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, ...

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When does a continuous function defined on a non-compact closed and bounded convex set has a fixed point?

Is there any result in fixed point theory which will give the existence of a fixed point for a continuous function defined on a non-compact, closed and bounded convex set?
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34 views

An equation with multiple solutions: finding the maximum of the function of the solutions.

Possibly, this is a bad (stupid) question, but sometimes some discussion helps. I have a fixed point equation (involving $\tanh$). I would like to derive the dependency of some function of the fixed ...
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22 views

A problem similar to Banach fixed point theorem

a) Let $(X,d)$ be a complete metric space and let $T: X \to X$. Prove that if there exists a natural $n$ such that $T^n(x)$ (composition of $T$ $n$ times) is a contraction then $T(x)$ has a unique ...
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1answer
35 views

Metric Geometry determining fixed points

Let $f\colon\mathbb{R}\to\mathbb{R}$ be given by $f(x)= e^{-x}$. Show that $f$ has a fixed point and determine what it is.
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fixed point iteration code [closed]

So as I was working out my math with the following equations using the fixed point iteration, I stumbled to wonder if one can ever write a C program, MATLAB (or the likes) for such math. So my ...
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44 views

Proof that a continuous function from the unit ball to itself without fixed points implies existence of retract from unit ball to unit sphere

Assume $f:B_{1}\to B_{1}$ (where $B_{1}$ is the closed unit-ball in $\mathbb{R}^{n}$ ) is a continuous function that has no fixed points I need to construct a function $g:B_{1}\to B_{1}$ which ...
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72 views

Use the Mean Value Theorem to show that if $|f'(x)| ≤ C<1$, then $f$ has at most one fixed point

Use the Mean Value Theorem to show that: if $|f'(x)| ≤ C < 1$ $\forall x$, then $f(x) = x$ has at most one solution. So using the Mean Value Theorem I know that $$-1<-C\leq ...
3
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90 views

Maximum diameter for the preimage of a point when the degree is not 1 or -1

When the degree of a map $S^n \rightarrow S^n$ isn't 1 or -1, are there always two points that map to the same point and whose distance from each other can be bounded below by a function of the ...
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2answers
48 views

prove that $x \mapsto \mathrm e^{-x}$ has a unique fixed point on R

Can anybody prove $x \mapsto \mathrm e^{-x}$ has a unique fixed point on R using the fixed point iteration theorem?
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1answer
23 views

Characterizing conditions for $\tanh{(kx-b)}=x$ to have 1/2/3 fixed points.

I am trying to understand what are the conditions for $\tanh{(kx-b)}$ to have 1 or 2 or 3 fixed points. That is I am trying to characterize conditions on $k$ and $b$ for which equation ...
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45 views

Corollary of Banach fixed-point theorem

Let $(X, \left\lVert\cdot\right\rVert)$ be a Banach space. Let $A:X\to X$ be a linear map and $\nu\in \mathbb{N}$ such that $A^k:X\to X$ is a contraction for every $k>\nu$. Is it true that for ...
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1answer
19 views

Undamped Forces

I want to make sure I am doing this problem correctly, especially when it comes to drawing the potential function V(x). Consider the system of differential equations: $$\dot {x}=y$$ $$\dot ...
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1answer
47 views

The set of all fixed points of a continuous function $f:[0,1] \to [0,1]$ , satisfying $f \circ f=f$ , is a non-empty interval ? [closed]

Let $f:[0,1] \to [0,1]$ be a continuous function such that $f \circ f=f$ on $[0,1]$ , then is it true that the set $\{x \in [0,1] : f(x)=x \}$ is a non-empty interval ?
2
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1answer
22 views

give a counter example that $T^n$ is contraction will not imply that $T$ is contraction.

Let $T$ be a contraction map then $T^n$ is contraction.. We can prove this result by induction on n.. But the converse is not true... help me to give a counter example that $T^n$ is contraction will ...
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1answer
27 views

Fixed Point Iteration - Numerical Analysis

please help me solve the following question. Qsn: Solve the following system by Fixed Point Iteration. $$ x^2-2x+y^2-2y=3$$ $$x+y=-1$$ Progress: So I know that we have to assume one of the ...
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25 views

When does a continuous function defined on a closed and bounded convex set has a fixed point?

For a function $f$ defined from a domain $K$ to itself, we have a point $x$ in $K$ is said to be a fixed point of $f$ if $f$ maps $x$ to itself. When the domain K is a compact convex set with some ...
3
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68 views

Holomorphic function with a unique fixed point

Let $\omega \subset \mathbb C$ be a simple connected set and $\,f:\omega \to A$ is an analytic function where $A \subset \omega$ is compact. Show that $f$ has an unique fixed point. I think we can ...
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1answer
21 views

Proving Theorem: subspace of polynomials of degree two or less?

How can I prove that the set $S$ of polynomials of degree $2$ or less, whose coefficients sum to zero, is a subspace of all polynomials with degree $2$ or less? I know I need to show that $a+b+c=0$ ...
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123 views

Why use the Lefschetz Zeta function?

Given a compact, triangulable space $X$ and a continuous function $f: X\rightarrow X$, then we define the Lefschetz number $\Lambda_{f}$ by $$\Lambda_{f} = \sum_{k\geq0}(-1)^{k}Tr(f_{\ast}\vert ...
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Proof of Tarski's self-reference lemma

In http://www.math.hawaii.edu/~dale/godel/godel.html, Tarksi's self reference lemma is mentioned but the proof is omitted. Tarski's Self-Reference Lemma. For any formula $p(x)$ in an adequate ...
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3answers
44 views

What are the conditions to apply Brouwer fixed-point theorem (one dimentional case)?

Does this theorem work for a continuous function $f : ]a,b[ \rightarrow ]a,b[$ with $a, b \in \mathbb{\bar{R}}$ ? Thanks in advance.
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Determine $p$, $q$ and $r$ so that the order of the fixed point iteration for computing $a^{1/3}$ becomes as high as possible

So I'm given the following equation for computing $a^{1/3}$ $$x_{k+1}=px_k + \frac{qa}{x_k^2} + \frac{ra^2}{x_k^5}$$ and I have to find the p, q, and r so that this equation converges to $a^{1/3}$ as ...
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43 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 $\dot {y}$ ...
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1answer
69 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 ...
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1answer
16 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 ...
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1answer
54 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
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1answer
36 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 ...
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33 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 ...
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27 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 ...
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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: ...
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1answer
35 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. ...
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39 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.
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71 views

System of differential equations, phase portraits

Consider the system of differential equations: $$x'=y-x^2$$ $$y'=x-y$$ a. Determine the fixed points. So setting both equation equal to 0, I get: $y=x^2$ and $x=y$ So the only fixed points would ...
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1answer
42 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 ...
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1answer
48 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
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1answer
65 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 ...
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54 views

Stability of fixed points for a differential equation

Consider the differential equation $x'=x^2-9$ I am pretty lost on this problem.. a. find the stability type of each fixed point To find the fixed points, I set this equal to $0$, right? Would ...
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25 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 \} ...
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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 ≤ ...
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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 ...
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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 ...
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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) = ...
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1answer
57 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 ...
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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 ...
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14 views

Finding a fixedpoint

How would I go ahead and find a suitable starting point to use in the fixedpoint algorithm? I've already made the algorithm in MATLAB but I don't know how to find a suitable starting point. Thanks!
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1answer
19 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 ...
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118 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 ...
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50 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 ...
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36 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. ...
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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 ...