Unique Solutions to Power-Transformed Affine Systems

TL;DR

This paper characterizes the existence and uniqueness of positive solutions for power-transformed affine systems, which are relevant in economic, financial, and control applications, under specific conditions on the system parameters.

Contribution

It provides a simple, complete characterization of when positive solutions exist and are unique for these systems, extending understanding in this mathematical class.

Findings

Conditions on A and b ensure positivity of solutions

Complete criteria for existence and uniqueness of solutions

Applicable to economic, financial, and control problems

Abstract

Systems of the form $x = (A x^s)^{1/s} + b$ 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.