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

Marko Kostić

Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 41 (2016) no. 1.

Résumé

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.