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 $D_C^{\alpha }u(t)=Au(t)+f(t),u(0)=x,0<\alpha \le 1,(\ast )$ where $D_C^{\alpha }u(t)$ is the derivative of the function $u$ in the Caputo’s sense, $A$ is a linear operator in a Banach space $\text{X}$ that may be unbounded and $f$ satisfies the property that ${\lim }_{t\to \mathrm{\infty }}(f(t+1)-f(t))=0$ which we will call asymptotic $1$-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 $A$ for all asymptotic mild solutions of Eq. (*) to be asymptotic $1$-periodic, or there exists an asymptotic mild solution that is asymptotic $1$-periodic.