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.