Note on Vector Translation with Respect to Thompson’s Metric

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.