Is it possible to prove $\varphi(t),\psi(t)=O(t^{-\varepsilon})$ as $t\to\infty$, based on the following differential inequality?

Question

Let $\varphi$ and $\psi$ be two nonnegative functions in $C^1[0,\infty)$, such that $\varphi(0)=\psi(0)=0$, $c_1\varphi\leq\psi\leq c_2\varphi$ for some constants $c_2>c_1>0$. For constants $\varepsilon_0\in(0,1],\ M>0$ and $0<\alpha<1$, $\varphi$ and $\psi$ satisfy the following differential inequality: \begin{equation} \frac{\mathrm{d}}{\mathrm{d} t}\psi(t)\leq\begin{cases} -\varphi(t)+\alpha\Big(\frac{1}{n}\sum\limits_{k=1}^{n}\varphi(k)\Big)+M(1\wedge t^{-\varepsilon_0}) &n\leq t< n+1,\ n\in\mathbb{Z}_{>0}\\ -\varphi(t)+M(1\wedge t^{-\varepsilon_0}) &0\leq t<1. \end{cases} \end{equation} Then is there a constant $C>0$ independent of $t$ such that $\varphi(t),\psi(t)\leq Ct^{-\varepsilon}$ for some $0<\varepsilon<\min\{1-\alpha, \varepsilon_0\}$? Or is there any counter example?