On a fractional differential equation with infinitely many solutions

TL;DR

This paper identifies specific conditions under which a fractional differential equation with Caputo derivative admits infinitely many solutions starting from zero, highlighting the complex solution structure of such equations.

Contribution

It provides new restrictions on fractional differential equations that guarantee the existence of infinitely many solutions, advancing understanding of their solution space.

Findings

Infinite solutions exist under certain restrictions

Conditions involve the structure of the function g

Results apply to Caputo fractional derivatives

Abstract

We present a set of restrictions on the fractional differential equation $x^{(\alpha)}(t)=g(x(t))$, $t\geq0$, where $\alpha\in(0,1)$ and $g(0)=0$, that leads to the existence of an infinity of solutions starting from $x(0)=0$. The operator $x^{(\alpha)}$ is the Caputo differential operator.