Pseudo Asymptotic Solutions of Fractional Order Semilinear Equations

Abstract

Using a generalization of the semigroup theory of linear operators, we prove existence and uniqueness of mild solutions for the semilinear fractional order differential equation $${D}^{\alpha+1}_t u(t) + \mu {D}_t^{\beta} u(t) - Au(t) = f(t,u(t)), t\in (0,\infty), \alpha \in (0,\infty), \alpha \leq \beta \leq 1, \, \mu \geq 0, $$ with the property that the solution can be written as $u=f+h$ where $f$ belongs to the space of periodic (resp. almost periodic, compact almost automorphic, almost automorphic) functions and $h$ belongs to the space $ P_0(\mathbb{R}_{+},X):= \{ \phi\in BC(\mathbb{R}_{+},X) \, :\,\, \lim_{T \to \infty}\frac{1}{T} \int_{0}^{T}||\phi(s)||ds=0 \}$. Moreover, this decomposition is unique.