Self-similar Differential Equations

Abstract

Differential equations where the graph of some derivative of a function is composed of a finite number of similarity transformations of the graph of the function itself are defined. We call these self-similar differential equations (SSDEs) and prove existence and uniqueness of solution under certain conditions. While SSDEs are not ordinary differential equations, the technique for demonstrating existence and uniqueness of SSDEs parallels that for ODEs. This paper appears to be the first work on equations of this nature.