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.


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
Full Text: DOI


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