equation (\[time-space\]) with $g_n$ in place of $g$, that is, $$\label{approx}
  (I^{1-\beta}u_n)(t)=\int^t_0\Delta u_n (s)dt+\sum_{k=1}^{\infty} (I^{1-\gamma} \int^{\cdot}_0 g^k_n(s)\, dW^k_s)(t).$$ Then by Lemmas \[lemma main estimate\] and \[lemma formula\] $$\label{eqn 7.03.2}
  \|u_n\|_{{\mathbb{H}}^2_2(T)}\leq N \|g_n\|_{{\mathbb{H}}^{\sigma_0}_2(T,\ell_2)},$$ $$\|u_n-u_m\|_{{\mathbb{H}}^2_2(T)}\leq N \|g_n-g_m\|_{{\mathbb{H}}^{\sigma_0}_2(T,\ell_2)}.$$ Thus $u_n\to u$ in ${\mathbb{H}}^2_2(T)$ for some $u$. Letting $n\to \infty$ in (\[approx\]) and using Remark \[remark 1\], we see that $u$ is a solution of (\[time-space\]). Also we easily get (\[main estimate1\]) from (\[eqn 7.03.2\]). The theorem is proved.

The following lemma is taken from [@YGD Corllary 2].

(Gronwall’s lemma) \[gronwall\] Suppose $b>0$ and $a(t)$ is a nonnegative nondecreasing locally integrable function on $[0,T)$, and suppose $\eta(t)$ in nonnegative locally integrable on $[0,T)$ with $$\eta(t)\leq a(t)+b\int^t_0 (t-s)^{\beta-1}\eta(s)ds, \quad \forall t<T.$$ Then it holds that $$\eta(t)\leq a(t)E_{\beta}(b \Gamma(\beta)t^{\beta}).$$

**SPDE of divergence form type**
================================

In this section, we study the equation of divergence type $$\begin{aligned}
                 \nonumber
  \partial^{\beta}_t u&=&D_i\left[
a^{ij}u_{x^j}+b^iu+f^i(u)\right]+cu+h(u)
\\
    &&+ \sum_{k=1}^\infty\partial^{\gamma}_t \int^t_0 (\sigma^{ijk}u_{x^ix^j}+\mu^{ik} u_{x^i}
    +\nu^ku +g^k(u))\, dW^k_s
    \label{eqn 7.04.5}
 \end{aligned}$$ where the coefficients $a^{ij}, b^i, c, \sigma^{ijk}, \mu^{ik}, \nu^k$ are functions depending on $(\omega,t,x)$ and the functions $f^i, h, g^k$ depend on $(\omega,t,x)$ and the unknown $u$.

For a $\ell_2$-valued continuous function $v$ in ${\mathbb{R}}^d$, we define the space $C^{\alpha}(\ell_2)$, $\alpha\in [0,1]$ by the 