How to determine all the fixed points of the discrete dynamical system?

Question

Determine all fixed points of the discrete dynamical system $$x_{n + 1} = e^{x_{n}} - 2 .$$

Determine if they are asymptotically stable.

How can I find the fixed points?

Thank you in advance.

Answer 1

Hint: Draw the graphs of $e^x$ and $2+x$. Show (using e.g. the IVT and convexity of exp) that there are precisely two intersections and study the derivative of $e^x$ at each intersection point (I don't think there is any closed form expression for these points).

Answer 2

You have an iteration of the form $$ x_{n+1} = f(x_n) $$ with $$ f(x) = e^x - 2 $$ A fixed point $x_n$ would fullfil $$ f(x_n) = x_n \quad (*) $$ A graphical solution is to draw the graphs of $f$ and $\DeclareMathOperator{id}{id}\id$, where $\id(x) = x$ is the identity function.

Where those graphs intersect, the condition $(*)$ holds, that intersection point is a fixed point.