How do I set up Implicit Function Theorem to verify this function is a $C^{r}$ diffeomorphism?

Suppose I have a $C^{r}$ diffeomorphism $F: \mathbb{R}^{n} \mapsto \mathbb{R}^{n}$ that I write as $x_{k+1} = F(x_{k})$, with $x \in \mathbb{R}^{n}$. Note that here $x_{k+1}$ is the image of $x_{k}$ under $F$ (i.e. we consider a dynamical system). Suppose I give a small $O(\epsilon)$ perturbation to $F$, and the resulting perturbed map is written in an implicit form as:

$$x_{k+1, \epsilon} = F(x_{k}) + \epsilon G(x_{k}, F(x_{k}), \epsilon)$$

for some $C^{r}$ function $G$ of both the preimage $x_{k}$, the image of $x_{k}$ under the unperturbed function $F$, and small parameter $\epsilon$.

How do I set up implicit function theorem to indeed verify that the perturbed map is again a $C^{r}$ diffeomorphism? Or do I use inverse function theorem? Also, on what does the upper bound on $\epsilon$ depend on, in order to have a diffeomorphism?

Edit. Can I simply define a new function $\Phi: \mathbb{R} \times \mathbb{R}^{n} \mapsto \mathbb{R}^{n}$ with $\Phi(\epsilon, x) \mapsto (F + \epsilon G)$ (here I suppressed arguments in $F$ and $G$) and notice that $D_{x}\Phi(0,x)$ is clearly invertible at all $x \in \mathbb{R}^{n}$ and then state that implicit function theorem says there is a unique $C^{r}$ function $x(\epsilon)$ defined for small $\epsilon$ ?

Another edit. I suppose I could apply inverse function theorem, but then I run into the problem of showing that a small linear perturbation of a linear isomorphism is also an isomorphism - is this true?

• It seems you are using $n$ for two different purposes. – Maybe the problem occurred in your study of a dynamical system; but for the purposes of this question you may as well write $y=F(x)$ and forget about the second $n$. – Or am I missing something? – Christian Blatter May 22 '18 at 15:45
• Sorry for ambiguity - indeed, I use $n$ to denote $n$-dimensional space, and also $n$ is used as a subscript indeed in the sense of dynamical systems (so $x_{n+1}$ is the image of $x_{n}$ under $F$). Note I use a subscript $\epsilon$ in $x_{n+1, \epsilon}$ to denote that it is the image under perturbed $F_{\epsilon}$ @ChristianBlatter – Alex May 22 '18 at 16:29
• I have just edited my question and changed the subscript from $n$ to $k$ @ChristianBlatter – Alex May 22 '18 at 16:31