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Questions tagged [fixedpoints]

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function. That is to say, c is a fixed point of the function f(x) if and only if f(c) = c. A set of fixed points is sometimes called a fixed set.

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Fixed point with time dependence

If I have a growth rate of $ \frac{dN}{dt} = (1+cos(\alpha t))(r(c-N)-\mu N) $ and I used the normal method of finding a fixed point (by making the differential equal to 0) I would get $ N = \...
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Global fixed point (function)

So suppose that we have an operator $T:C[0,\infty)\to C[0,\infty)$ such that for all $M\in\mathbb{R}^+$ the restriction of $T|_{C[0,M]}$ maps into $C[0,M]$ and has a unique fixed point, then is that ...
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Finding if a fixed point is attractor or repulsor without differentiation.

Given the function $F(x)=\frac{\pi}{2}\sin(x)$. Find the fixed points and, if they exist, determine if the points are attractors or repulsors without differentiation. I already found the fixed points ...
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Closed graph and fixed points

I’m currently trying to understand the following Proposition from a paper i’m reading: Prop.: Let $X$ and $Y$ be two Hausdorff topological linear spaces. Let $H:X \times Y \rightarrow Y$ be a ...
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Distance between a point and a line in space with unknown line equation

We have $A(-2,3,1)$ and we have to find the distance from $A$ to line which contains point $P(-3,5,2)$ and this line makes equal points with coordinate axis. I know how to solve this, I need the ...
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Doubts in demonstrating Tarski's fixed-point theorem

The book I'm using enunciates: $\textbf{Theorem:}$ Let X be set and $\sigma:P(X) \to P(X)$ increasing function $(x_1 \subset x_2 \subset X \Rightarrow \sigma(x_1) \subset \sigma(x_2) \subset X)$ then ...
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Fixed points of coupled ordinary differential equations

I am trying to find the fixed points of a set of coupled differential equations, which are as follows: $y_1^\prime = y_2$, $y_2^\prime = -\sin(y_1)$ My first attempt involved integrating $y_2^\prime$,...
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Does Lyapunov Stability imply Attractivity for intervals on the real line?

For intervals on a real line, I have found a result which states that for a continuous map, attracting fixed points are Lyapunov stable. However, I found no result about the converse. So, is the ...
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Univalent Mapping - Uniqueness of Fixed Point on the Positive Orthant

I am facing a n-dimensional fixed-point problem descending from a game theoretic problem given by the set of equations $$ \forall j \in n: R_{j} (\vec{x}_{-j}) - x_{j} = W \left( A_{j} \exp \left(-\...
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Holomorphic Map of Open Unit Disk into Itself

I'm currently trying to work through the following problem: Let $\mathbb{D}=\{z\in\mathbb{C}~:~|z|<1\}$ denote the open unit disk. Suppose $f:\mathbb{D}\to\mathbb{D}$ is a holomorphic function ...
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What are the necessary conditions for a polynomial Q(X) such that the roots of Q(X) - X are equal to the real roots of a polynomial P?

If $P(X), Q(X) ∈ ℝ[X]$ , and $P(X) | P( Q(X) ) $ , what could be the necessary conditions for $Q(X)$ such that the set of the real roots of $P(X) $ to be equal to the set of the real roots of $Q(X) - ...
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How to show that $F(g)(x) = \int_{0}^x \cos(\frac{g(t)}{2}) dt$ has a unique fixed point?

Let $X = C([0,1],\mathbb{R})$ equipped with the sup norm $d(f,g) = \sup_{x \in [0,1]} \,\{|f(x)-g(x)|\}$ for each $f,g \in X$. Define $F: X \to X$ by $$ F(g)(x) = \int_{0}^x \cos\left(\frac{g(t)}{2}\...
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Counting numbers of Fixed point of Zeta function by Argument Principle

This is my first post about this topic and now I am trying to evaluate the integral, $$N=\frac{1}{2\pi i}\oint_{|z-1|=1}\frac{\zeta'(z)-1}{\zeta(z)-z}dz+1$$ $\zeta-$is the Riemann Zeta function. I am ...
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Is there a way of proving that a function has a particular number of fixed points.

From my understanding, a function is said to have a fixed point if $f(x) = x$. Is there a way for finding how many fixed points a function has?
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Fixed Point Iteration $x^3 - 3 = 0$

I am having trouble solving $x^3 - 3 = 0$ using the fixed point iteration method. It is advised in the problem to put $g(x)$ in a form similar to $g(x) = x + c(x^2 - 5)$ for $x^2 - 5 = 0$ but I am ...
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Prob. 7 (b), Sec. 28, in Munkres' TOPOLOGY, 2nd ed: A shrinking self-map of a compact metric space has a unique fixed point

Here is Prob. 7, Sec. 28, in the book Topology by James R. Munkres, 2nd edition: Let $(X, d)$ be a metric space. If $f$ satisfies the condition $$ d\big( f(x), f(y) \big) < d(x, y) $$ for ...
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Convergence to a fixed point [duplicate]

Let $f : [a,b] \rightarrow [a,b]$ be a continuous function s.t. $f'(x)$ is defined on $(a,b)$ and $\left\lvert f'(x)\right\rvert \leqq t$ where $0<t<1$. Prove that for any point $x_0$ in $[a,b]$ ...
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Phase Portrait Using Polar Coordinates

I converted a system to polar coordinates and got: $$r'=r^2 \sin \theta \\ \theta'=r^2\cos\theta $$ Now I have to graph the phase portrait near the fixed point at (0,0) and don't know where to begin. ...
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Modifying the “base” of Veblen's hierarchy to exceed $\Gamma_0$

The Veblen hierarchy is usually defined with $\varphi_0(x) = \omega^x$. As a result, we can define the Feferman–Schütte ordinal as the first fixed point of the function $\varphi_\alpha(0) = \alpha$. ...
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How to find all fixed points for this problem?

Find all fixed points of the below function $$f:X\rightarrow X$$. $$X=R^N$$ and $$d(x,y)\equiv\Vert x-y\Vert_2$$ and $$f(x)\equiv Ax$$ where $$A\equiv \begin{bmatrix} 1 & 1 \\ 0 &1 \\ ...
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Locally convergence for Fixed -point iterations

Taking the following iteration $$x_{n+1}=-\ln(x_n),\quad x_0\in]0,+\infty[$$ Study the convergence of this iteration. By applying Fixed-point theorem, I found that the function is not contraction on $...
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Prove a Lemma Involving Asympotically Stability

I am trying to prove the following Lemma: Lemma: Suppose that the point $x^*$ is a fixed point of $x(n + 1) = f(x(n))$ (1) while also an asymptotically stable(unstable) fixed point with respect ...
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How to show that one not monotonous f doesn't have fixpoints?

I have a question about fixed points If I have one function $f$ (that is not monotonous!) I would like to demostrate that this function hasn't fixed points. I need to find a funciton $f$ for which ...
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Dynamical systems with forced oscillation

I want to work out the dynamics of this equation (by that I mean find the fixed points of the system and then the stability of the fixed points): $\frac{dN}{dt}= \sin(\alpha t)\left(rN\left(c-N\right)...
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44 views

Showing uniqueness of a fixed point on $[0,1]$

Given $g(x)=-x\sin^2(\frac{1}{x})$ for $0<x\leq1$ My attempt: let fixed point given by $g(x)-x=-x\sin^2(\frac{1}{x})-x=0$ $$0=-x\left(\sin^2\left(\frac{1}{x}\right)+1\right) $$ Therefore only for ...
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Different definition of Veblen functions

Consider the Veblen hierarchy, where $\psi_0(x) = \omega^x$ and $\psi_1(x)$ is the x'th fixed point of $\psi_0$, $\psi_2(x)$ is the x'th fixed point of $\psi_1(x)$, and so on. We eventually get to $\...
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Fixed points of ordinal exponentiation for bases besides $\omega$

The first fixed point of the map $x \to \omega^x$ is the first epsilon number $\epsilon_0$, which is the supremum of $\omega, \omega^\omega, \omega^{\omega^\omega}, ... = \omega^{\omega^{\omega^{.^{.^{...
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Let $u'(t)=Au^2-Bu$. Find conditions on A, B to guarantee global solution.

I'm taking a real variable course and we have just covered the Banach Contraction Principle. Our professor sometimes makes problems up on the spot for us to try and figure out together. This is one ...
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Fixed point of Riemann Zeta function

I have been looking for fixed points of Riemann Zeta function and find something very interesting, it has two fixed points in $\mathbb{C}\setminus\{1\}$. The first fixed point is in the Right half ...
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Is the least inaccessible cardinal equivalent to the first aleph fixed point? [duplicate]

Let $I$ be the least / first inaccessible cardinal. As inaccessible cardinas are all aleph fixed points, and they are regular, so each inaccessible cardinal is an aleph fixed point after the previous ...
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Fixed points of a differential equation

My assignment is to determine fixed points of the differential equation $$\frac{dN}{dt} = \left(\frac{aN}{(1+N)} - b -cN\right)N$$ where $a,b,c > 0$ and find out their stability. I do ...
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Guesses on fixed point existence

Let $\mathcal{X} \subset \mathbb{R}^n$ be a finite set and the mapping $\Phi : \mathcal{X} \rightarrow \mathcal{X}$ be defined as follows: $\Phi(x) := \{y \in \mathcal{X} \mid J(y,x) \leq J(z,x), \, \...
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Difference between Stable, unstable sets , manifolds and subspaces?

I am confused about the difference between the stable sets, unstable sets ; stable and unstable manifolds; and stable and unstable subspaces of a fixed point $p$ of a map $f : M \rightarrow M$ where $...
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Number of lines formed from given points

If these are $n$ points out of which $m$ are collinear then the total number of lines are $$= \binom{n}{2}-\binom{m}{2}+1$$ According to what I have thought, when we subtract $m\choose{2}$ we ...
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A composition of projections with three fixed points — is it necessarily the identity?

We are given a line $l$. The line is mapped onto itself through a series of projections that involve other lines and -- importantly! -- conics. In the end, points $A$, $B$, and $C$ on $l$ appear to be ...
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How to find fixed point without graph approach?

The function tan(x)=x has 1. Unique fixed point 2. No fixed point 3. Infinite many fixed points 4. More than one but finitely many fixed points. My attempt: I have solved this problem by graphical ...
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Are the fixed point sets homeomorphic?

Assume $f:G\rightarrow H$ is a continuous epimorphism of topological groups and $K\leq H$ is a subgroup of $H$. One can consider the actions of subgroups $K$ and $f^{-1}(K)$ on $H/K$ and $G/f^{-1}(K)$ ...
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Shortest length of wire required to connect points

There are 2 sets , $A=\{a_1,a_2,a_3..... ,a_n\} $ and $B=\{b_1,b_2,b_3.. b_n\}$ . All elements of these sets are real numbers. Each point in set $A$ need to be connected to one and only one point in ...
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How to find g(x) and aux function h(x) when doing fixed point interation?

I'm learning fixed point iteration (first and second form). My teacher said there are two forms: g(x) = x - f(x) g(x) = x - h(x)f(x) where ...
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Derivative of an infinite composition of functions

Let $f(x)=g(g(g(...g(x))))$, where the function $g$ is applied to $x$ and infinite amount of times. I am assuming that $x$ is real. What is special about points at which $df/dx=0$? Are they related to ...
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If $g:X \to [0,\infty)$ is defined by $g(x)=d(x,f(x))$, prove that $g$ is uniformly continuous on $X$.

Let $f:X \to X$ be a function on a compact metric space $(X,d)$ such that $d(f(x),f(y)) < d(x,y)$ for all $x\neq y$ a) If $g:X \to [0,\infty)$ is defined by $g(x)=d(x,f(x))$, prove that $g$ is ...
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Parabola always through two points.

I need to draw parabola through two points, but also need to be able to change the width of the parabola. In more details let's say I have parabola: $f ( x ) = - \frac { ( x - 1 ) ^ { 2 } } { W } + 1$...
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$B _{\ell ^{2}} ^{+}$ with the norm $\Vert\vert \cdot \Vert\vert _{\sqrt{2}}$ don't have normal structure

Consider the space $\ell ^{2}$ with the standard norm \begin{align*} \Vert x \Vert _{2} = \left( \sum _{i =1} ^{\infty} x _{i} ^{2} \right) ^{1/2} \end{align*} and we define the equivalent norm \...
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Existence of fixed simple closed curve by polynomials

As the problem mentioned in the title, I wonder that if there exists a simple closed curve on the complex plane which is not circle that can be fixed by a non-linear polynomial with complex ...
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Fixed-point iteration: $\sqrt{\varepsilon} = \sqrt{c - \varepsilon} \tan (a \sqrt{c - \varepsilon})$

Let $a, c > 0$. Use the Banach fixed-point theorem to show that $$\sqrt{\varepsilon} = \sqrt{c - \varepsilon} \tan (a \sqrt{c - \varepsilon})$$ has at least one solution $\varepsilon \in (0, c)$ ...
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Fixed point function

Given a function $g(x)$ defined over $\Omega = [a,b]$ with the following properties: $\Omega$ stable by $g$ : $g(\Omega) \subset \Omega \iff \forall x \in \Omega, g(\Omega) \in \Omega$ $\exists K,K&...
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System of equations $a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2)$

Given a positive real number $t$, find the number of real solutions $a, b, c, d$ of the system $$a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2) = t$$ I have a solution Let $f(x)=\frac t{1-x^2}$ ...
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23 views

How to deal with an integro-differential equation of this form - fixed points?

I've encountered an integro-differential equation of the following form: $$ \frac{dx(t)}{dt} = \int_0^t ds\ f_{1}(s) - \int_0^t ds\ f_{2}(s) x(t - s) $$ The functions $f_{1}(t)$ and $f_{2}(t)$ are ...
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2answers
64 views

Showing lim xn = c

Can anybody help with with (3) My Solution for (1) and (2) Put $g(x):=f(x)-x$, which is still continuous on $[a,b]$ and differentiable on $(a,b)$. Observe that $x\in [a,b]$ is a fixed point for $f$ ...
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1answer
45 views

Finding a common Lipschitz constant for a family of contractions

Let $(X,d)$ be a complete metric space and $f:[0,1] \times X \to X$ a family of functions such that $f(t,\cdot)$ is a contraction for every $t \in [0,1]$. Further assume that $f(\cdot,x)$ is ...