On the Existence and Uniqueness of Solutions of Certain Classes

Abstract

In this paper, we investigate the existence and
uniqueness of solutions for the following abstract multi-term
fractional differential equation:
\begin{align}\label{FDEab}
\begin{split}
& {\mathbf D}_{t}^{\alpha_{n}}u(t)+ \sum
\limits_{i=1}^{n-1}A_{i}{\mathbf D}_{t}^{\alpha_{i}}u(t)= 0,
\quad t > 0, \\
& u^{(k)}(0)=u_k,\quad k=0,\cdot \cdot \cdot, \lceil \alpha_{n}
\rceil -1,
\end{split}
\end{align}
where $n\in {\mathbb N}\setminus \{1,2\},$ $A_{1},\cdot \cdot \cdot
,A_{n-1}$ are closed linear operators on a sequentially complete
locally convex space $X,$ $0 = \alpha_{1}<\cdot \cdot
\cdot<\alpha_{n},$ and ${\mathbf D}_{t}^{\alpha}$ denotes the Caputo
fractional derivative of order $\alpha$ (\cite{baj}). Plenty of
various examples illustrates our abstract theoretical results
obtained throughout the paper.