# Existence and uniqueness of fixed point

Let $f:\mathbb{R} \to \mathbb{R}$ be a differentiable function. Suppose $|f'(x)|\le r<1,\forall x\in \mathbb{R}$ and for some $r\in \mathbb{R}$.Then by contraction mapping theorem f has a unique fixed point in $\mathbb{R}$. Now suppose the inequality changes as $f'(x)\le r<1, \forall x\in \mathbb{R}$ and for some $r\in \mathbb{R}$. Then is it true that $f$ has at least one fixed point. What about uniqueness if the existence is true? Conversely if there is unique fixed point for such a differentiable function, is it necessary that $f'(x)\le r<1, \forall x\in \mathbb{R}$ and for some $r\in \mathbb{R}$?

• If $f(x)=x$ and $y=f(y)$, $x\neq y$, then by the MVThm there is a point $z$ between $x$ and $y$ with $f'(z)=1$. Nov 13, 2022 at 12:48

The function $g(x) := x- f(x)$ is differentiable and $$g'(x) = 1 - f'(x) \geq 1 - r \qquad \forall x\in\mathbb{R}.$$ Since $1-r>0$, this proves that $g$ is strictly increasing (hence it can have at most one zero) and that $$\lim_{x\to -\infty} g(x) = -\infty, \qquad \lim_{x\to +\infty} g(x) = +\infty,$$ so it has at least one zero (being continuous). The unique zero of $g$ is the unique fixed point of $f$.

Both existence and uniqueness still hold. You can easily see geometrically it by noticing that $f$ will always be increasing less than $id(x)=x$ and a fixed point is the same as an intersection of the graph of $f$ with the diagonal of $\mathbb{R}^2$ (which is the graph of $id$).

Formally, let $x\in\mathbb{R}$ and suppose $f(x)>x$. Let $k=\frac{f(x)-x}{1-r}$, which solves the equation $$f(x)+kr = x+k\ .$$ Then $$f(x+k) = f(x)+\int_x^{x+k}f'(t)dt\le f(x)+kr = x+k\ .$$ By the intermediate value theorem, it follows that $f$ has a fixed point. A similar proof gives a fixed point if $f(x)<x$, proving existence.

For uniqueness, suppose $x_0<x_1$ are two distinct fixed points. Then $$f(x_1) = f(x_0)+\int_{x_0}^{x_1}f'(t)dt \le x_0+(x_1-x_0)r<x_1\ ,$$ giving a contradiction. Thus the fixed point must be unique.

Notice that this works for $f:\mathbb{R}\to\mathbb{R}$, not for general functions in metric spaces.

• I cannot notice "f will always be increasing" Jul 16, 2017 at 16:11
• @supremum Read all the sentence, please, not just some part of it. Jul 16, 2017 at 16:15
• isn't f(x)=-2x satisfy those conditions still being decreasing and not comparable with Id(x) in general. Jul 16, 2017 at 16:18
• @supremum "increasing less than" does not preclude the function from being decreasing, it only says that it will never grow faster than it. Jul 16, 2017 at 16:19
• I have a small complaint about your proof : one does not have $f(x+k)=f(x)+\int_x^{x+k} f'(t) \,\mathrm{d}t$ if $f$ is only assumed to be differentiable. You should replace this argument by one based the mean value theorem. Other than that, it should be good. Jul 16, 2017 at 16:25

$f: x\mapsto \sin (x)$ has a unique fixed point $x=0$ but $f'(2\pi)=1$.

$g:x\mapsto 2x$ has one fixed point but $g'(x)=2$.