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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Is every convergent limit of an iteration a fixed point as well?

Let $f(x)$ be a function and suppose $\lim_{n \to \infty}f^n(a)=L$ for some $a$ in the domain of $f$. What are the sufficient conditions for $L$ being a fixed point of $f$? Is the continuity of $f$ ...
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112 views

Proof for the existence of a second fixed point in Poincaré's last geometric theorem

In "Geometry and Billiards" by S. Tabachnikov the author proves Poincaré's last geometric theorem: "An area-preserving transformation of an annulus that moves the boundary circles in opposite ...
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179 views

Cone metric spaces and fixed point theory

These days cone metric spaces, as a generalization of metric spaces, started to be very interesting... A special interest for this space is its application to the fixed point theory. On the last ...
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90 views

Prove that $\exists\ C$ such that $T: C \to C$ with $C=\overline{\text{conv}}T(C)$

I have a problem: Let $K$ be a nonempty, closed, bounded and convex subset of reflective Banach space $X$. Suppose that $$T:K \to K$$ is nonexpansive. Prove that $\exists\ C$ such that ...
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Is there a method for finding the fixed point of logarithmic functions?

I am faced with this function (warning, I am not good at math) $x(t+1)=0.5 \ln x(t)+1$ initial condition = 1 . I know the fixed point is 1 because $0.5 \ln (1)+1=1$ but I wanted to know the ...
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75 views

Fixed points of multivariable calculus

I have discrete case. $z=1-x-y$; $x=a_1(x^{2}+2yz)+a_2(z^{2}+2xy)+a_3(y^{2}+2xz)$; $y=b_1(x^{2}+2yz)+b_2(z^{2}+2xy)+b_3(y^{2}+2xz)$; $z=c_1(x^{2}+2yz)+c_2(z^{2}+2xy)+c_3(y^{2}+2xz)$; where ...
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189 views

How to find fixed point

I have recurrence equations of \begin{align*} x_{n+1} &:= 0.5\cdot (x_{n}^2+2\cdot y_{n}-2\cdot x_{n}\cdot y_{n}-2\cdot y_{n}^2) \\ y_{n+1} &:= -0.8\cdot (x_{n}^2+2\cdot y_{n}-2\cdot ...
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160 views

Prove the Contraction Mapping Theorem.

Prove the Contraction Mapping Theorem. Let $(X,d)$ be a complete metric space and $g : X \rightarrow X$ be a map such that $\forall x,y \in X, d(g(x), g(y)) \le \lambda d(x,y)$ for some ...
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27 views

Problems finding fixed point.

$x_n=f(x_{n-1})={ax_{n-1} +e^{-x_{n-1}}\over 1+a}$ for $a>0$ Setting $f(x)=x$. The problem is while trying to find the fixed point of $x={ax +e^{-x}\over 1+a}$ I only get $x=e^{-x}$. What's the ...
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50 views

Applying Schauder fixed point theorem to a map (explanation needed)

Let $F:L^2(\Omega) \to L^2(\Omega)$ be continuous map. Let $D$ be a function space. Since $F(L^2(\Omega)) \subset D$, and $D \subset L^2(\Omega)$ is a compact embedding, $F$ is a compact operator ...
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43 views

Transforming Nested Fixed-Point Formulas into Infinitary Logic Formulas with Finitely many Variables

There is a definition (actually a description of how it could be defined) of a fixed-point logic formula. The formula is in inflationary fixed point logic (IFP) in this case but it could also be ...
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50 views

Is there a name for this theorem about the convergence of a function?

Let $f(x)$ be a continuous function over $\mathbb{R}$ such that for all $a < b$, we have $a < f(a) < b$. Then, for any $x < b$, the sequence $\{t_n\}$ defined by $t_0 = x, t_n = ...
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33 views

How to find fixed point for Hamilton equations

I am learning the dynamical system. I start with Taylor-Greene-Chirikov map as follow \begin{eqnarray*} && I_{n+1} = I_n + K\sin(\theta_n), \\ && \theta_{n+1} = \theta_n + I_{n+1} ...
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99 views

Application of Brouwer fixed point theorem, why is compactness not required here??

Define a map $K:X_n \to X_n$ where $X=\text{span}(v_1, ..., v_n)$ where $v_i$ are basis functions some Hilbert space $H$. So $X_n$ is finite-dimensional. $B_r(0)$ denotes the ball of radius $r$ ...
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39 views

Trouble with fixed point iteration.

Find the fixed point(s) of $g(x) = (1/2)x^2 + (1/2)x$. Does the fixed point iteration(s) converge(s) to the the fixed point(s) if you start with a close enough approximation? Then choose $x_0 ...
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20 views

The number of periodic orbits

Let $\{a_n\}_{n=2}^\infty$ be a sequence of nonnegative integers. How to construct a continuous map $f$ on the 2-dimensional closed disk such that the number of $n$- periodic orbits is $a_n$? So far, ...
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268 views

Fixed Point Iteration, does it converge?

Find the fixed point(s) of $g(x) = x^2 + 3x - 3$. Does the fixed point iteration(s) converge(s) to the fixed points if you start with a close enough first approximation? I set $g(x) = x$ and got ...
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27 views

How to linearize and solve ODE $\dot{z}_n = \sum_{m} i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}$ for $z_n\approx 0$?

I came across a physical system which obeys the following ODE $$\frac{d z_n}{dt} = \sum_{m=1}^N i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}, \qquad n\in\{1,2,\dots,N\}$$ where $z_n \equiv z_n(t)$ are ...
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66 views

why do we take this interval?

I am looking at an exercise,where given that $a_{n}=\sqrt{2+\sqrt{2+...+\sqrt{2}}}$,I have to show that $\lim_{n \to \infty}a_{n}=2$. We find that $a_{0}=0,a_{1}=\sqrt{2},a_{2}=\sqrt{2+a_{1}} \text{ ...
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105 views

Applying the Banach's Contraction Principle

I have a problem: For a system of linear equations: $$x_i=\sum_{j=1}^{n}a_{ij}x_j+b_i,\ \ i=1,2, \ldots , n \tag 1$$ Prove that, if $$\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^2 \le q<1$$ then ...
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83 views

Successive approximation (Banach's fixed point theorem)

Let (X,d) be a complete metric space and $T\colon X\to X$. Moreover it exists a $n\in\mathbb{N}$, so that the n-th power of $T$ is $q$-contractive. Yesterday my task was to Show, that $T$ ...
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85 views

If $T^n$ is $q$-contractive, $T$ exactly has one fixed point

Consider a complete metric space $(X,d)$ and $T\colon X\to X$. Suppose there exists $n\in\mathbb{N}$ such that the n-th power of $T$ is $q$-contractive. Show that then $T$ has exactly one ...
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142 views

Proving $\displaystyle\lim_{n\to\infty}a_n=\alpha$ $a_1=\frac\pi4, a_n = \cos(a_{n-1})$

Let $a_1=\frac\pi4, a_n = \cos(a_{n-1})$ Prove $\displaystyle\lim_{n\to\infty}a_n=\alpha$. Where $\alpha$ is the solution for $\cos x=x$. Hint: check that $(a_n)$ is a cauchy ...
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171 views

Fix-Point Theorem Proof.

Firstly, the assignment: Let $a,b \in\mathbb{R}$ and $a < b$. Furthermore let $f: [a,b] \rightarrow [a,b]$ be monotone increasing. Show that if $x:= \mathbf {sup}\{y \in [a,b] \| \ y ≤ f(y)\}$ ...
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208 views

Proof that the solution to cosx = x, is the limit of a recursive sequence.

So I've got this question. Exists a sequence $a_n$ such that: $$a_0 = \frac \pi4, a_n=\cos\left(a_{n-1}\right)$$ Prove that $\lim_{n\rightarrow\infty} a_n = \alpha$ Where $\alpha$ is the solution to ...
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828 views

Fixed-Point Theorem Proof

Merry Christmas everybody. Let $a,b\in\mathbb{R}$ and $a<b$. Prove that: If $f: [a,b] \rightarrow [a,b]$ is continuous, then there is a fixed-point in $f$. So basically, if f is continous I ...
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Fixed point of a shrinking map. Proof.

Could you please verify my proof? Let $f:X \to X$ be a shrinking map on a compact metric space $(X,d)$. In other words: $d(f(x),f(y)) < d(x,y)$ for all $x,y \in X$. Prove: $f$ has a unique ...
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Using Banachs fixpoint-theorem to show the uniqueness of a solution of a non-linear integral equation

Show, that under the conditions (1) $-\infty<a<b<\infty, f\in C([a,b]\times\mathbb{R}), F\in C([a,b]\times [a,b]\times\mathbb{R})$ (2) $\exists~1>c\geq 0~\forall x\in [a,b]~\forall ...
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34 views

Question about $\Sigma_{i=1}^n (a_i^z-a_i^{-z})$

Let $z$ be a complex number and $n$ a positive integer. Let $a_n$ be a sequence of $n$ real numbers such that $a_n > 1$ for every $n$. Define $f_n(z;a_1,a_2,...,a_n)=\Sigma_{i=1}^n ...
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60 views

Fixed points of complex rational functions dilemma.

There's a theorem about fixed points of complex rational functions which states that those with degree $d$ have $d+1$ fixed points. the case where denominator has greater degree makes sense and it's ...
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Brouwer Fixed Point Theorem via the Jordan Curve Theorem

There is a proof of the Brouwer Fixed Point Theorem via the Jordan Curve Theorem ? The Brouwer Fixed Point Theorem. Let $B=\{x\in \mathbb R^2 :∥x∥≤1\}$ be the closed unit ball in $\mathbb R^2$ . ...
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How many distinct fixpoints does $\sum_{i=1}^n 2 \sinh(a_i z)$ have?

Let $n$ be a positive integer. Let $z$ be a complex number. Let $a_1,a_2,...,a_n$ be a given sequence of distinct reals larger than $\ln(2)$. How many distinct solutions are there to $\displaystyle ...
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169 views

Check Points are line, triangle, circle or rectangle

How to determine geometric properties of four distinct points in a plane (x1,y1), (x2,y2), (x3,y3), (x4,y4) represented in the 2-D Cartesian coordinate system, whether these four points are on a ...
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Regarding fixed point

Please help! I am stuck at this question: Let X be compact and $f : X \rightarrow X$ be continuous. If $f$ has no fixed point then there exist $\delta > 0$ such that $d(a,f(a)) \geq \delta$ for ...
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48 views

Does Particular Point Topology has Fixed Point Property?

A Particular Point Topology is not compact, is path-connected but what about Fixed Point Property? Does it have fixed point property? If so how? I have been told that it should have Fixed Point ...
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Brouwer Fixed Point Theorem $f(S^1)\subset B$

I have a question about the Brouwer Fixed Point Theorem: Theorem 1.(Brouwer Fixed Point Theorem) Let $B=\{x\in \mathbb R^2 :∥x∥≤1\}$ be the closed unit ball in $\mathbb R^2$ . Any continuous ...
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Prove that a real variable function that satisfies certain conditions has a fixed point.

Let $f: \mathbb R \to \mathbb R$ be a differentiable function such that $f(0)=1$ and $|f'(x)|<\dfrac{1}{2}$. $i)$ Prove that there exists $x_0 \in [0,2] :f(x_0)=x_0$. $ii)$ Let ...
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Application of fixed point theorem in $R^n$

Let $A=(a_{ij}) \in \mathbb R^{n \times n}$ a matrix such that $|a_{ij}|<\frac{1}{n}$ for every $i,j$. Prove that $I-A$ is invertible. My attempt at a solution: $I-A$ is invertible $\iff$ ...
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203 views

Function iteration and intervals of attraction for fixed points

I am currently studying iteration sequences and I am a bit hung up on one specific bit which involves determining intervals of attraction of fixed points. I've been given a graphical method to ...
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111 views

Proving a particular function is surjective on a Banach space.

Let $(E,\|x\|)$ and let $f: E \to E$ such that $f+Id$ is a contraction ($Id$ is the identity map). Prove that $f$ is surjective and prove that, if $f$ is linear, then $f$ is a homeomorphism. The ...
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509 views

Must a holomorphic function from $D(0,1)$ to $D(0,1)$ have a fixed point?

Must every holomorphic function $f:D(0,1)\longrightarrow D(0,1)$ have a fixed point? I know that any holomorphic function with two fixed points is the identity: $f=Id$, but I can't find out an ...
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Two indexes $c$ and $b$, such that $Dom(\varphi_a)=\{b\}$ and $Dom(\varphi_b)=\{c\}$

Problem: Assume $\{\varphi_i\}_{i=1}^\infty$ is an effective enumeration of the computable functions. Find two indexes $c$ and $b$, such that $Dom(\varphi_c)=\{b\}$ and $Dom(\varphi_b)=\{c\}$. ...
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126 views

Fixed point in plane transformation.

Some one give me a idea to solve this one. It's a problem from Vladimir Zorich mathematical analysis I. Problem 9.c from 1.3.5: A point $p \in X$ is a fixed point of a mapping $f:X \to X$ if ...
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75 views

Fixed point iteration for $\sqrt[3]{a}$

So I'm given the scheme for computing $\sqrt[3]{a}$ $$x_{k+1}=px_k + \frac{qa}{x_k^2} + \frac{ra^2}{x_k^5}$$ and I have to find the p, q, and r so that this scheme is as fast as possible. Any hints ...
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71 views

Number of Fixed points of an odd degree polynomial

Let $p(x)$ be a polynomial of degree $2n+1$ with real coefficients. then $p(x)$ has (I) exactly $2n+1$ fixed points (II) at least one fixed point (III) at most one fixed point (Iv) $n$ fixed ...
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1answer
96 views

Getting a root of a continuously differentiable function by Banach's Fixed Point Theorem.

Banach Fixed Point Theorem: Consider a metric space $X = (X, d)$, where $X\neq \varnothing$. Suppose that $X$ is complete and let $T: X \to X$ be a contraction on $X$. Then $T$ has precisely one ...
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78 views

Reference request: generalized Lefschetz-Hopf fixed point theorem

Let $X$ be a smooth projective variety over $\mathbb{F}_p$. A basic (and very important) theorem is that we have equality $$ \# X(\mathbb{F}_{p^n}) = \sum_{i=0}^{2\dim X} (1)^i ...
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90 views

On the Proof of the Perron-Frobenius Theorem.

The Perron-Frobenius theorem states that a square matrix with nonnegative entries has a real nonnegative eigenvalue. One possible proof uses the Brouwer fixed point theorem, and every proof I've seen ...
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111 views

Showing a sequence $x_{n+1} = Tx_n$ forms a contraction mapping

I want to show that the sequence given by $$x_{n+1} = Tx_n = x_n-\frac{(x_n^2-2)}{x_n+x_{n-1}}$$ forms a contraction mapping. That is $$|Tx_1-Tx_2|\leq c|x_1-x_2|.$$ Where $c$ is to be determined. I ...
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2answers
82 views

Continuous mapping on unit ball such that $T(\Bbb x_0)=0$

got a question from a course in functional analysis. " Let $T:\{\Bbb x\in\Bbb R: ||\Bbb x||\leq 1\}\to\Bbb R^n$ be a continuous mapping. Moreover assume that $\langle T(\Bbb x),\Bbb x\rangle>0$ ...