have $$\begin{aligned}
    \mathcal{M}_{41}&\leq& C \mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^t (t-s)^{-1-\alpha}\Big|\int_{s}^{t} S_{t-r} g(u^{N,n}_r) \mathrm{d}B^{H,N,n}_r  \Big| \mathrm{d}s \Big)^2\mathrm{d}t\Big] \cr
    &&+C\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^t (t-s)^{-1-\alpha}\Big|\int_{0}^{s}(S_{t-r}-S_{s-r}) g(u^{N,n}_r) \mathrm{d}B^{H,N,n}_r  \Big| \mathrm{d}s\Big)^2\mathrm{d}t\Big] \cr
    &\leq&
    C\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^t \frac{\Lambda_{\alpha,B^{H,N,n}}^{0,t}\int_{s}^{t}[(r-s)^{-\alpha}+(t-r)^{-\alpha}](1+|u^{N,n}_r|)\mathrm{d}r}{(t-s)^{1+\alpha}}
    \mathrm{d}s\Big)^2\mathrm{d}t\Big]\cr
    &&+
    C\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^t \frac{\Lambda_{\alpha,B^{H,N,n}}^{0,t}\int_{s}^{t}\big(\int_{s}^{r}\frac{|u^{N,n}_r-u^{N,n}_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\big)\mathrm{d}r}{(t-s)^{1+\alpha}}
    \mathrm{d}s\Big)^2\mathrm{d}t\Big]\cr
    &&+C\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^t\frac{\Lambda_{\alpha,B^{H,N,n}}^{0,t}\int_{0}^s  \frac{r^{-\alpha}+(s-r)^{-\alpha}}{(s-r)^{\beta}}(1+|u^{N,n}_r|)\mathrm{d}r}{(t-s)^{1+\alpha-\beta}}  \mathrm{d}s\Big)^2\mathrm{d}t\Big]\cr
    &&+C\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^t\frac{\Lambda_{\alpha,B^{H,N,n}}^{0,t}\int_{0}^s  (s-r)^{-\beta}(\int_{s}^{r}\frac{|u^{N,n}_r-u^{N,n}_q|}{(r-q)^{1+\alpha}}\mathrm{d}q)\mathrm{d}r}{(t-s)^{1+\alpha-\beta}}\mathrm{d}s\Big)^2\mathrm{d}t\Big]\cr
    &\leq&
    C_N\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^t\int_{0}^{r} \frac{(r-s)^{-\alpha}+(t-r)^{-\alpha}}{(t-s)^{1+\alpha}}\mathrm{d}s(1+|u^{N,n}_r|)
    \mathrm{d}r \Big)^2\mathrm{d}t\Big]\cr
    &&+
    C_N\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^t\int_{0}^{r} (t-s)^{-1-\alpha}\mathrm{d}s\Big(\int_{0}^{r}\frac{|u^{N,n}_r-u^{N,n}_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)\mathrm{d}r
    \Big)^2\mathrm{d}t\Big]\cr
    &&+C_N\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^t \int_{r}^t \frac{(s-r)^{-\beta} r^{-\alpha}+(s-r)^{-\alpha-\beta}}{(t-s)^{1+\alpha-\beta}}\mathrm{d}s(1+|u^{N,n}_r|)\mathrm{d}r\Big)^2\mathrm{d}t\Big]\cr
    &&+C_N\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^t \int_{r}^t \frac{(s-r)^{-\beta}}{(t-s)^{1+\alpha-\beta}}\mathrm{d}s\Big(\int_{0}^{r}\frac{|u^{N,n}_r-u^{N,n}_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)\mathrm{d}r\Big)^2\mathrm{d}t\Big]\cr
    &\leq& C_{N} \mathbb{E}\Big[\int_{0}^{T} \int_{0}^t (t-r)^{-\alpha}[ r^{-\alpha}+(t-r)^{-\alpha}](1+|u^{N,n}_r|^2)\mathrm{d}r\mathrm{d}t\Big]\cr
    &&+C_{N} \mathbb{E}\Big[\int_{0}^{T}\int_{0}^t \Big(\int_{0}^{r}\frac{|u^{N,n}_r-u^{N,n}_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)^2\mathrm{d}r\mathrm{d}t\Big]\cr
    &\leq& C_{N} \int_{0}^{T} (1+\mathbb{E}[\|u^{N,n}\|_{\alpha,t}^2])\mathrm{d}t.\end{aligned}$$

For $\mathcal{M}_{42}$, by Lemma \[itofbm2\] and Hölder’s inequality, we have $$\begin{aligned}
    \mathcal{M}_{42} 
    &\leq&
    C\mathbb{E}\Big[\sup _{t \in[0, T]} \Big( \Lambda_{\alpha,B^{H,N,n}}^{0,t}\int_{0}^t(r^{-\alpha}+(t-r)^{-\alpha})(1+|u^{N,n}_r|)\mathrm{d}r
    \Big)^2\Big]\cr
    &&+
    C\mathbb{E}\Big[\sup _{t \in[0, T]} \Big( \Lambda_{\alpha,B^{H,N,n}}^{0,t}\int_{0}^{t}\int_{0}^{r}\frac{|u^{N,n}_r-u^{N,n}_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\mathrm{d}r
    \Big)^2\Big]\cr
    &\leq & C_{N} \int_{0}^{T}(1+\mathbb{E}[\|u^{N,n}\|^2_{\alpha, t}])\mathrm{d}t\cr
    &&+ C_{N}  \mathbb{E}\Big[\int_{0}^{T} \Big(\int_{0}^{t}\frac{|u^{N,n}_t-u^{N,n}_s|}{(t-s)^{1+\alpha}}\mathrm{d}s\Big)^2\mathrm{d}t\Big].\end{aligned}$$

The obtained results of $\mathcal{M}_1,\mathcal{M}_{22},\mathcal{M}_{32}$ and $\mathcal{M}_{41}$ yield that $$\begin{aligned}
    \mathbb{E}\Big[\int_{0}^{T} \Big(\int_{0}^{t}\frac{|u^{N,n}_t-u^{N,n}_s|}{(t-s)^{1+\alpha}}\mathrm{d}s\Big)^2\mathrm{d}t\Big]\leq C+ C_{N} \int_{0}^{T} (1+\mathbb{E}[\|u^{N,n}\|_{\alpha,t}^2])\mathrm{d}t.\end{aligned}$$ Thus, we have $$\mathcal{M}_{42}\leq C+C_{N} \int_{0}^{T} (1+\mathbb{E}[\|u^{N,n}\|_{\alpha,t}^2])\mathrm{d}t.$$

Finally, by Gronwall’s lemma, we have $$\begin{aligned}
    \mathbb{E}[\|u^{N,n}\|^2_{\alpha, T}] \leq C_{N}.\end{aligned}$$ This completes the proof of Lemma \[lemboundN\].

\[lemboundNm\] Under the assumptions of Theorem \[mixuniq\]. Then, there exists a constant $C_{N,R}$, such that $$\begin{aligned}
\label{bounduNm}
    \mathbb{E}\Big[\|u^{N,n}-u^{N,m}\|^2_{\alpha,T}\mathbf{1}_{D_{T}^{N,R}}\Big]  \leq C_{N,R}\mathbb{E}\Big[\Big(\sum_{i=1}^{\infty}\sqrt{\lambda_{i}}\|\beta_{i}^{H,N,n}-\beta_{i}^{H,N,m }\|_{\alpha,0,T}\Big)^2\Big],
    \end{aligned}$$ where $D_{T}^{N,R}:=\{\|u^{N,n}\|_{\alpha,T} \leq R, \|u^{N,m}\|_{\alpha,T} \leq R\}$.

From (\[map1\]), we have $$\begin{aligned}
    &&\mathbb{E}\Big[\|u^{N,n}-u^{N,m}\|^2_{\alpha,T}\mathbf{1}_{D_{T}^{N,R}}\Big] \cr
    &\leq& C \mathbb{E}\Big[\Big\|\int_{0}^{\cdot} S_{\cdot-r} (f(u^{N,n}_r)- f(u^{N,m}_r)) \mathrm{d} r\Big\|^2_{\alpha, T}\mathbf{1}_{D_{T}^{N,R}}\Big] \cr
    &&+C\mathbb{E}\Big[\Big\|\int_{0}^{\cdot} S_{\cdot-r} (\sigma(u^{N,n}_r)-\sigma(u^{N,m}_r)) \mathrm{d}W_r\Big\|^2_{\alpha, T}\mathbf{1}_{D_{T}^{N,R}}\Big] \cr
    &&+C\mathbb{E}\Big[\Big\|\int_{0}^{\cdot} S_{\cdot-r} g(u^{N,n}_r) \mathrm{d}B^{H,N,n}_r-\int_{0}^{\cdot} S_{\cdot-r} g(u^{N,m}_r) \mathrm{d}B^{H,N,m}_r\Big\|^2_{\alpha, T}\mathbf{1}_{D_{T}^{N,R}}\Big] 