a strict contraction on any lower (or initial) set $${C\cap x-C}$$ of the cone and derive an explicit formula for the Lipschitz constant. We apply our results to Stein equations, Riccati equations, and Ferrante–Levy equations on normal cones of Banach spaces to establish the existence, uniqueness and continuous dependence on parameters of positive solutions.

### Journal

PositivitySpringer Journals

Published: Feb 24, 2011

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