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Almost automorphic mild solutions to fractional differential equations. (English) Zbl 1166.34033
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.

MSC:
34G20Nonlinear ODE in abstract spaces
26A33Fractional derivatives and integrals (real functions)
43A60Almost periodic functions on groups, etc.; almost automorphic functions
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References:
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