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

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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.

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Correspondence to Jimmie Lawson.

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Lawson, J., Lim, Y. A Lipschitz constant formula for vector addition in cones with applications to Stein-like equations. Positivity 16, 81–95 (2012). https://doi.org/10.1007/s11117-011-0112-1

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