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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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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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 ...
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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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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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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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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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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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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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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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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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$\phi : S^n \to S^n$ 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 < ...
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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 ...
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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 ?
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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 ...
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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 $(k-1)/k$, ...
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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 ...
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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, ...
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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 ...
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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 ...
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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: ...
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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 ...
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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 ...
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Closest fixed point to a convex set

Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$. Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ ...
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About the compactness condition in Schauder fixed point theorem

The theorem is Let $X$ be a locally convex topological vector space, and let $K ⊂ X$ be a non-empty, compact, and convex set. Then given any continuous mapping $f: K → K$ there exists $x ∈ K$ ...
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Fixed point property in topology

I have a few questions concerning relating the fixed point property for a space $X$ (every continuous map from $X$ to $X$ has at least one fixed point) to some concepts in topology. a). I know that a ...
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Find out Fixed Points

Consider a set $M$ of all possible square matrices of dimension $k$ over a finite field $F_p$. Consider a map $f$ defined on $M$ as $f(X)=X^2+C$ where $X \in M$ and C is an arbitrary fixed matrix from ...
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Diffeomorphism and hyperbolic points

Suppose $f$ is a diffeomorphism.Prove that all hyperbolic periodic points are isolated. I tried using the mean value theorem using two diferent periodic points (assuming the periodic points arent ...
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Fixed element $u \in U$ and subsets difference only at the $u$ point.

I have a task with a tip to it. The task: Let U - is not empty ultimate multitude. Prove, that number of subsets of the multitude U, with even power, same as how many subsets with odd power. ...
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Firm non-expansiveness in the context of proximal operators

$\newcommand{\prox}{\operatorname{prox}}$ Probably the most remarkable property of the proximal operator is the fixed point property: The point $x^*$ minimizes $f$ if and only if $x^* = \prox_f(x^*) ...
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Fixed point of projected operator

Let $X \subset \mathbb{R}^n$ be a compact convex set and let $f: X \rightarrow X$ be Lipschitz continuous and such that $$ \left( f(x) - f(y) \right)^\top \left( x-y\right) \leq 0 $$ for all $x, y ...
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A technical step in the proof of Atiyah-Bott fixed point formula

From John Roe, Elliptic operators, topology and asymptotic methods, page 135. Here Roe tried to prove that $$ \DeclareMathOperator{\Tr}{Tr} \sum_{q}(-1)^{q} ...
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Number of limit cycles: Counterexample of the extended Bendixson-Dulac criterion?

The problem concerns the number of limit cycles in the vector field of coupled differential equations (ODEs) in two dimensions, i.e. $$ \ \dot{x} = X(x,y)\\ \dot{y} = Y(x,y) $$ Specifically, let $$ \ ...