In this paper, we establish some new properties of $(\omega ,c)$-periodic and asymptotically $(\omega ,c)$-periodic functions, then we apply them to study the existence and uniqueness of mild solutions of these types to the following semilinear fractional differential equations: (1) $\{\begin{array}{rl}{}^c{D}_t^{\alpha }u(t)& =Au(t){+}^c{D}_t^{\alpha -1}f(t,u(t)),1<\alpha <2,t\in \mathbb{R},\\ u(0)& =0\end{array}$(1) and (2) $\{\begin{array}{rl}{}^c{D}_t^{\alpha }u(t)& =Au(t){+}^c{D}_t^{\alpha -1}f(t,u(t-h)),1<\alpha <2,t,h\in {\mathbb{R}}_{+},\\ u(0)& =0\end{array}$(2) where ${}^c{D}_t^{\alpha }(\cdot )(1<\alpha <2)$ stands for the Caputo derivative and A is a linear densely defined operator of sectorial type on a complex Banach space $\mathbb{X}$ and the function $f(t,x)$ is $(\omega ,c)$-periodic or asymptotically $(\omega ,c)$-periodic with respect to the first variable. Our results are obtained using the Leray–Schauder alternative theorem, the Banach fixed point principle and the Schauder theorem. Then we illustrate our main results with an application to fractional diffusion-wave equations.

在本文中，我们建立了一些新的性质$(\omega ,C)$-周期性和渐近性$(\omega ,C)$-周期函数，然后我们应用它们来研究以下半线性分数阶微分方程的这些类型的温和解的存在性和唯一性：(1)$\{\begin{array}{rl}{}^C{丁}_{吨}^{\alpha }你(吨)& =A你(吨){+}^C{丁}_{吨}^{\alpha -\mathrm{1个}}F(吨,你(吨)),\mathrm{1个}<\alpha <\mathrm{2个},吨\in \mathbb{R},\\ 你(0)& =0\end{array}$(1)和(2)$\{\begin{array}{rl}{}^C{丁}_{吨}^{\alpha }你(吨)& =A你(吨){+}^C{丁}_{吨}^{\alpha -\mathrm{1个}}F(吨,你(吨-H)),\mathrm{1个}<\alpha <\mathrm{2个},吨,H\in {\mathbb{R}}_{+},\\ 你(0)& =0\end{array}$(2)在哪里${}^C{丁}_{吨}^{\alpha }(\cdot )(\mathrm{1个}<\alpha <\mathrm{2个})$代表 Caputo 导数，A是复 Banach 空间上扇形类型的线性稠密定义算子$\mathbb{X}$和功能$F(吨,X)$是$(\omega ,C)$-周期性或渐近地$(\omega ,C)$-关于第一个变量的周期性。我们的结果是使用 Leray–Schauder 替代定理、Banach 不动点原理和 Schauder 定理获得的。然后我们通过对分数阶扩散波动方程的应用来说明我们的主要结果。