We’re rewarding the question askers & reputations are being recalculated! Read more.

Questions tagged [fixed-point-theorems]

Fixed-point theorem is a result about existence of fixed points, i.e. points fulfilling $F(x)=x$, under some conditions on the function $F$. Results of this type appear in many areas of mathematics, e.g. functional analysis (Banach), algebraic topology (Brouwer), lattice theory (Knaster-Tarski, Kleene) etc.

Filter by
Sorted by
Tagged with
22
votes
2answers
10k views

Prove the map has a fixed point

Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$ such that $\rho (Ax,Ay) < \rho(x,y)$ for $x\neq y$. Prove A have a unique fixed point in $K$. The ...
2
votes
1answer
839 views

If $T^n$ is $q$-contractive, $T$ exactly has one fixed point

Consider a complete metric space $(X,d)$ and $T\colon X\to X$. Suppose there exists $n\in\mathbb{N}$ such that the n-th power of $T$ is $q$-contractive. Show that then $T$ has exactly one fixed point $...
7
votes
1answer
2k views

Every increasing function from a certain set to itself has at least one fixed point

I need a hint for the following question: Let $S$ be a nonempty ordered set such that every nonempty subset $E\subseteq S$ has both a least upper bound and a greatest lower bound. Suppose $f:S \...
3
votes
2answers
876 views

Check Points are line, triangle, circle or rectangle

How to determine geometric properties of four distinct points in a plane (x1,y1), (x2,y2), (x3,y3), (x4,y4) represented in the 2-D Cartesian coordinate system, whether these four points are on a line,...
4
votes
2answers
590 views

To prove : If $f^n$ has a unique fixed point $b$ then $f(b)=b$

If $f: \mathbb R \to \mathbb R$ be a function such that for some $n_o \in \mathbb N$ , the $n_o$th iterate of $f$ has a unique fixed point $b$ , then how to prove that $f(b)=b$ ? I cant think of ...
20
votes
2answers
3k views

Generalization of “easy” 1-D proof of Brouwer fixed point theorem

In topology class, before we learned the homotopy-based proof of Brouwer's fixed point theorem, the professor mentioned the easy proof of the one-dimensional case: just draw the graph $f(x) = x$ in $[...
3
votes
2answers
880 views

Contraction and Fixed Point [duplicate]

How do I show that for $T: X \rightarrow X$ where X is complete and $T^m$ is a contraction that T has a unique fixed point $x_0 \in X$. I know there exists $\lambda_1 \in (0,1)$ for $x, y \in X$ ...
7
votes
1answer
2k views

If $f^N$ is contraction function, show that $f$ has precisely one fixed point. [duplicate]

If $f$ is a mapping of a complete metric space $(X, d)$ into itself and $f^N$(composite $f$ for $N$ times) is a contraction mapping for some positive integer $N$, then $f$ has precisely one fixed ...
3
votes
2answers
462 views

On Tarski-Knaster theorem

Let $P(X)$ denote the power set of $X$ (the set of all subsets of $X$) with a partial order given by inclusion. If $F: P(X) \to P(X)$ is monotone (order preserving), then $F$ has a fixed point. How ...
11
votes
3answers
2k views

Why is convexity a requirement for Brouwer fixed points? Shouldn't “no holes” be good enough?

Brouwer's fixed point theorem: Every continuous function $f$ from a convex compact subset $K$ of a Euclidean space to $K$ itself has a fixed point. I am wondering why the word "convex" is in there....
8
votes
0answers
305 views

does this Newton-like iterative root finding method based on the hyperbolic tangent function have a name?

I've recently discovered that modifying the standard Newton-Raphson iteration by "squashing" $\frac{f (t)}{\dot{f} (t)}$ with the hyperbolic tangent function so that the iteration function is $$N_f (...
8
votes
1answer
956 views

Continuous function on closed unit ball

Take a continuous mapping $f: \bar{B^{n}} \rightarrow \bar{B^{n}}$, where $\bar{B^{n}}$ is a closed unit ball in $\mathbb{R}^{n}$. Assume that $f(x) \neq x$ for every $x \in \bar{B^{n}}$. Define ...
1
vote
2answers
698 views

Bourbaki-Witt fixed point theorem: two questions

Consider the following theorem: Let $f\colon E \to E$ have the propery that $f(x)\geq x$, where $(E,\leq)$ is a non-void partially ordered set with the property that every totally ordered subset of $...
4
votes
2answers
676 views

Generalization of Banach's fixed point theorem

I wanted to show that if $f:X\to X$ is a function from a complete metric space to itself and if $f^k$ is a contraction, then $f$ has a unique fixed point (say $p$) and for any $x$ in $X$ $f^n(x)\...
4
votes
5answers
1k views

Picard's existence theorem, successive approximations and the global solution

Picard's existence theorem states that if $U$ is an open subset of $\mathbb{R}^2$ and $f$ is a continuous function on $U$ that is Lipschitz continuous with respect to the second variable then there is ...
3
votes
2answers
282 views

Periodic orbits

Let $x\in\mathbb R$ be a periodic point with lenght 2 of the recursion $x_{n+1}=f(x_n)$ My book about Dynamic systems says that this recursion has a fixed point now, because a periodic point with ...
2
votes
1answer
750 views

Proof: Contraction Mapping and Cauchy Sequence

Question: Let $(X, d)$ be a metric space, let $f : X → X$ be a contraction, and let $a_o \in X$. Let $a_1 = f(a_o)$ and $a_{n+1} = f(a_n)$ for $n \geq 1$. Prove that $(a_n)$ is a Cauchy ...
1
vote
2answers
276 views

Apply Banach's fixed point theorem

Let $$T:f\mapsto (x\mapsto \frac{2}{5}\int_0^1 (x^2+t^5)f(t) dt + \sin(x))$$ for any $x\in[0,1]$, $f\in C([0,1])$. I want to show that that there is a uniqu $\tilde{f}$ that solves that equation $f(...
0
votes
3answers
944 views

Prove there is no contraction mapping from compact metric space onto itself

This question is from Foundations of mathematical analysis by Richard Johnsonbaugh The thing with this question is that there is a question that seems to prove the opposite claim Prove the map has a ...
0
votes
2answers
146 views

Fixed point iterations for real functions - depending on $f'(x)$?

Let $f$ be a differentiable real function. In many situations a solution of $f(x)=x$ can be found as limit of the recurrent sequence determined by some initial value and the recurrence $x_{n+1}=f(x_n)$...
35
votes
2answers
1k views

Homeomorphic to the disk implies existence of fixed point common to all isometries?

A fellow grad student was working on this seemingly simple problem which appears to have us both stuck. (The problem naturally came up in his work so isn't from the literature as far as we know). Let ...
18
votes
2answers
2k views

Contraction mapping in an incomplete metric space

Let us consider a contraction mapping $f$ acting on metric space $(X,~\rho)$ ($f:X\to X$ and for any $x,y\in X:\rho(f(x),f(y))\leq k~\rho(x,y),~ 0 < k < 1$). If $X$ is complete, then there ...
5
votes
2answers
204 views

Existence of infinite iteration of functions $f_\infty$?

Given a sequence of functions $\{f_n\}$ satisifying an iterated relation such as $f_n(x)=g(x+f_{n-1}(x))$ $f_n(x)=g(xf_{n-1}(x))$ $f_n(x)=g(x/f_{n-1}(x))$ Where $g:=f_1$ is ...
14
votes
1answer
495 views

Algebraic fixed point theorem

I was wondering if there are some "algebraic" fixed point theorems, in group theory. More precisely, given a group $G$ and a group morphism $f : G \to G$, what conditions on $G$ and $f$ should we ...
5
votes
1answer
1k views

Showing that $f$ has exactly one fixed point

Let $\gamma$ be the circle $\{z \in \mathbb{C}: \lvert z\rvert=1 \}$. Suppose $f$ is a function analytic on an open set containing $\gamma$ and its interior and that $\lvert\, f(z)\rvert<1$ for ...
3
votes
2answers
2k views

Proving and understanding the Fixed point lemma (Diagonal Lemma) in Logic - used in proof of Godel's incompleteness theorem

http://en.wikipedia.org/wiki/Diagonal_lemma I am wondering about the proof of the "Fixed-Point Lemma" $\text{Mod } \Sigma$ is the class of all models of $ \Sigma$. $\text{Th Mod } \Sigma$ is the set ...
8
votes
3answers
2k views

Mapping homotopic to the identity map has a fixed point

Suppose $\phi:\mathbb{S}^2\to\mathbb{S}^2$ is a mapping, homotopic to the identity map. Show that there is a fixed point $\phi(p)=p$.
4
votes
3answers
311 views

Iterates of $f_b(x) = x - \log_b(x) $ - for $\log(b) \approx 0.399$: convergence to accumulation points or chaos?

In a previous question I arrived at the family of functions (depending on a real parameter b for the base of exponentiation/logarithm): $$ f(x) = x - \log_b(x) \qquad \qquad b \gt 0$$ with the ...
9
votes
3answers
595 views

In this case, does $\{x_n\}$ converge given that $\{x_{2m}\}$ and $\{x_{2m+1}\}$ converge?

I'm playing around with a sequence $\{x_n\}$ defined by $$ x_{n+1}=\frac{\alpha+x_n}{1+x_n}=x_n+\frac{\alpha-x_n^2}{1+x_n}. $$ Here $\alpha\gt 1$, and $x_1\gt\sqrt{\alpha}$. I'm trying to compute $\...
7
votes
2answers
786 views

Proof $\lim\limits_{n \rightarrow \infty} {\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}=2$ using Banach's Fixed Point

I'd like to prove $\lim\limits_{n \rightarrow \infty} \underbrace{\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}_{n\textrm{ square roots}}=2$ using Banach's Fixed Point theorem. I think I should use the ...
5
votes
1answer
1k views

In a complete lattice every monotone function has a fixpoint (Knaster–Tarski Theorem)

L is a complete lattice, so every subset has a supremum and infimum. In addition, there exists a function $f:L \rightarrow L$ such that $a \leq b$ implies $f(a) \leq f(b)$. Prove that there exists ...
4
votes
2answers
2k views

Using the Banach Fixed Point Theorem to prove convergence of a sequence

Use the Banach fixed point theorem to show that the following sequence converges. What is the limit of this sequence? $$\left(\frac{1}{3}, \frac{1}{3+\frac{1}{3}}, \frac{1}{3+\...
4
votes
2answers
537 views

Showing that a function $f$ has a unique fixed point in a metric space.

Let $(X, d)$ be a compact metric space, and suppose $f : X → X$ satisfies $$d(f(x), f(y)) < d(x, y)$$ for all $x \neq y \in X$. Show that f has a unique fixed point. All I've gotten it so far ...
1
vote
3answers
2k views

Prove that $f$ has a fixed point . [duplicate]

For $f:[a,b]\rightarrow [a,b]$ is a continuous . Prove that $f$ has a fixed point . Is that true if we change $[a,b]$ by $[a,b)$ or $(a,b)$.
4
votes
0answers
722 views

Equivalence of Brouwers fixed point theorem and Sperner's lemma

I'm looking for a proof that Brouwer's fixed point theorem implies Sperner's lemma. All proof I've found just prove that there must be at least one completely colored n-simplex, not that there must be ...
4
votes
1answer
88 views

Fix point of squaring numbers mod p

Take the set of integers $\{0, 1, .., p-1\}$, square each element, you get the (smaller) set of quadratic residues. Repeat until you get a fix point set. The size of this set is a function of $p$. ...
3
votes
2answers
484 views

Confused about a version of Schauder's fixed point theorem

I have read this: We have a map $S:W_0 \to W_0$. Moreover $W_0$ is not empty, convex, and weakly compact in $W$. Thus we can apply Schauder's fixed point theorem: Schauder's fixed point ...
3
votes
3answers
1k views

Analogue to Fixed Point Theorem for Compact metric spaces

If $\omega:H \rightarrow H$ (into) is continuous and $H$ is compact, does $\omega$ fix any of its subsets other than the empty set?
3
votes
1answer
571 views

Extinction probabilities of binomial tends to Poisson distribution

I am stuck on exercise 11.2 From Grimmett's probability on graphs. Here is a link to the pdf on his website. Consider a branching process whose family-sizes have the binomial distribution bin$(n, \...
2
votes
1answer
342 views

Brouwer's fixed point theorem implies Sperner's lemma

I'm looking for a proof that Brouwer's fixed point theorem implies Sperner's lemma. The proof should be understandable by an undergraduate. Thanks in advance!
1
vote
1answer
120 views

Studying Muller's example of IVP with unique solution whose Picard iterates do not converge.

I have read about the following example from Muller: $(M) \begin{cases} x' = f(t,x) \\[1mm] x(0) = 0 \end{cases}$ where $f: \mathbb{R}\times\mathbb{R} \rightarrow \mathbb{R}$ is the function: $f(t,...
1
vote
1answer
155 views

The composition of non-linear entire functions with no fixed points has infinitely many fixed points.

The entire [non-linear] functions $f$ and $g$ do not have fixed points. Show that $f \circ g$ has infinitely many fixed points. How do you prove this statement? Or if it is not true as stated, what ...
1
vote
1answer
146 views

Edelstein Theorem

Let (M,d) be a compact metric space and $d(f(x),f(y)) < d(x,y) $ for all $ x\neq y$ Prove that if $f$ is a continuous fuction then there is a unique $x_0 \in M $ such that $f(x_0)=x_0$ I know that ...
1
vote
1answer
94 views

Brouwer fixed-point theorem on a function

For $f:\mathbb R^n \rightarrow \mathbb R^n \quad \exists R \gt 0 $ that $f$ is continous on $\bar B_R(0):= \{ x \in \mathbb R^n : \Vert x\Vert_2 \leq R \} \subset \mathbb R^n$ . Let also $\langle f(x)...
1
vote
4answers
390 views

Fix-Point Theorem Proof.

Firstly, the assignment: Let $a,b \in\mathbb{R}$ and $a < b$. Furthermore let $f: [a,b] \rightarrow [a,b]$ be monotone increasing. Show that if $x:= \mathbf {sup}\{y \in [a,b] \| \ y ≤ f(y)\}$ ...
0
votes
1answer
202 views

Fixed points of contractions in metric spaces

How do I prove that all contractions on a complete, non-empty metric space has exactly one fixed point? What I know: I know that all contractions are continuous and that completeness of $A$ means ...
0
votes
1answer
123 views

Iterative Convergence Formulation for Linear Fractional Transformation with Rational Coefficients

All numbers discussed here are rational (we have not yet constructed the real numbers). Let $S > 0$, $K > 0$ with $K^2 > S$ Set $$F(x) = \frac{S + Kx}{K + x}$$ Let $p > 0$ such that $p^...
-2
votes
1answer
155 views

Fixed point of interior of closed disk

Let $D = \{(x,y)\in \mathbb{R}^2: x^2 + y^2 \leq 1 \}$. Let $A \subset \mathrm{int}D$. Let $A$ be connected and compact and let $D \setminus A$ be connected. Let $f:A \longrightarrow A$ be a ...
4
votes
2answers
10k views

When does Newton-Raphson Converge/Diverge?

Is there an analytical way to know an interval where all points when used in Newton-Raphson will converge/diverge? I am aware that Newton-Raphson is a special case of fixed point iteration, where: $...
10
votes
1answer
505 views

Fixed points in computability and logic

I asked this question on CS.SE, too: https://cstheory.stackexchange.com/questions/27322/fixed-points-in-computability-and-logic I would like to understand better the relation between fixed point ...