Let $f: \mathbb{R}^m \times \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a continuous differentiable function. For each $x \in \mathbb{R}^m$ consider $f_x:\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $f_x(y)=f(x,y)$. Suppose there exists $a \in \mathbb{R}^m$ such that $f_a$ have unique fixed point $b$ ($f_a(b)=b$), and $Df_a(b)$ has no eigenvalue $1$. Prove that, $x$ sufficiently close to $a$, the function $f_x$ has unique fixed point close to $b$.
I know that $Df_a(y_0) = D_Yf(a, y_0)$ only.