A note on one-dimensional time fractional ODEs

TL;DR

This paper revisits fundamental properties of one-dimensional time fractional ODEs using a specific Caputo derivative definition, establishing comparison principles, asymptotic behaviors, and stability results with simplified proofs.

Contribution

It introduces new comparison principles, asymptotic analysis, and simplified proofs for stability in time fractional ODEs based on a particular Caputo derivative definition.

Findings

Established generalized comparison principles.

Determined full asymptotic behaviors of solutions.

Provided simplified proofs for monotonicity and stability.

Abstract

In this note, we prove or re-prove several important results regarding one dimensional time fractional ODEs following our previous work \cite{fllx17}. Here we use the definition of Caputo derivative proposed in \cite{liliu17frac1,liliu2017} based on a convolution group. In particular, we establish generalized comparison principles consistent with the new definition of Caputo derivatives. In addition, we establish the full asymptotic behaviors of the solutions for $D_c^{\gamma}u=Au^p$. Lastly, we provide a simplified proof for the strict monotonicity and stability in initial values for the time fractional differential equations with weak assumptions.