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Zbl 1166.34033
Araya, Daniela; Lizama, Carlos
Almost automorphic mild solutions to fractional differential equations.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 69, No. 11, A, 3692-3705 (2008). ISSN 0362-546X

Authors' abstract: We introduce the concept of $\alpha$-resolvent families to prove the existence of almost automorphic mild solutions to the differential equation $$D^\alpha_t u(t) = Au(t) + t^n f(t), 1 \leq \alpha \leq 2, n\in \Bbb Z$$ considered in a Banach space $X$, where $f: R \rightarrow X$ is almost automorphic. We also prove the existence and uniqueness of an almost automorphic mild solution of the semilinear equation $$D^{\alpha}_t u(t) = Au(t) + f(t, u(t)), \quad 1 \leq \alpha \leq 2$$ assuming $f(t, x)$ is almost automorphic in $t$ for each $x \in X$, satisfies a global Lipschitz condition and takes values on $X$. Finally, we prove also the existence and uniqueness of an almost automorphic mild solution of the semilinear equation $$D^{\alpha}_t u(t) = Au(t) + f(t, u(t), u'(t)),\quad 1 \leq \alpha \leq 2$$ under analogous conditions as in the previous case.
[Nasser-eddine Tatar (Dhahran)]
MSC 2000:
*34G20 Nonlinear ODE in abstract spaces
26A33 Fractional derivatives and integrals (real functions)
43A60 Almost periodic functions on groups, etc.

Keywords: Almost automorphic function; resolvent family; fractional derivative

Cited in: Zbl 1221.34015

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