Is the following estimate of this integral correct?

Question

Let $\epsilon>0$ and suppose that $U_t^\epsilon, V_t^\epsilon, \Gamma^{2\epsilon}_t$ are some real functions of time (depending on $\epsilon$) defined on $[0,T]$, $T < \infty$ and assume that for $p >1$ $$|U_t^\epsilon|^p \leq C (1+\int_0^t |V_r^\epsilon|^p dr)$$ for every $t \in [0,T]$, for some $C>0$ independent of $\epsilon$ and that $\Gamma^{2\epsilon}_t \leq C$ for some $C>0$ independent of $\epsilon$.
Assume moreover that $G(x,y)$ is a sublinear function.
Now define \begin{align*}|V_t^\epsilon| & = \int_0^t \frac{1}{\epsilon} e^{-\frac{2 \delta}{\epsilon} (t-s)} |G(U_s^\epsilon, \Gamma^{2\epsilon}_s)| ds \\ & = \int_0^{t/\epsilon} e^{-2 \delta (t / \epsilon - s)} |G(U_{\epsilon s}^\epsilon, \Gamma^{2\epsilon}_{\epsilon s})| ds \end{align*} Is then the following calculation that I've done correct?
By Holder's inequality, some changes of variables and the sublinearity of $G$ have: \begin{align*} |V_t^\epsilon|^p & \leq \left( \int_0^{t / \epsilon} e^{-\frac{ \delta p}{ (p-1)} (t / \epsilon - s)} ds \right)^{p-1} \int_0^{t / \epsilon} e^{-\delta p (t / \epsilon - s)} |G(U_{\epsilon s}^\epsilon, \Gamma^{2\epsilon}_{\epsilon s})| ^p ds\\ &\leq \left( \int_0^{t / \epsilon } e^{-\frac{ \delta p}{ (p-1)} \sigma} d\sigma \right)^{p-1} \int_0^{t } \frac{1}{\epsilon} e^{-\delta p (t - s)/ \epsilon} (1+|U_s^\epsilon|^P+ |\Gamma^{2\epsilon}_s|^p ) ds \\ & \leq C+ C \int_0^{t } \frac{1}{\epsilon} e^{-\delta p (t - s)/ \epsilon}\int_0^s |V_r^\epsilon|^p drds \\ & =C + C \int_0^{t } \left( \int_0^{(t-r)/ \epsilon} e^{-\delta p \sigma} d\sigma \right) |V_r^\epsilon|^p dr \\ & \leq C + C \int_0^{t } |V_r^\epsilon|^p dr \end{align*} for some $C>0$ independent of $\epsilon$.