Differential equations with a difference quotient

Abstract

The purpose of this paper is to study a class of ill-posed differential equations. In some settings, these differential equations exhibit uniqueness but not existence, while in others they exhibit existence but not uniqueness. An example of such a differential equation is, for a polynomial $P$ and continuous functions $f(t,x):[0,1]\times [0,1]\to \mathbb{R}$, $$ \frac{\partial}{\partial t} f(t,x) = \frac{ P(f(t,x))-P(f(t,0))}{x}, \quad x>0. $$ These differential equations are related to inverse problems.

Keywords

• Ill-posed
• differential equation with difference quotient
• existence without uniqueness
• uniqueness without existence
• inverse problem