Inequality for integro-differential system


Reading a paper about homogenization of stochastic coefficients [A. Gloria et al., Invent. math. (2015)], I found the following lemma that gives an estimate of the solution for a given ODE system. The lemma is:

Let $ 1\le p,\gamma<\infty$ and $ a(t),b(t)\ge 0$ . Suppose that there exists $ C_1 <\infty$ such that for all $ t\ge0$ , \begin{gather} \tag{1}\label{eq1} a(t) \le C_1 \left( (t+1)^{-\gamma} + \int_{0}^{t} (t-s+1)^{-\gamma} b(s) \,ds \right), \ b(t)^p \le C_1\left(-\frac{d}{dt}a(t)^p\right). \end{gather} Then there exists $ C_2<\infty$ depending only on $ C_1,p$ and $ \gamma$ such that \begin{equation} \tag{2}\label{eq2} a(t) \le C_2 (t+1)^{-\gamma}. \end{equation}

My question is: Is it possible to prove a similar result replacing \eqref{eq1} by \begin{equation*} a(t) \le C_1 \left( t^{-\gamma} + \int_{0}^{t} (t-s)^{-\gamma} b(s) \,ds \right)? \end{equation*} How would \eqref{eq2} change?