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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
\[ D^\alpha_t u(t) = Au(t) + t^n f(t), 1 \leq \alpha \leq 2, n\in \mathbb Z \]
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:

34G20 Nonlinear differential equations in abstract spaces
26A33 Fractional derivatives and integrals
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
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