Differential Equations with a Difference Quotient

TL;DR

This paper investigates a class of ill-posed differential equations related to inverse problems, highlighting issues of existence and uniqueness, and providing insights into their mathematical properties.

Contribution

It introduces and analyzes a new class of ill-posed differential equations involving difference quotients, exploring their uniqueness and existence properties.

Findings

Differential equations exhibit uniqueness without existence in some cases.

In other cases, they exhibit existence without uniqueness.

The equations are connected to inverse problems.

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 $f(t,x):[0,1]\times [0,1]\rightarrow \mathbb{R}$, \begin{equation*} \frac{\partial}{\partial t} f(t,x) = \frac{ P(f(t,x))-P(f(t,0))}{x}, \quad x>0. \end{equation*} These differential equations are related to inverse problems.