Asymptotically Almost Periodic and Asymptotically Almost Automorphic Solutions of Abstract Degenerate Multi-Term Fractional Differential Inclusions With Riemann-Liouville Derivatives

Marko Kostić

Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 44 (2019) no. 1.

Résumé

In this paper, we investigate the functional analytical approach for seeking of solutions to the following abstract multi-term fractional differential inclusion: \[ {\mathcal B}D_t^{lpha_{n}} u(t)+um_{j=1}^{n-1}{{\mathcal A}_{j}}D_t^{lpha_{j}}u(t)ı{\mathcal A}D_t^{lpha}u(t)+f(t), \quad tı(0,au),ag{*} \] where $n\in{\mathbb N}\setminus\{1\}$, ${\mathcal A}$, ${\mathcal B}$ and ${\mathcal A}_{j}$ are multivalued linear operators on a complex Banach space $X$ $(1\leq j\leq n-1)$, $0\leq\alpha_1\cdots\alpha_{n}$, $0\leq\alpha\alpha_n$, $0\tau\leq \infty$, $f(t)$ is an $X$-valued function, and $D_{t}^{\alpha}$ denotes the Riemann--Liouville fractional derivative of order $\alpha$ (see Ph.D. Thesis by E. Bazhlekova, Eindhoven University of Technology, 2001). We introduce and analyze several different types of solutions and degenerate $k$-regularized $(C_{1},C_{2})$-existence and uniqueness (propagation) families for $(*)$. Asymptotically almost periodic and asymptotically almost automorphic solutions of $(*)$ are sought in the case that ${\mathcal B}=I$ $($the identity operator on $X)$, $A_{j}\in L(X)$ for $1\leq j\leq n-1$ and ${\mathcal A}$ is a convenable chosen translation of a $C$-almost sectorial multivalued linear operator.