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, Kleene)...

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Proving a function to be a contraction

I'm studying contractions and I am trying to understand under which conditions a function is actually a contraction. I have understood that a continuous function $f: [a,b]\rightarrow\mathbb{R}$ is a ...
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Whether a continuous function has fixed point or not when the domain and range are not $[0,1]$

Which of the following is false $?$ $A.$ Any continuous function from $[0,1]$ to $[0,1]$ has a fixed point. $B.$ Any homeomorphism from $[0,1)$ to $[0,1)$ has a fixed point. $C.$ Any bounded ...
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The conjugate of a complex function

This may be a stupid question. I am trying to explain/prove the following: Let $f$ be a complex rational function of degree $d ≥ 2$ with fix point $z_0 \neq 0$. It is always possible to conjugate $f$...
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unique fixed point problem

Let $f: \mathbb{R}_{\ge0} \to \mathbb{R} $ where $f$ is continuous and derivable in $\mathbb{R}_{\ge0}$ such that $f(0)=1$ and $|f'(x)| \le \frac{1}{2}$. Prove that there exist only one $ x_{0}$ such ...
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Does any continuous function from the open unit interval $(0,1)$ to itself has a fixed point?

I know that the result is true for closed interval $[0,1]$ by using intermediate value property. But in the case where we consider open interval $(0,1)$ does the solution change?
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Stating if a function being a contraction

I have to state if a function $f:\mathbb{R}\rightarrow\mathbb{R}$ is a contraction on the interval $I\subset \mathbb{R}$ and say if it admits fixed points. The function is $f(x)=\left\{ ...
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$r\rightarrow1/r$ invariant

(Not sure the tags are appropriate, but can't think of better ones. Please suggest better.) Suppose you have a function $f(x,y,z,...;g(r))$ with the requirement that $r\rightarrow1/r$ leaves $f$ ...
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Problem in Topology related to continuous functions

Let $\mathbb{R-Q}$ be the subspace of $\mathbb{R}$ with the usual metric. Is there a function, $f:\mathbb{R-Q} \rightarrow \mathbb{R-Q}$ such that f is continuous and $f$ does not have a fixed ...
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Lefschetz number of a transformation of the sphere.

In differential topology the Lefschetz number of an automorphism of a compact manifold is the oriented intersection number of the graph of that automorphism with the diagonal. I would like a proof or ...
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Edelstein Fixed Point Theorem

Let $(M,d)$ be a metric space, $M$ compact. If $f:M \to M$ is continuous and weakly contractive (i.e. $d(f(x), f(y)) < d(x,y) , \forall x,y \in M$), then $\exists x_0 \in M $ unique s.t $f(x_0)=...
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Proof there is no fixpoint

I want to proof that a certain function (in this case tan(x) on (0, 1/4)) doesn't have a fixpoint. Is there a general approach to try that? For example an "iff" fixpoint theorem?
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Existance of a fixed point of a bijective, smooth function:

Let $[a,b],\,\, [c,d]$ be two bounded closed intervals of $\mathbb{R}$ such that $$[a,b]\cap[c,d]\not=\emptyset$$ and $f:[a,b]\to[c,d]$ be a bijective, smooth function. We know that, if $[a,b]=[c,d]...
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60 views

Fixed point iteration methods lead to division by zero

So I need to prove that for $f(x) = ((x - x^*)^2)h(x)$ assuming $f''(x) \neq 0$ and assuming $f(x)$ be twice continuously differentiable, that Newton's method converges linearly in this case. So $g(x) ...
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Weak contraction mapping preserves order?

Suppose I have a function $\varphi$ on the unit interval such that $|\varphi (x)-\varphi (y)|<|x-y|$. Is it true that if $0 \leqslant c \leqslant 1$ then $\varphi (0) \leqslant \varphi (c) \...
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4answers
223 views

Fixed point iteration convergence of $\sin(x)$ in Java [duplicate]

Is it mathematically correct to say that $\sin(x)$ converges to zero as $x$ approaches $0$? If the $\sin(x)$ iteration is done starting at $\dfrac{\pi}{2}$ in Java, for $10^9$ iterations, the result ...
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What is meant when saying a function has a solution?

I have to code the solutions to these questions in Python. But before even thinking about coding, I have to understand what I have to do here. Table 1: Observed frequency distribution of Y $$\begin{...
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Solution of an equation involving Banach fixed point.

I find this queistion in my textbook: QUESTION) Let $T \in \mathcal{L}(\mathcal{B})$, where $\mathcal{B}$ is a Banach Space. If exists $n$ with $||T^{n}|| < 1$, show that, given $\eta \in \mathcal{...
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Newton's Method and Banach Fixed Point Theorem

This might be a dumb question, but I want to see how Newton's method can be understood in the context of Banach fixed point theorem (BFPT): A closed subset of a complete metric space is complete. If $...
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showing $Ax=b$ has a unique solution by finding the fixed points of a function

Let $n ∈ \mathbb N$. Consider an $(n×n)$-matrix A with real components and a column vector $b ∈ \mathbb R^n$. They give rise to an affine transformation $T : \mathbb R^n → \mathbb R^n$ with $T(x) = Ax+...
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Criteria for being below the least fixed point

Given a complete partial order $(D,\, {\leq})$ and a Scott-continuous function $F\colon D \rightarrow D$ and some fixed point $X = F(X)$. Are there any criteria that ensure that $X \leq \mathsf{lfp} ...
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Banach Fixed Point Theorem and the function $f(x)=x+e^{-x}$

let $X = [0,\infty)$ equipped with the standard metric $d(x,y) = |x-y|$. Let $f: X \rightarrow X$ be defined by $f(x)=x+e^{-x}$ Explain why this function doesn't contradict Banach's Fixed Point ...
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Fixed-Point Applications

Let $1<a<e^{1/e}$, and define $f(x)=a^x$ (b) Show that if $x_1$ is any point in the interval $(1,e)$ and $p$ is a fixed point of $f(x)$ in the interval $(1,e)$, then $$|f(x_1) - p | <|x_1 - ...
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“Comparing” fixed point Theorems.

In class, I saw Banach's (Picard) fixed point theorem: Given a complete metric space and a contractive mapping, it admits a unique fixed point. And Brouwer's: Given a continuous function in ...
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Free $\mathbb{Z}_{2}$ action on the plane

Motivated by the following question we ask: Is there a free action of $\mathbb{Z}_{2}$ by homeomorphism on $\mathbb{R}^{2}$? Lie groups with no free $\mathbb{Z}/2\mathbb{Z}$ action
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Application of Banach Fixed Point Theorem to sequences

I have the following question Show that there is unique real bounded sequence $(a_n: n \in \mathbb{N})$ such that $$a_n = \frac{n+1}{n}+\sum^\infty_{m=1}\frac{\sqrt{3a^2_{m+n}+1}}{4m^2}$$ for all $n \...
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What is the relationship between eigenvector and computing PageRank?

I read several papers about PageRank and didn't get stuck in understanding the idea of PageRank because it is simple, but I got stuck in computing PageRank, those papers talk about some mathematical ...
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Every finite group of congruences of the $n$-dimensional Euclidean space has a fixed point

Is there an "absolute" proof of the fact that if $G$ is a finite group of congruences of the $n$-dimensional Euclidean space, then $G$ has a fixed point?
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Second order DE with Banach fixed point theorem

Let's consider such a problem: $$\left\{\begin{matrix} &u''-u=f(u)+g(x) & \\ &u'(0)=u'(1)=0 & \end{matrix}\right.$$ for $x\in(0,1)$, where $u=u(x)$ isn't known and $f \in C(\...
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Fixed Point Iteration Method - Starting Point

If $g(x):=21^{1/2}x^{-1/2}$, then $21^{1/3}$ is a fixed point of $g$. Question: Using the Fixed Point Iteration Method, are there conditions on the starting point $x_0$ in order for the method to ...
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Prove that there is a unique $\phi \in C([a,b])$ such that $\phi (x) = A(x) + \int_a^xK(x,y)\phi(y)dy$

Contraction Map Question I was reviewing another question on here (linked above), and I wanted to expand on it with another question (I hope this is the correct way to do such a thing). Here is my ...
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Set of all contraction maps

Consider $(X, d)$ be a compact metric space, and let $Con(X)$ denote the set of all contraction maps on $X$. We shall define the “distance” between two maps $f, g ∈ Con(X)$ as follows, $$d_{Con(X)}(f, ...
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Contractive composite function

A question arises when we are dealing with a function that is not known to be contraction but the composites of that function to itself is a contraction (it is called eventually contractive). More ...
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Is fixed point property a topological property?

Is fixed point property a topological property? I already came up with some examples, and think the answer may be yes. But I don't know theoretical proof to get that.
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Fixed point in a special continuous mapping

Show that a continuous mapping $f:[0,1]\to [0,1]$ which satisfies $f(f(x))=x$ for each $x \in [0,1]$, and for which $f(x)$ does not equal x for at least one x in [0,1], must have exactly one fixed ...
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Contraction Map (Fredholm integral type)

I have been stuck on this problem for a little while. I think the proof might be similar to that of proving the Fredhom integral of the second type, however I am not sure. I am a little confused on ...
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How can I find the roots of x^2+px+q=0 by using contractive mapping theorem?

Basically, Contractive mapping theorem guarantees that if F is contractive on some interval, then there exists a unique fixed point. However, $x^2+px+q=0$ has two different solutions as long as $p^2&...
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Continuous mapping and fixed points

Does a continuous mapping $f\colon \mathbb R \to \mathbb R$ which satisfies $f(f(x))=x$ for each $x \in \mathbb R$ necessarily have a fixed point?
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Diffeomorphism on $S^2$ to $S^2$ with at least two fixed points

Let $f:S^2\to S^2$ diffeomorphism orientation-preserving such that for every fixed point of $f$, $1$ is not an eigenvalue of $df_p$. Then $f$ admits at least two fixed points. I'd to construct a ...
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Convergence of autonomous system / time-scale separation

Let $x \in \Delta_n$, where $\Delta_n = \{u \in \mathbf{R}^n_{\ge 0} \mid \sum u_i = 1\}$ is the probability simplex. Suppose that I have an (autonomous) dynamical system $$ \dot{x} = A(x) \cdot x $$ ...
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Convergence of fixed point iteration when $g'(p)=1$.

I am dealing with a function $f(x)=e^{-\frac{1}{x^2}}$, which has a root $p$ of infinite multiplicity at 0. I am struggling with the convergence rate of the resulting standard Newton fixed point ...
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Inverse function theorem - good proof

I am looking for a reference which give a full demonstration of the inverse function theorem (let say in Banach spaces) where we can have estimates of the bounds of the neighbourhoods that we build to ...
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Is this map defined from an ODE a contraction?

Let $f:\mathbb{R}^+\to \mathbb{R}$ be a $T$ periodic function, i.e. $f(t+T) = f(t)$ for all $t\in \mathbb{R}^+$. Let $\sigma\in \mathbb{R}^+$. Define a weighted $\mathcal{l}_2$ norm, $\left\| \cdot \...
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Fixed point problem with matrix

I am looking for guidance for the following fixed point problem. $A$ is an $n\times n$ matrix. The rows of $A=(A_{i})_{i\in N}$ are in the set $A_{i}\in\mathcal{A}$, where $\mathcal{A}$ is the set ...
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Fixed Point equivalence proof

Given two finite-state LTSs $L$ and $M$ (where $M$ is a "specific" subgraph of $L$) and the Hennessy-Milner-Logic monotone interpretation function $\Theta_F(S)$, where $F$ is the HML formula and $S$ ...
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If $X$ is a compact space and $A:X\rightarrow X$ is such that $\rho(Ax,Ay)<\rho(x,y)$, $x\neq y, \Rightarrow A $ have only one fix point on X.

I have problems with this demostration can anybody help me please? If $X$ is a compact space and $A:X\rightarrow X$ is such that $\rho(Ax,Ay)<\rho(x,y)$, $x\neq y, \Rightarrow A $ have only one ...
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Compare two fixed points component-wisely

Tarski fixed point theorem conclude that a isotone from a complete lattice to itself have a largest fixed point. Now suppose I have two isotone on defined on the same lattice, and one isotone is ...
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Polynomials in two variables over $\mathbb{F}_p$ fixed by $\operatorname{SL}_2(\mathbb{F}_p)$

Let $A=(a_{ij})\in\operatorname{SL}_2(\mathbb{F}_p)$. Consider the ring map $A:\mathbb{F}_p[x,y]\to\mathbb{F}_p[x,y]$ defined by $$A(x)=a_{11}x+a_{21}y$$ $$A(y)=a_{12}x+a_{22}y$$ and extended ...
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Prove that the equation $\cos x - x -1/2 = 0$ has a unique real solution

Prove that the equation $\cos x - x -1/2 = 0$ has a unique real solution. My solution: I was starting with a function $F(x) = \cos x - 1/2$ and the interval $[0,\pi/4]$ and trying to show that the ...
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3 fixed points proof

The problem: Suppose $g(x)$ is continously differentiable and has exactly 3 distinct fixed points $r_1 < r_2 < r_3$ with $\lvert g \prime (r_1) \rvert = .5$ and $\lvert g \prime (r_3) \rvert = ....
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Proof that the function on $C[0,b]$ is a contraction

Question : Let $a$,$b$ be real numbers with $0\lt b\lt 1$ . Consider the subset $X\subset C[0,b]$ consisting of the functions $f$ s.t $f(0)=a$ .Then $X$ is closed in $C[0,b]$. ...