equivalently $$\begin{aligned}
\label{map2}
u_t^{N,n}&=& S_t u_{0}+\int_{0}^{t} S_{t-s} f^{N,n}(u^{N,n}_s) \mathrm{d} s+\int_{0}^{t} S_{t-s} \sigma(u^{N,n}_s) \mathrm{d}W_s,\end{aligned}$$ where $f^{N,n}(u):=f(u)+g(u) \sum_{i=1}^{\infty} \sqrt{\lambda_{i}}e_i  \frac{\mathrm{d}}{\mathrm{d}s} \beta_{i}^{H,N,n}(s)$ is a random drift.

Such equations were studied in [@da2014stochastic Section 7]. To proceed, it is easy to obtain $$\begin{aligned}
\label{lambeta}
    \Big|\sum_{i=1}^{\infty} \sqrt{\lambda_{i}}\frac{\mathrm{d}}{\mathrm{d}s} \beta_{i}^{H,N,n}(s)\Big|&\leq& \sum_{i=1}^{\infty} \sqrt{\lambda_{i}} \Big|\frac{\mathrm{d}}{\mathrm{d}s} \beta_{i}^{H,N,n}(s)\Big|\cr
    &\leq & n \sum_{i=1}^{\infty} \sqrt{\lambda_{i}} \Big| \beta_{i}^{H,N}(s)-\beta_{i}^{H,N}((s-\frac{1}{n})\vee 0)\Big|\cr
    &\leq & n^\alpha \sum_{i=1}^{\infty} \sqrt{\lambda_{i}} \Lambda^{0,s}_{\alpha} (\beta_{i}^{H,N})\cr
    &\leq & C_{n,N},\end{aligned}$$ where $C_{n,N}$ is a constant dependent on $n$ and $N$. Thus, by (\[lambeta\]), we obtain the function $f^{N,n}$ satisfies Lipschitz and growth conditions. Then, by [@da2014stochastic Theorem 7.4], there exists a unique mild solution $u^{N,n}$ to Eq.(\[map2\]).

To obtain the convergence of approximations, we give the following two key lemmas.

\[lemboundN\] Under the assumptions of Theorem \[mixuniq\], there exists a constant $C_{N}$, such that $$\begin{aligned}
        \mathbb{E}[\|u^{N,n}\|^2_{\alpha, T}] \leq C_{N}.
    \end{aligned}$$

We start with $$\begin{aligned}
    \mathbb{E} [\|u^{N,n}\|^2_{\alpha, T}]& \leq &
    C \mathbb{E}[\|S_{\cdot} u_{0}\|^2_{\alpha, T}]+C \mathbb{E}\Big[\Big\|\int_{0}^{\cdot} S_{\cdot-s} f(u^{N,n}_s) \mathrm{d} s\Big\|^2_{\alpha, T}\Big]\cr
    &&+C\mathbb{E}\Big[\Big\|\int_{0}^{\cdot} S_{\cdot-s} \sigma(u^{N,n}_s) \mathrm{d} W_s\Big\|^2_{\alpha, T}\Big]\cr
    &&+C\mathbb{E}\Big[\Big\|\int_{0}^{\cdot} S_{\cdot-s} g(u^{N,n}_s) \mathrm{d} B^{H,N,n}_s\Big\|^2_{\alpha, T}\Big]\cr
    &=:&\mathcal{M}_1+\mathcal{M}_2+\mathcal{M}_3+\mathcal{M}_4,\end{aligned}$$ where $B^{H,N,n}:=\sum_{i=1}^{\infty} \sqrt{\lambda_{i}} e_{i} \beta_{i}^{H,N,n}$.

Since $\alpha<\beta$, $u_0\in V_\beta$, it is easy to obtain $$\begin{aligned}
    \mathcal{M}_{1}&\leq &C\mathbb{E}\big[\sup _{t \in[0, T]} |S_t u_{0}|^2\big]+ C\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^{t} \frac{|(S_t-S_s)u_{0}|}{(t-s)^{1+\alpha}} \mathrm{d} s\Big)^2\mathrm{d}t\Big] \cr
    & \leq& C+ C\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^{t} |u_{0}|_{{\beta}}(t-s)^{\beta-1-\alpha} \mathrm{d} 