Abstract degenerate multi-term fractional differential equations with Riemann-Liouville derivatives

Marko Kostić

In this paper, we investigate the following abstract multi-term fractional differential equation: \[%\begin{equation}abel{R-L} BD_t^{lpha_{n}} u(t) + um imits_{j=1}^{n-1}A_{j}D_t^{lpha_{j}} u(t)= AD_t^{lpha} u(t)+f(t), \quad t ı (0,au), \]%\end{equation} where $n\in {\mathbb N}\setminus \{1\},$ $A,$ $B$ and $A_{1}, \ldots ,A_{n-1}$ are closed linear operators on a complex Banach space $E,$ $0 \leq \alpha_{1}<\cdot \cdot \cdot<\alpha_{n},$ $0\leq \alpha<\alpha_{n},$ $0<\tau \leq \infty,$ $f(t)$ is an $E$-valued function, and $D_{t}^{\alpha}$ denotes the Riemann-Liouville fractional derivative of order $\alpha$ $($see $[$E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, Eindhoven, $2001])$. We introduce and further analyze some new types of degenerate $k$-regularized $(C_{1},C_{2})$-existence and uniqueness $($propagation$)$ families for the previous equation.