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, ...

learn more… | top users | synonyms

1
vote
0answers
57 views

Nullclines for differential equations

Consider the system of differential equations $$\dot {x}=y-x^2$$ $$\dot {y}=x-y$$ a. Determine the fixed points (1,1) (0,0) b. Determine the nullclines and the signs of $\dot {x}$ and $\dot {y}$ ...
3
votes
1answer
75 views

Only one fixed point for $f:\bar{\mathbb{D}}\rightarrow\bar{\mathbb{D}}$ on the boundary.

We know for Brouwer theorem that $f$ (continuous bijective function) have a fixed point. My questions are: 1) Is there a function with only one fixed point $x_0\in Int(\bar{\mathbb{D}}) $ (open ...
1
vote
1answer
37 views

Differential equations, stability of fixed points

Consider the differential equations: $$\dot{x}=x^2-9$$ $$\dot{x}=x(x-1)(2-x)=-x^3+3x^2-2x$$ a. Find the stability type of each fixed point. (I am not sure about the stability of the points. Do I ...
2
votes
1answer
76 views

Fixed Point Equivalences of Axiom of Choice

Axiom of Choice has many known equivalences. Also there are many known fixed point theorems (unproved statements) which provide useful information about existence of fixed points for particular ...
3
votes
1answer
52 views

Contraction vs. iterated function convergence

Let $X$ be a Banach space with norm $|x|_X$. (For example $X=\mathbb R$) We assume that a function $F: X \rightarrow X$ is Lipschitz continuous but not a contraction map, hence: $$ | F(x) - F(y) |_X ...
0
votes
1answer
38 views

Computing the fixed point for $cosx$

While studying about Compiler Design I came with the term 'fixed point'.I looked in wikipedia and got the definition of fixed point but couldn't get how fixed point is computed for $cosx$ as said in ...
0
votes
1answer
31 views

Fixed Point Summary

I know someone has given me resources for this before but I can't seem to find them... Would someone please summarize stable vs unstable, attracting vs repelling, and node, saddle,etc fixed points? I ...
-1
votes
1answer
57 views

Fixed points and stability of them

Find the fixed points and classify them for the system of equations: $$x'=v$$ $$v'=-x+wx^3$$ $$w'=-w$$ $$0=v$$ $$0=-x+wx^3$$ $$1=wx^2$$ $$0=w$$ is the only fixed point (0,0,0)?? jacobian: ...
0
votes
1answer
95 views

System of differential equations, phase portraits and stability of fixed points

Consider the system of differential equations: $$x'=-x-y+4$$ $$y'=3-xy$$ a. Find the fixed points. $x'=-x-y+4$ $x+y=4$ $x+3/x=4$ x=3,x=1 $y'=3-xy$ $y=3/x$ fixed points: (1,3), (3,1) b. ...
2
votes
2answers
88 views

History of fixed point theory

I am looking for encyclopedic references for fixed point theory and its applications. What is the best reference for this subject? thank you.
0
votes
1answer
69 views

System of differential equations, phase portrait

Consider the system of differential equations: $$x'=-2x+y-x^3$$ $$y'=-y+x^2$$ a. Determine the fixed points. (Am I correct in thinking that to determine the fixed points, I must set x' and y'=0? I'm ...
1
vote
1answer
87 views

Show that operator T is a contraction mapping

I want to check whether the operator T defined as: $Tf(x) = \beta \left\lbrace \sum_{\theta} \mu_\theta \left[ h_\theta(x) + \int f(x') Q_\theta(x,dx') \right]^\alpha \right\rbrace^{1/\alpha} $ is a ...
3
votes
1answer
90 views

Role of determinant of the matrix of any Homology group.

I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ...
0
votes
2answers
134 views

Stability of fixed points for a differential equation

Consider the differential equation $x'=x^2-9$ a. find the stability type of each fixed point To find the fixed points, I set this equal to $0$, right? Would someone mind explaining why I do ...
1
vote
2answers
37 views

On subgroups of isometries and their respective fixed-points

I am working on the following problemset: Let $G < \DeclareMathOperator{\Iso}{Iso}\Iso(E)$ be a finite subgroup of the isometries in the euclidean plane. Denote: $$G = \{g_1, \ldots , g_n \} ...
6
votes
2answers
85 views

Path Connectedness and fixed points

We have the following given to us, Let $α, β \colon [0, 1] \to [0, 1]$ be (not necessarily continuous) functions such that $α(x) ≤ β(x)$, for all $x ∈ [0, 1]$. The set $K = \{\,(x, y); α(x) ≤ y ≤ ...
1
vote
0answers
71 views

Find Lyapunov function for $\dot{x} = -\sin(x)$

$$\dot{x} = -\sin(x)$$ Find the fixed points and also find out if it is attractive or repelling Find Lyapunov function for each of the attractive fixed points. I thought: Fixed points are ...
1
vote
1answer
33 views

Find fixed points and give the nullclines of the following system ($\dot{x} = \(y)$ and $\dot{y} = \cos(x)$)

Given the following system: $\dot{x} = \sin(y)$ $\dot{y} = \cos(x)$ Find the fixed points and check their stability Give the nullclines So I thought: fixed points are ...
2
votes
3answers
101 views

Validity of this proof that any continuous function with domain and range in [0,1] must have a fixed point.

The following proof was given in a solutions manual to a question asking to prove that a continuous function with domain and range in $[0,1]$ must have a fixed point: Consider the function $F(x) = ...
3
votes
1answer
416 views

Leray-Schauder fixed point theorem

I know the proof of the Schauder fixed point theorem which states Schauder fixed point theorem : If $D$ is a non-empty , convex and compact subset of Banach space $B$ and $T:D \to D$ a ...
7
votes
0answers
195 views

Fixed points in computability and logic

I asked this question on CS.SE, too: http://cstheory.stackexchange.com/questions/27322/fixed-points-in-computability-and-logic I would like to understand better the relation between fixed point ...
1
vote
1answer
30 views

Real 2D Analysis Question using Brouwer's Fixed Point Theorem

My question is as above. Currently I am stuck at the very start, part (i)! I can't come up with an appropriate $f$, even though I've been thinking about it for ages! If someone would be able to ...
5
votes
1answer
143 views

Composition of projections has a fixed point in a Hilbert space

Let Let H be a Hilbert space with an inner product ⟨⋅,⋅⟩ : H×H→R, and induced norm $∥⋅∥ : H→R_+$ Let $C_1$ and $C_2$ be closed, convex, nonempty, disjoint subsets of $H$ with at least one of ...
1
vote
0answers
60 views

Does this function have a unique fixed point?

Consider the following equations in fixed-point form: $$\mathbf{x}=F(\mathbf{x})=[f_1(\mathbf{x}),f_2(\mathbf{x}),\cdots,f_n(\mathbf{x})]^T$$ where the function $f_i$ is defined as ...
1
vote
0answers
94 views

Newton method and the Banach fixed-point theorem

I try to combine the Newton method and the Banach fixed-point theorem but I have still some questions: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous. ...
1
vote
0answers
27 views

A proviso in l'Hospitals rule

L'Hospital's Rule, which states that: $\displaystyle\lim_{x\to a}\frac{f(x)}{g(x)} = \displaystyle\lim_{x\to a}\frac{f'(x)}{g'(x)}$ can be applied when: (1) f, g are differentiable, (2) g'(z) ≠ 0 ...
0
votes
0answers
20 views

Can Banach fixed-point theorem be generalized to the case where each term depends on multiple previous terms in recurrence sequence

Banach fixed-point theorem http://en.wikipedia.org/wiki/Banach_fixed-point_theorem I'm wondering if the theorem still applies if the recurrence relation is, for example, ...
1
vote
1answer
95 views

Banach fixed point theorem (application)

Let $X$ and $Y$ be Banach Spaces, and $T: X \to Y$, where T is continuous, linear and bijective, and let $S: X \to Y$ (where $S$ is continuous and linear) with $|S|\cdot|T^{-1}|< 1$. Show that ...
5
votes
2answers
151 views

Fun with Newton's Method - Infinitely many cycles

I'd like to preface this problem by saying that I have absolutely no clue if it is solvable or not. This is just the result of some musings, and I'm looking for either some guidance, or to be pointed ...
0
votes
1answer
53 views

Fixed Point Theorem in finite dimensional Euclidean space

A fixed point theorem says that: "any continuous mapping of $\mathbb{R}^n$ into a bounded subset of $\mathbb{R}^n$ has a fixed point". So consider $f: \mathbb{R}^n \rightarrow X \subset ...
7
votes
2answers
610 views

There exists x on closed interval such that f(x)=x

If $f$ is a continuous function on a closed interval, how can I show that there exists some $x$ on $f$ that $f(x)=x$? I know it will require the Intermediate Value Theorem. Initially I thought of ...
0
votes
0answers
44 views

Stability of a fixed point

Determine the stability of all the fixed points of the following functions: So basically I understand how to find fixed points and whether they are attracting/repelling...but I am confused on how to ...
0
votes
1answer
55 views

$\phi : S^n \to S^n$ with no fixed point

The question is as follows: "Find a continuous map from $S^1$ to $S^1$ with no fixed points. What about for $n > 1$?" I want to write $S^n = \{(1, \theta_1, \dots, \theta_n) | 0 \leq \theta_i < ...
1
vote
0answers
27 views

Question about Computability

Q:Suppose $U(n,x)$ is Gödel Universal Function, show that there is $n$ such that $U(n,x)=n+x$ for all $x$ I did some proof but I am mot sure if I am right. Let's consider a computable binary function ...
1
vote
3answers
105 views

Proof regarding a fixed point [duplicate]

Show that for any strictly increasing function $f:[0,1]\to[0,1]$ there is a fixed point such that $f(x)=x$. ( The function isn't necessarily continuous) . Any ideas ?
1
vote
1answer
52 views

If $f:[a,b]\to[a,b]$ is a nondecreasing function then $\exists x_0\in[a,b]$ such that $f(x_0)=x_0$

I got this problem: Prove that if $f:[a,b]\to[a,b]$ is a nondecreasing function then $\exists x_0\in[a,b]$ such that $f(x_0)=x_0$ (i.e. $f$ has a fixed point). (Hint: set $A=\{x\in[a,b]|x\leq ...
0
votes
0answers
39 views

Newton map homework help

Let $p(x)=(x-x_0)^kg(x)$, with k>1 and $g(x_0)\ne0$. Assume that $p(x)$ and $g(x)$ have at least two continuous derivatives. Show that the derivative of the Newton map for $p$ at $x_0$ is $(k-1)/k$, ...
0
votes
1answer
53 views

Logistic Maps Homework Help

Consider the logistic map $G(x) = 4x(1-x)$. Let $q_0=0<q_1<q_2<q_3...<q_7$ be the eight points left fixed by $G^3$. Determine which are the two fixed points and which other points are ...
0
votes
1answer
33 views

Stability of a Fixed Point Homework Help

Determine the stability of all the fixed points of the following function: $f(x)=2\sin(x)$. I've found the fixed points. They are $x=0$ and $x=1.895$. How can I determine the stability now?
0
votes
1answer
43 views

Non-uniform contraction

Let $(X,d)$ be a metric space. A map $f: X \longrightarrow X$ is called contracting if there exists a $\lambda < 1$ such that for any $x, y \in X$ $$d(f(x),f(y)) \leq \lambda d(x,y)$$ It is well ...
4
votes
1answer
58 views

Is this a valid way to show $\chi(SL_n(\mathbb{R}))=0$?

Why does $\chi(SL_n(R))=0$? I'm going about it like this. Let $X:=SL_n(R)$. Define a map $f:X\to X$ such by $A\mapsto BA$, where $B$ is the identity matrix, except with an extra $1$ in the upper ...
0
votes
0answers
66 views

Finding the proper g(x) for fixed point iteration on $2\sin{\pi x} + x = 0$

After spending over an hour trying to get this problem I realize my trig is weak. I found: Fixed point iteration .Numerical method. The selected solution is informative, but lacking detail to really ...
5
votes
1answer
69 views

Show that there exists $\xi\in [a,b]: f(\xi)=\xi$.

Let $a,b\in\mathbb{R},~a<b$ and consider $f\colon[a,b]\to [a,b]$ continuous. Show that $f$ has a fixed point. i.e. that there exists a $\xi\in [a,b]$ with $f(\xi)=\xi$. My idea is to ...
1
vote
1answer
29 views

Fixed points and dynamics of $x-x^2$, $x+x^3$, $x-x^3$ and $\tan x$

For each of the following functions, show that $0$ is a fixed point, and the derivative of the function is $1$ at $x=0$. Describe the dynamics of points near $0$. Is $0$ attracting, repelling, ...
1
vote
1answer
66 views

Fixed Points and Graphical Analysis

For the following functions (i) find the fixed points and (ii) use the graphical analysis to determine the dynamics for all points on R: $f(x)=2sin(x)$. I have no problem finding the fixed points, ...
0
votes
1answer
56 views

Fixed points and periodic orbits of $F(x)=x^2-1.1$

A question asked me to find the fixed points of $F(x)=x^2-1.1$, then use the fact that these points were also solutions of $F^2(x)=x$ to find the cycle of the prime period 2 for F. How do I go about ...
3
votes
3answers
185 views

Fixed point combinator and functions with no fixed point

In lambda calculus the fixed point combinator is defined as: It is very easy to see how $Yg =g(Yg)$ for any $g$ by using $\beta$-reduction. At the same time I wonder what is the meaning of ...
5
votes
0answers
67 views

Limit of a Discrete Dynamical System, Part 2

In my previous post (i.e., Limit of a Discrete Dynamical System) the following system was considered: ...
2
votes
0answers
24 views

Proof of equality of least fixed point of two continuous function

The question is simple: given a set U, a continuous function (Scott continuity) $f \colon \mathcal{P}(U) \to \mathcal{P}(U)$ and function $g(X) := f(f(X))$, prove that $g$ is continuous and its least ...
2
votes
0answers
51 views

Reference request for the proof of the Brodskii–Milman fixed point theroem for isometries

Can any one help me to access the paper M.S Brodskii and D.P Milman, On the center of a convex set, Dokl. Akad. Nauk SSSR 59 (1948) 837–840 in Russian? or to prove the theorem If $K$ is a ...