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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 t α u(t)=Au(t)+D t α-1 f(t,u(t)),t,

where A is a linear sectorial operator densely defined in a Banach space X of type ω<0, D t α is the fractional derivative in the Riemann-Liouville sense with 1<α<2, and f(t,u) is an almost automorphic function in t 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:
34A08Fractional differential equations
43A60Almost periodic functions on groups, etc.; almost automorphic functions
34G20Nonlinear ODE in abstract spaces