time interval $[0,T]$, where $T >0$ is arbitrary but fixed. Let $A$ be the infinitesimal generator of an analytic semigroup $S$ on $V$ and assume that $-A$ has discrete spectra $0 < \bar{\lambda}_1 < \bar \lambda_{2} < \cdots <\bar \lambda_{k}  < \cdots $ and $\lim_{k\rightarrow \infty} \bar \lambda_k =\infty$.

This paper firstly will prove an existence and uniqueness theorem for a class of mixed stochastic partial differential equations (SPDEs) driven by both fBm and Bm with a given initial value $u_0$, which is given by $$\begin{aligned}
\label{spde-uniq}
\mathrm{d} u_t =(A u_t+f(u_t))\mathrm{d}t+\sigma(u_t)\mathrm{d}W_t+g(u_t) \mathrm{d}B^H_t,
\qquad
0 \le  t \le T.\end{aligned}$$ Precise conditions on the nonlinear coefficients $f,\sigma, g$ will be given in Section \[se-3\].

The idea of this part is based on a pathwise approach developed by Zähle [@Zahle1998], who defined the stochastic integral with respect to fBm based on a sort of generalized integration by parts formula with respect to fractional derivatives. Nualart and Răşcanu [@Rascanu2002] and Garrido-Atienza et al. [@Random2010garr] investigated stochastic differential equations (SDEs) in finite dimension. Infinite-dimensional equations were treated with the same success as finite-dimensional ones, e.g. Tindel et al. [@tindel2003stochastic] and Garrido-Atienza et al. [@GarridoAtienza2010dcds; @Garrido-Atienzalu2010]. Pathwise solutions of this kind of equation without Bm term 