On asymptotic periodic solutions of fractional differential equations and applications

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form D C α u ( t ) = A u ( t ) + f ( t ) , u ( 0 ) = x , 0 > α ≤ 1 , ( ∗ ) D^{\alpha }_Cu(t)=Au(t)+f(t), u(0)=x, 0>\alpha \le 1, ( *) where D C α u ( t ) D^{\alpha }_Cu(t) is the derivative of the function u u in the Caputo’s sense, A A is a linear operator in a Banach space X \mathbb {X} that may be unbounded and f f satisfies the property that lim t → ∞ ( f ( t + 1 ) − f ( t ) ) = 0 \lim _{t\to \infty } (f(t+1)-f(t))=0 which we will call asymptotic 1 1 -periodicity. By using the spectral theory of functions on the half line we derive analogs of Katznelson-Tzafriri and Massera Theorems. Namely, we give sufficient conditions in terms of spectral properties of the operator A A for all asymptotic mild solutions of Eq. (*) to be asymptotic 1 1 -periodic, or there exists an asymptotic mild solution that is asymptotic 1 1 -periodic.