paper is organized as follows. In Section 2 we present some preliminary results on the fractional derivatives and in Section 3 we introduce stochastic Banach spaces and few key estimates. Our main results for (\[2014.2\]) and (\[eqn 2014.1\]) are presented and proved in Section 4 and Subsection 5.1, respectively. Subsection 5.2 contains an application to an equation driven by space-time white noise.

We close this section with some notation. As usual, ${\mathbb{R}}^{d}$ stands for the Euclidean space of points $x=(x^{1},...,x^{d})$. For $i=1,...,d$, multi-indices $\alpha=(\alpha_{1},...,\alpha_{d})$, $\alpha_{i}\in\{0,1,2,...\}$, and functions $u(x)$ we set $$u_{x^{i}}=\partial u/\partial x^{i}=D_{i}u,\quad
D^{\alpha}u=D_{1}^{\alpha_{1}}\cdot...\cdot D^{\alpha_{d}}_{d}u,
\quad|\alpha|=\alpha_{1}+...+\alpha_{d}.$$ We also use the notation $D^m_x$ for a partial derivative of order $m$ with respect to $x$. By $C^{\infty}_0({\mathbb{R}}^d)$ we denote the collection of smooth functions having compact support in ${\mathbb{R}}^d$. For $p\geq 1$, let $$L_p=L_p({\mathbb{R}}^d):=\{ u : {\mathbb{R}}^d \to {\mathbb{R}},  \| u \|^p_{L_p}:=\int_{{\mathbb{R}}^d} |u(x)|^p dx < \infty   \}$$ and we use the notation $(f,g)_{L_2}:= \int_{{\mathbb{R}}^d} f(x)g(x) dx$. We denote $${\mathcal{F}}(g)(\xi):=\frac{1}{(2\pi)^{d/2}}\int_{{\mathbb{R}}^d}e^{-i\xi\cdot
x}g(x)dx \quad \text{and}\quad {\mathcal{F}}^{-1}(f)(\xi):=\frac{1}{(2\pi)^{d/2}}\int_{{\mathbb{R}}^d}e^{i\xi\cdot
x}f(x)dx$$ the Fourier transform of $g$ in ${\mathbb{R}}^d$ and the inverse Fourier transform of $f$ in ${\mathbb{R}}^d$, respectively.

If we write $N=N(a,b,\cdots)$, this means that the constant $N$ depends only on $a,b,\cdots$. Throughout this paper, for functions depending on 