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Almost automorphic solutions to a class of semilinear fractional differential equations. (English) Zbl 1192.34006

The authors consider the existence of almost automorphic mild solutions to the fractional differential equation
\[ D^{\alpha}_{t}u(t)=Au(t)+D^{\alpha-1}_{t}f(t,u(t)),\quad t\in \mathbb R, \]
where \(A\) is a linear sectorial operator densely defined in a Banach space \(X\) of type \(\omega<0\), \(D^{\alpha}_{t}\) is the fractional derivative in the Riemann-Liouville sense with \(1<\alpha<2\), and \(f(t,u)\) is an almost automorphic function in \(t\in\mathbb R\) satisfying some Lipschitz type conditions.
These results go back to the reviewer’s paper [“Existence and uniqueness of almost automorphic mild solutions of some semilinear abstract differential equations”, Semigroup Forum 69, No. 1, 80–86 (2004; Zbl 1077.47058)]. The technique used is the Banach fixed point theorem. An example is given to illustrate the abstract results.

MSC:

34A08 Fractional ordinary differential equations
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
34G20 Nonlinear differential equations in abstract spaces

Citations:

Zbl 1077.47058
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References:

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