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Given a map $F:X \to X$ on a complete metric space $(X,d)$, and let $K<1$ such that:

$$ d(F(x), F(y)) \le K d(x,y), \quad \forall x,y \in X $$

then the contraction mapping theorem tells us that $F$ has a unique fixed point, and we can iteratively solve for this fixed point.

My question is, if we take $K=1$, then it is no longer a contraction, I've seen this being called a 'non-strict' contraction. I'm wondering if there are any results regarding this case and fixed points? Do they exist but aren't unique, or do they not exist at all?

WeakLearner
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    In general, no (see Severin Schraven's answer). If, however, $X$ is compact and we have $d(F(x),F(y))<d(x,y)$ for any $x,y\in X$ (without Lipschitz property), consider $f \colon X \to [0, \infty)$ defined as $f(x):=d(x,F(x))$. $f$ is continuous (can you prove that?), so it attains its global minimum somewhere in $X$, at $x^$, say. Suppose $f(x^)>0$. Is that possible? – user539887 Apr 22 '18 at 09:26
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    See also https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem, https://en.wikipedia.org/wiki/Schauder_fixed-point_theorem – Lutz Lehmann Apr 22 '18 at 09:29
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    A small remark: I forgot to mention in my comment that $d(F(x),F(y))<d(x,y)$ for any $x, y \in X$ with $x \ne y$. – user539887 Apr 22 '18 at 11:18

2 Answers2

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In general, they do not need to admit fixed points. Take for example any translation (different from the identity) in $\mathbb{R}^n$. It has Lipschitz constant equal to 1, but has no fixed point.

On the other hand, the identity also has Lipschitz constant equal 1, and there are quite a lot of fixed points.

Let $X$ any set and endow it with the discrete metric, then any injective map $f: X \rightarrow X$ has Lipschitz constant equal 1. Thus, you can have any number of fixed points you like (just fix the points of your choice and make sure that you have an bijection without fixed points on the complement. You can actually do that, see for example here Existence of a bijective function with no fixed points).

Severin Schraven
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  • @user539887 : To be more specific, with $x_{n+1}=\sin(x_n)$ set $u_n=x_n^{-2}$ then you get $$u_{n+1}=u_n/(1-x_n^2/6+O(x_n^4))^2=u_n+1/3+O(u_n^{-1})$$ so that $u_n\approx u_0+n/3$ or $$x_n\approx x_0/\sqrt{1+nx_0^2/3}$$ which is far away from being geometrically convergent. – Lutz Lehmann Apr 22 '18 at 11:00
  • @user539887 Thanks for pointing out my terrible mistake. The problem was, that of course we could have $x=y$ in my argument and for this case the inequality $d(F(x),F(y))<d(x,y)$ does not apply. – Severin Schraven Apr 22 '18 at 11:10
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The class of mappings you're referring to in your definition have many names, but are most commonly referred to as ``nonexpansive mappings,'' and the fixed point theory for nonexpansive mappings has a wonderful history. In order to pose thoughtful questions for nonexpansive mappings, one must restrict the underlying spaces or sets to a much more restrictive class than simply complete metric spaces. For instance, Brouwer's Theorem tells us that if $C \subseteq \mathbb{R}^n$ is compact and convex and $T : C \to C$ is continuous, then it has a fixed point. Nonexpansive mappings are necessarily continuous, so they would have fixed points on compact, convex sets in $\mathbb{R}^n$ as well.

In order to make this more interesting, one must move into infinite dimensional spaces. Indeed, a version of Brouwer's Theorem still holds (the Schauder-Tychonoff Theorem) in infinite dimensional spaces, but a remarkable thing happens when you move to infinite dimensions: norm-compactness is no longer equivalent to norm-closed and norm-bounded. Hence, you have a wonderful gray area full of intriguing examples and theorems to explore. Namely, you have the question ``if $(X,\|\cdot\|)$ is a Banach space and $C \subseteq X$ is closed, bounded, and convex, then does every nonexpansive $T : C \to C$ necessarily have a fixed point?''

The answer is rich, complicated, and depends very deeply on the geometry/topology of the underlying space $X$ (or the underlying set $C$). For an extensive history of the topic (up to approximately the year 2000), see The Handbook of Metric Fixed Point Theory (Kirk and Sims, eds.); for a briefer history, see Topics in Metric Fixed Point Theory (Goebel and Kirk).

TM Gallagher
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