Questions tagged [fixed-points]

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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What are the fixed points of the arithmetic derivative over the non-negative integers?

I just watched When the derivative of a number is not zero -- The arithmetic derivative by Michael Penn. I like to explore things visually and computationally, so I found this recursive implementation ...
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Calculate Intersection point of 2 Lines with angle [closed]

I would like to calculate an intersection point of two lines in a 2D area. I think it should be really simple but i cannot figure it out. I have two points P1(x,y), P2(x,y) and 2 angles alpha and beta....
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Proof for fixed point under specific circumstances

Prove that every $f \in C(I, \mathbb{R})$ with $I := [-1, 1] \subset \mathbb{R}$ and $f(I) \subseteq I$ has a fixed point. This would be true if $f$ is a contraction on $I$, since then Banach's fixed ...
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Fixed points and stability of 2 differential equations

I need help checking this answer please Question: $\frac{ds}{dt} = -R_0su = f(s,u)$ $\frac{ds}{dt} = R_0su -(S_0 + 1)u= g(s,u) $ Constraint:$s+u=1$ Answer: $FP_1 = (1,0)$ $R_0su -(S_0 + 1)u =0$ $R_0(...
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Are these steady states of non linear dynamic system actually steady states?

I have the following non linear dynamic system in discrete time: \begin{equation} x_{t+1} = \frac{1}{1 + \exp\left(- \beta \left( 2 d \left(c + \frac{(1 - c)}{1 + a (1 - x_{t}) d}\right) - b - d \...
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Existence of approximate fixed point

In K.Urai's paper "Fixed point theorems and the existence of economic equilibria based on conditions for local directions of mappings", he claimed the following in Lemma 17: Statement: Let $...
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Stability of fixed points

Hi I was exploring the bifurcation diagram of the logistic map when I came across the concept of fixed points and their stability. However, I don't really understand how it is stable when |f′(x)|<...
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When is the minimum equal to the fixed point for this function?

Consider a smooth function $f:R^n\times R^n\rightarrow R$ such that $g(y) = \text{argmin}_x f(x, y)$ is a contraction with the unique fixed point $x^*=g(x^*)=\text{argmin}_x f(x, x^*)$. My question is,...
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Prove the $k$-th power of the logistic map with parameter $\mu = 4$ has $2^k$ fixed points

I'm trying to solve an exercise in which I need to prove that the logistic map with parameter $\mu = 4$, $F_4:[0,1]\to[0,1]$, $F_4(x) = 4x(1-x)$, satisfies that for every positive integer $k$, $F_4^k$ ...
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Continuous function on the unit interval with commuting compositions

Let $I$ be the unit interval and $f, g:I\to I$ be continuous functions. Assume that $f\circ g = g\circ f$. This post shows that if $f$ and $g$ in addition are assumed to be increasing then $f$ and $g$ ...
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Finding fixed points algebraically

I am working on the following problem: Consider the function $f: [-1,1] \mapsto \mathbb{R}, f(x) = x(x-1)(x+1)$ What are the fixed points of $f$? What are the fixed points if we extend the domain to $[...
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Fixed Points of Weighted Sum of Two Mappings

I do not have a background in functional analysis but I am basing something on tuition here and would really appreciate some help with this problem. I have two mappings $F_1: X\rightarrow X$ and $F_2: ...
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Showing that a function has a fixed point in an interval

I want to show that a function has a fixed point in a given interval. Here is the setup: Let $f : [1,3] \mapsto \mathbb{R}$ be defined by $f(x) = 3^{g(x)}$ with $g(x) = \frac{3}{2} - |x - \frac{x^2}{2}...
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Contraction Mapping and Fixed Point with two different distance metrics

I have been looking at the fixed point theorems that use the contraction-mapping and all seem to use the same distance metric for the input and output spaces. If we have a differentiable mapping $f: X ...
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Given a random permutation $\sigma$ of a set $\{1,...,n\}$. Assume we want to compute $X$=# of integers $i$ with $\sigma(i)=i$. What is $E[X]$?

Given a random permutation $\sigma$ of a set $\{1,...,n\}$. Assume we want to compute $X$=# of integers $i$ with $\sigma(i)=i$. What is the expected value of $X$? Potential Solution Part 1 I had an ...
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Check my example: does a function satisfying properties $1-4$ exist?

As an exercise, I need to find a $C^2$ function $f:\mathbb{R}^*\to\mathbb{R}$ such that $\eta>0$ exists such that $$ 1.\quad |f^{\prime}(x)|\le \eta\quad \mbox{ as } x\to +\infty;$$ $$ 2.\quad f^{\...
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How to find fixed points for an equation with $\ln$ term

Find fixed points $$x'= M + x - \ln(1+x)$$ First step:1 $$0 = M + x - \ln(1+x)$$ I know when $M > 0$, there are no fixed points.            when $M < 0$, there are two fixed points  but I do not ...
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Why Dottie$=2\sqrt{I^{-1}_\frac12(\frac 12,\frac 32)-I^{-1}_\frac12(\frac 12,\frac 32)^2} = \sin^{-1}\big(1-2I^{-1}_\frac12(\frac 12,\frac 32)\big)$?

Introduction: For some background information on the Dottie Number D, see the great posts at: What is the solution of $\cos(x)=x$? Some definitions: The “solution” to Kepler’s equation is Kepler E: ...
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Proving a vector field without explicit inverse surjective

Consider the vector field given by \begin{align} F: \mathbb{R}^3 \mapsto \mathbb{R}^3, (x,y,z)\mapsto (y+e^z,z+e^x,x+e^y). \end{align} I want to show that $F$ is surjective/onto. After some trying, I ...
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Stability of Banach Contraction Principle w.r.t parameters

I am curious about the following question, originated in little discussion I had with a colleague this afternoon. Let $X,Y$ be complete metric spaces and consider a map $F: X \times Y \to Y$ such that ...
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Is a fixed point possible at a discontinuity in the vector field?

Consider a dynamical system of the following form: $$ \dot{x} = x(1-x)-xy $$ $$ \dot{y} = y\left(1-\frac{y}{x}\right) $$ Would the point $(0,0)$ be a fixed point of the system? Clearly, the vector ...
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Fixed point property of Mobius strip

Does the mobius strip has fixed point property? If it isn't what is the map from mobius strip to itself witch lacks fixed point?
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Relation between a fixed point and being a well-order

I've been trying to prove the following, but with no particular success: Given a linear order $\leq$ on $A$, define $\pi:2^A\to 2^A$ by $X\mapsto\{y\in A: (\forall x < y)(x\in X) \}$. Let $A_0$ be ...
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Composition of quasicontraction mapping

From Ciric fixed-point theorem, we know that if $T:X\rightarrow X$ is a quasicontraction mapping i.e., $d(T(x),T(y)) \le \alpha \max (d(x,y),d(T(x),y),d(x,T(y)),d(x,T(x)),d(y,T(y)),~\alpha \in [0,1)$ ...
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When the graph of $y = f(x)$ is reflected in the line $y = x$, the number of invariant points is

What are the invariants? Would the invariant points be where the points of reflected graph and original intersect?
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Conformal map that fixes 2 points on upper half-lane

Lets denote the upper half-plane by $\mathbb{H}$ and the ball centered at origin as $\mathbb{D}$. My question is what is wrong with my reasoning because I think something should be wrong but I can't ...
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Set theory, fixed points

Let $f : \mathcal{P}(B) \to \mathcal{P}(B)$ be a monotonic function and $I$ - its least fixed point. Prove that: if $I \subseteq A \subseteq B$ and $f_A : \mathcal{P}(A) \to \mathcal{P}(A)$, $f_A(X) =...
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Which of the following iterative formulae could be used to solve the equation $x^3-2x^2-x+1=0$ using the fixed point iteration method?

Which of the following iterative formulae could be used to solve the equation $x^3-2x^2-x+1=0$ using the fixed point iteration method? $$x=x^3-3x^2-1$$ $$x=\sqrt[3]{3x^2+x-1}$$ $$x=\pm\sqrt{\frac{x^3-...
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1 vote
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Tarski's Fixed Point but with an Order Reversing Function

I have read the Tarski's fixed point theorem for powersets. It requires an order-preserving/monotonic function. I wonder if there are results (with strengthened conditions) for order-reversing ...
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Prove that if $x^*$ is an equilibrium point of $y(n+1)=f^2(y(n))$, then it need not be equilibrium of $x(n+1)=f(x(n))$

This question is related to Problem 13 in section 1.5 in Elaydi's "An Introduction to Difference Equations": Prove that if $x^*$ is an equilibrium point of $x(n+1)=f(x(n))$, then it is an ...
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Prove points close to attracting fixed points converge to the attracting fixed point

I am trying to prove that if $x$ is an attracting fixed point of $f(x)$, then if $x_{n+1}$ is "close enough" to $x$. Then $x_n = f(x_{n+1})$ is even "closer" to x and as n goes to ...
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Multiplier of fixed point $\infty$ as limit

For $R: \mathbb{C}_\infty \rightarrow \mathbb{C}_\infty$ a rational map with $R(\infty) = \infty$ I know that the Multiplier of $\infty$ under R is: $$ m(R,\infty) = (\phi \ \circ \ R \ \circ \ \phi^{-...
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Non-trivial solution of $x = r\sin\pi x$

To compute the fixed points of a sine map, I need to solve $$x = r\sin\pi x$$. The question asks me to find the value of r for which the non-trivial fixed point (a second solution of the above ...
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Number of points in line segment

What is the number of points in a line segment? In schools we are taught that a line segment is made up of infinitely many points. Let us suppose that there is a line segment $AB$ of length $a$ units ...
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Convergence of fixed points of a sequence uniformly convergent.

Let $(M,d)$ a metric space, a function $f:M\to M$, is a contraction if there exists $k\in [0,1)$, such that $d(f(x),f(y))\leq k\,d(x,y)$. If $(M,d)$ is complete, one can show that $f$ has a unique ...
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Let $f:[-4,\infty)\rightarrow S$ be a continuous function. Then which of the following is/are true?

Let $f:[-4,\infty)\rightarrow S$ be a continuous function. Then which of the following is/are true? If $S$ is closed, then $f$ has a fixed point. If $S=(-2,\epsilon)$, then $f$ has a fixed point. ...
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Affine map without fixed points is injective

My question is: given an affine endomorph map $f$ in such way that there no exists fixed points (that is, $f(x)\neq x$ for all $x$). Then, can we assert that $f$ is injective? Thanks in advance!
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Show that there exists a unique $z_0\in B(0,1)$ s.t $f(z_0)=z_0$.

Let $\varepsilon >0$ and let $f:B(0,1+\varepsilon )\rightarrow B(0,1)$ be a holomorphic function. Show that there exists a unique $z_0\in B(0,1)$ s.t $f(z_0)=z_0$. For this problem, I don't want to ...
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2 votes
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Missing parentheses in $s(k (s I I))(s(\lambda y. s(k y))(\lambda y. s I I)$ leads to interesting error in an nLab page. Need a double check.

I think I found an error in the nLab page on partial combinatory algebra in the Example combinators section: Finally, consider the classical construction of the fixed-point combinator, $Y = \lambda y....
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$f$ is a fixed-point of $h$. $\Theta h$ is also a fixed-point of $h$. Can we conclude that $f$ equals $\Theta h$?

I’m following along to a derivation of the function that returns the sum of the first $n$ natural numbers in this brilliant.org article on Lambda Calculus. To get to the derivation, either Go to ...
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Idempotency and Fixed-point combinators

In the $\lambda$-calculus, for a fixed-point combinator $P$, we have $Pf = f(Pf)$ for all functions $f$. Thus we could always expand $(Pf)$ as follows: $$Pf = f(Pf) = f(f(Pf)) = f(f(f(Pf))) = f(f(f(…f(...
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In lambda calculus, how many fixed-point combinators are there? [closed]

In lambda calculus, how many fixed-point combinators are there? I am familiar with Curry’s paradoxical combinator a.k.a. the $Y$-combinator and Turing’s fixed-point combinator, $\Theta$, which are ...
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3 votes
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Lambda Calculus functions that are fixed-points of themselves

In general, a fixed-point of a function, $f$, is a point $x$ in the domain of $f$ such that $f(x) = x$. In lambda calculus, a function can take a function, including itself, as input, so we can even ...
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1 vote
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Showing that an involution of the projective line with a fixed point fixes exactly two points

My idea to show this is to notice that a projective transformation is in general given by the form $\phi(z)=\frac{az+b}{cz+d}$, then $\phi(z)=z$ gives a second order equation with at most two ...
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Find the equilibrium point for a single neural network layer

Given a matrix $\mathbf{W}\in\mathbb{R}^{n\times n}$ and a vector $\mathbf{b}\in\mathbb{R}^{n}$, find the equilibirum for the following single-layer neural network: $$ f(\mathbf{x})=\text{ReLU}(\...
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differentiable function only at one fixed interior point with $|f'(x_0)|<1$

If $x_0$ is a interior point of $A\subset\mathbb{R}, f:A\to \mathbb{R}, f(x_0)=x_0, f$ differentiable in $x_0$ and $|f'(x_0)|<1$. Does exists a neighborhood $V$ of $x_0$ st $\lim_{n\to \infty} f^n(...
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There exists $L$ so that $\|x-\tilde{x}\|\leq \frac{1}{1-L}\cdot \sup_{z\in D} \|\phi(z) -\tilde{\phi}(z)\|\,$?

Let $x=\phi(x)$ and $\tilde{x}=\tilde{\phi}(\tilde{x})$ be two fixed point equations, which meet the requirements of the Banach fixed point theorem in $D\subseteq \mathbb R^n$. Show, that there exists ...
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Is there any noncompact manifold with the fixed point property?

This question refers to manifolds without boundary. I was thinking about Brouwer´s fixed point theorem when this came to mind, and I found the name of the property ("fixed point property"), ...
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5 votes
1 answer
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Attracting fixed point of $f$ if and only if repelling fixed point of $f^{-1}$

While working on some dynamical system problems, I came across an interesting exercise that has left me stumped for quite a few weeks now. I have a solution, but I don't think it holds enough water ...
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3 votes
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Question about integral operator being a contraction

I've been told that if $$T(f)(x)=g(x)+c\int_0^xK(x,y)f(y)dy$$ for some $c\in\mathbb{R}$ and $f,g\in C[0,1]$ ($g$ fixed) and $K\in C([0,1]\times[0,1])$, then there exists $n\in\mathbb{N}$ such that $T^...
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