Unique Solutions to Power-Transformed Affine Systems

Systems of the form $x=(A{x}^s{)}^{1\mathrm{/}s}+b$ arise in a range of economic, financial and control problems, where $A$ is a linear operator acting on a space of real-valued functions (or vectors) and $s$ is a nonzero real value. In these applications, attention is focused on positive solutions. We provide a simple and complete characterization of existence and uniqueness of positive solutions under conditions on $A$ and $b$ that imply positivity.