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)...

learn more… | top users | synonyms

1
vote
0answers
26 views

Theorem about number of crossing-points between a function and a line

Assume $f(x)$, with $x \in [a,b]$. Take $u$ so that $f(a)<u<f(b)$. By the Intermediate value theorem, we know that $f(x)$ crosses $u$ at least once. My question is, given some extra information ...
1
vote
1answer
38 views

Can somebody explain the notation $f \in C^4$

To give some context the full question is: Suppose $f \in C^4$ in a interval containing the root $\alpha$ and that Newton’s method gives a sequence of iterates $\{x_k\}$, $k = 0, 1, 2, \dots$ which ...
2
votes
1answer
96 views

Brouwer's fixed-point theorem and the intermediate value theorem?

In one dimension, Brouwer's fixed-point theorem (BPFT) can be proved easily based on the Intermediate Value Theorem (IVT). Is the inverse also true? I.e, is it possible to prove the IVT directly from ...
3
votes
1answer
117 views

Show that the comb space has fixed-point property

By "comb space", I mean the space $X=([0,1] \times \{0\}) \cup (K \times [0,1])$, where $K=\{ \frac{1}{n} : n \in \mathbb{N}^+ \}$, without the leftmost vertical line segment. How to prove that this ...
1
vote
1answer
88 views

Application of the Banach fixed point theorem

Let $a > 0$. We consider the function: $f: (0, \infty) \to (0, \infty)$, defined by $f(x) = \frac{1}{2}(x + \frac{a}{x})$. Let $(x_n)_{n \in \mathbb{N}_0}$ be defined by: $x_0 \in (0, \infty)$, $...
-1
votes
1answer
149 views

Translate this proof from German to English

I need your help to translate some exercises from German to English. I will attach like images. Thanks :) Satz 3. Es sei $(X,d)$ ein ultrametrischer Raum. $X$ ist genau dann transvollständig, wenn ...
2
votes
0answers
34 views

Show that F can have at most two fixed points

Consider a function $F:\mathbb{R}^n\to\mathbb{R}^n$, where $F=(F_1,...,F_n)$. Suppose that $F$ is strictly quasi-concave, and that for all $i=1,...,n$ the function $F_i:\mathbb{R}^n\to\mathbb{R}$ is ...
1
vote
1answer
91 views

Brouwer's fixed point continuous function

Can anyone point me out the continuous functions without brouwer fixed point's for the following sets $$A = \{x \in \mathbb{R}^2 | x_1,x_2 \geq 0 \text{ and }x_1^2+x_2^2 = 1 \}$$ $$B = \{x \in \...
2
votes
0answers
62 views

What are conditions for Tarski's fixed point theorem so that greatest and least fixed points are not the same?

Tarski's fixed point theorem claims that for any monotone function on a complete lattice there are least and greatest fixed points (or more generally, that the set of fixed points is a complete ...
0
votes
1answer
54 views

What is the difference between a period $n$ point, and a point of least period $n$?

What is the difference between a period $n$ point, and a point of least period $n$? Simply what is the definition of the two of them, and how do they differ. I think I have a rough idea of one but ...
0
votes
1answer
89 views

Spectral radius and convergence of fixed point iteration

Let $F: \mathbb{R}^n \to \mathbb{R}^n$ be a differentiable map. -editted- Let $x^\star$ be a fixed point of $F$. Then, is it true that the fixed point iteration $x_{n+1} = F(x_n)$ converges locally ...
2
votes
1answer
407 views

How to find periodic points non-algebraically

For example, $f(x) = x - x^2$ by observation has a fixed point at $x = 0$, and an eventually periodic point at $x = 1$ (that goes to $0$). For the second iteration, to see if there's a periodic point ...
4
votes
1answer
40 views

Prove there exists $a \in E$ such that $a = f(a)$, assuming $d(f(x), f(y)) \le Kd(x,y)$ with $K<1$

Let $f: E \rightarrow E$, $E$ a complete metric space. Assume that there exists $K$ such that $0 < K < 1$ and $d(f(x), f(y)) \le Kd(x,y)$ for all $x,y \in E$. Prove that there exists $a \in E$ ...
3
votes
1answer
104 views

No clear analytic method to prove unique maximum? ($2^{-x}+2^{-1/x}$)

Prove that $f(x) = 2^{-x}+2^{-1/x}$ has the unique local maximum $(1,1)$ for $x>0$. Do not use computer software. Proving that $(1,1)$ is a maximum is easy, but I'm having trouble with the ...
0
votes
2answers
60 views

Solve for y in sin(y) = cos(y) using a fixed point procedure

I'm reading an programming book that uses a lot of math equations and formulas as coding examples. In one lesson, it demonstrates finding the fixed point for $\sin(x) + \cos(x)$ by repeatedly calling ...
1
vote
1answer
50 views

Metric Fixed Point Theory

I am learning Metric Fixed Point Theory by Mohammed A Khamsi and William A Kirk. I need help in understanding a step in the proof of the following theorem(Chapter 3, Theorem 3.2, Page No. 43): ...
1
vote
1answer
43 views

Newton Iteration Function

So I'm having trouble figuring this problem out so if someone can help me out that'd be great. Find all the fixed points for the associated Newton iteration function for $$ f(x) = \frac{x}{(x-1)^n} $$...
1
vote
0answers
34 views

Show that the fix points of a function couldn't be in the interior

I want to solve the following problem: Show that the fix points of a function $f:\mathbb B^n\rightarrow \mathbb B^n$ could possibly not be in the interior. By this, Show that the Brouwer fixed-point ...
1
vote
2answers
130 views

Is there a metric proof for the Caristi theorem?

I'm writing a paper about hyperconvexity in metric spaces and came across Caristi's theorem: Let $(X, d)$ be a complete metric space and $\phi\colon X \to \mathbb{R}^+$ a continuous function. An ...
5
votes
1answer
45 views

Function on a Power Set

Let $f\colon \mathcal{P}(A)\mapsto \mathcal{P}(A)$ be a function such that $U \subseteq V$ implies $f(U) \subseteq f(V)$ for every $U, V \in \mathcal{P}(A)$. Show there exists a $W \in \mathcal{P}(A)$ ...
1
vote
0answers
121 views

Fixed point theorem for multivalued functions on b-metric space

Hy guys! I'm studying some fixed point results for multi-valued function on a b-metric space $(X,d,s)$. I'm looking for the proof of a theorem wich is just a generalization of Nadler's results for ...
0
votes
1answer
22 views

Fixed point implication question

Suppose $f:[a,b] \to [a,b]$ is continuous and $f''>0$. Use the fundamental theorem of calculus to argue that if $f(x^*) = x^*$ and $f'(x^*) \geq 1$, then $f(x) > x$ for all $ x > x^*$. My ...
1
vote
1answer
46 views

Fixed point property of “3-star”

Let $X = (I_1\sqcup I_2 \sqcup I_3)/(0_1 \sim 0_2\sim0_3),(I_i=[0,1]_i)$. I spend much time to trying to prove that any continuous map $X\to X$ have fixed point, but with no results..
0
votes
1answer
109 views

System of equations and the Brouwer's Fixed-Point Theorem.

Let's consider the following system of equations: \begin{eqnarray}{ e^{xyz} = \frac{x}{\sqrt{e^{2xyz}+1}}\\ \cos(x+y+z) = \frac{y}{\sqrt{e^{2xyz}+1}}\\ \sin(x+y+z) = \frac{z}{\sqrt{e^{2xyz}+1}} }...
1
vote
1answer
26 views

If $\frac{dy}{dx}=A(x)y+B(x)$ and if $A(x)$, $B(x)$ are bounded and integrable, show the fixed point theorem solves the initial value problem

Given the linear differential equation $\frac{dy}{dx}=A(x)y+B(x)$, show that if $A(x)$ and $B(x)$ are bounded and integrable on $I=\{x|a \leq x \leq b\}$, then the fixed point theorem yields a ...
2
votes
1answer
53 views

If $f$ maps a complete metric space $S$ onto $S$ and $f^2=f \circ f$ is a contraction mapping, show $f$ has a unique fixed point.

If $f$ maps a complete metric space $S$ onto $S$ and $f^2=f \circ f$ satisfies the fixed point theorem given below, show $f$ has a unique fixed point. The following is the fixed point theorem: If f ...
3
votes
2answers
64 views

Given $\frac{dy}{dx}=x^2+y^2$ and initial condition $\varphi (0)=1$, find the first 6 terms in the Taylor expansion solution $y=\varphi (x)$

Given $\frac{dy}{dx}=x^2+y^2$ and initial condition $\varphi (0)=1$, use the method of reduction to an integral equation and successive approximation to find the first 6 terms in the Taylor expansion ...
2
votes
0answers
35 views

Using a fixed point theorem.

Let $x,y \in[0,1] $, consider the following system of equations: $$ ((x+y)/2)^n-x=0 $$ $$ {x^n \over x^n+y^n+1}-y=0 $$ where $ n \in N $ a) Transform the system of equations into equivalent fixed ...
1
vote
2answers
47 views

Show using the fixed point theorem that if $f(x)=\frac{1}{4}[1-x-\frac{1}{10}x^5]$ is defined on $I=\{x|0 \leq x \leq 1\}$ then it has a zero in $I$

The main idea here is to apply the fixed point theorem to $g(x)=f(x)+x$, in order to show that f has a zero in $I$. If $g$ has a fixed point (i.e. $g(x_0)=x_0$), then $f(x_0)=0$. I just don't see how ...
0
votes
1answer
23 views

Question about function compositions

Let us have $m,n$ positive integers, and suppose, that $ f o f ... f(m$ times$)$ and $f o f... f(n$ times$)$ have an $x$ fix point. For what $(m,n)$ positive integers will it be true, that $x$ is a ...
4
votes
1answer
105 views

What is the role of fixed point theorems in modern mathematics?

About Fixed Point Theorems, Wikipedia says: Results of this kind are amongst the most generally useful in mathematics. This seems an accurate statement: indeed, there are many journals ...
0
votes
0answers
65 views

Proof of Banach's homeomorphism theorem without the contraction map principle.

Let $E$ a Banach's space and $X\subset E$ open. The Banach's homeomorphism theorem tells us that if a function $F:X\to E$ is a contraction on $X$ then $(I+F):X\to E$ is a homeomorphism of $X$ onto ...
0
votes
0answers
64 views

How to get the proper fixed point iteration function?

When we find the approximated root of a function $f(x)$ in an interval $[a,b]$ from the fixed point iteration method, we derive a new function $g(x)$ which has a fixed point as a root of $f(x)$. Is ...
1
vote
0answers
66 views

Condition for maximizer of convex combination to be expansion mapping

I have $\Pi_n:\mathbb R^{n+1}\rightarrow \mathbb R$ and $F_n:\mathbb R^2\rightarrow \mathbb R$ with $$F_n(x,a)=\Pi_n(x,...,x,a)$$ $$f_n(x)=\operatorname{ArgMax}_{a\in\mathbb R}\{F_n(x,a)\} $$ such ...
0
votes
1answer
29 views

Fixed point location for functions

How are fixed points calculated? Are intersections of $ y = f(x) ,y = f^{-1} (x) $ graphs give real fixed points for all $f$ ?
1
vote
5answers
442 views

Can't there be more than one fixed points in a contraction? or none?

I was going through the contraction mapping theorem in my book where it says, that if $\phi: G\to G$ is a contraction, then $\phi$ has a unique fixed point $\alpha$ on $G$. Sequence {$x_n$}, $x_{n+1} ...
1
vote
0answers
44 views

Deriving the fixed point for $\omega$ (i.e. $\lambda x.xx$) and proving it to be so

I am studying the simply typed $\lambda$-calculus, and I am struggling a bit with really understanding fixed-points and the $\mathbf Y$ combinator. I have read or skimmed all the questions on here ...
3
votes
2answers
124 views

Example of a function, so that $g(x)\neq x$

I'm trying to find an example of a function $g:\mathbb{R}\to \mathbb{R}$ (or $g:[1,\infty) \to \mathbb{R}$), so that $$|g(x_1)-g(x_2)|<|x_1-x_2|$$ for all $x_1, x_2\in \mathbb{R}$ ( or $x_1,x_2\in ...
0
votes
1answer
38 views

Unclear passage of a theorem concerning compact operators (Schauder fixed point theorem)

I'm looking at this proof of Schauder theorem and I am struggling with a passage. This is my problem: Let $X$ be a Banach space, $K \subset X$ a convex, close and bounded set and $F:K \rightarrow K$...
2
votes
1answer
83 views

Which of the followings have a fixed point?

Consider the following sets : $$S=\left\{(x,y)\in \mathbb R^2:x^2+y^2=1\right\}.$$ $$D=\left\{(x,y)\in \mathbb R^2:x^2+y^2\le 1\right\}.$$ $$E=\left\{(x,y)\in \mathbb R^2:2x^2+3y^2\le 1\right\}.$$ ...
1
vote
1answer
32 views

Atiyah-Bott fixed point formula; signs

In classical paper by Atiyah-Singer on page 16 (or 560) stated formula $(3.1)$. It should give classical Lefschetz fixed-point formula if the operator is $d + d^* : \Omega^{even} \rightarrow \Omega^{...
2
votes
0answers
39 views

Noisy contraction mapping

I am trying to describe the behavior of the iterates of a contraction mapping when noise is added. Given a real valued random variable $X_{0}$ a sequence $\{Z_{n}\}_{n=0}^{\infty}$ of i.i.d real ...
1
vote
1answer
44 views

$f (x)=\exp(x/2)−25x^2$. Show that f on $ (4\log(20), \infty)$ has exactly one root.

Let $$f (x)=\exp(x/2)−25x^2$$ Show that $f$ on $ (4\log(20), \infty)$ has exactly one root $x^*$. (Note that log the natural logarithm) I'm struggling with this question, we were given a hint, ...
0
votes
1answer
21 views

Verify (without Banach Contraction Priciple), that the function g(x) = 1 + x - (1/8)x^3 has a unique fixed point

I know how to show that there exist a solution by the intermediate value theorem but I'm not sure how to show that the root is unique?
1
vote
0answers
41 views

How do I show that infinite application of this function gives a constant?

I want to show that g(x) returns the same value independent of x and hence is a constant. $$g(x) = \lim_{n \rightarrow \infty}(\underbrace{f \circ f \circ \cdots \circ f}_{n\text{ times}})(x)$$ ...
0
votes
2answers
329 views

Example of contraction mapping

Please give me some examples of contraction mapping on $(C[0,1]), \lvert \lvert \cdot \rvert \rvert_\infty)$ and $(C[0,1],\lvert \lvert \cdot \rvert \rvert_1) $. Note that : 1. $\lvert \lvert f \...
2
votes
1answer
157 views

If $f$ is a homeomorphism then any periodic point have period less or equal 2

How can one prove the followiong statment? Let $f:[0,1]\to [0,1]$ be a homeomorphism. If $x\in\operatorname{Per}(f)$ then the period of $x$ can't be greater than $2$, i.e, $f(x)=x$ or $f^2(x)=x$.
3
votes
0answers
92 views

Proof of a fixed point theorem on the disk

There is a very nice fixed point theorem which I'd have liked to give to my students : Let $n$, $m$ be two integers larger or equal to one. Let $B_n$ be the open unit ball in $\mathbb{R}^n$, and ...
0
votes
0answers
59 views

Find the fixed point of $\cos(x)$ (equivalently of $\cos(\cos(x))$) restricted to $[0,\frac \pi 2]$.

I've proved that $\cos(\cos(x))$ restricted to $[0,\frac \pi 2]$ is a contraction, which imply by Banach's fixed point theorem that it has a unique fixed point on this interval. I've also proved ...
1
vote
0answers
26 views

Suppose $S^m$ is a contraction on a complete metric space $(X,d)$. I want to show that this implies $S$ has a unique fix-point. [duplicate]

Let $(X,d)$ be a complete metric space and let $S: X \rightarrow X$ be a mapping. Suppose there exist $m \ge 1$ such that $\underbrace {S^m = S \circ S \circ \dots \circ S}_{\text {m times}}$ is a ...