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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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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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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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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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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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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156 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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A contraction approach on Collatz conjecture [closed]

Talking about Collatz like iterative sequences, If we define Collatz function as $f(x)=k(2.5x+1)+x/2$ where $k=x$%2 then according to Banach fixed point theorem(from wiki), "Every contraction ...
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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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41 views

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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31 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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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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129 views

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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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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117 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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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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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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79 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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54 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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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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Application of fixed point theorem for an integral equation.

Consider the following non-linear integral equation: $f(x)=\lambda \int_a^b K(x,y;f(y))dy + \phi(x)$ with $K$ and $\phi$ continuous functions , such that $K$ satisfies the Lipschitz condition on the ...
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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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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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This system is contractive?

I have a system which has a form of find point problem, described as following $$p_i=h_i(\mathbf{p})$$ where $$p_i\in[0,1]$$ is the $i$-th components of the $n$-dimensional column vector ...
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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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Are the solutions to $\zeta \left(\frac{1}{2}+i n a\right)=a$ the same as fixed points? $n=1,2,3,…$

Are the solutions to $\zeta \left(\frac{1}{2}+i n a\right)=a$ called fixed points? $n=1,2,3,...$ $\zeta$ is Riemann zeta function. Mathematica program for finding $\zeta \left(\frac{1}{2}+i 3 ...
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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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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 $\langle T(\Bbb x),\Bbb x\rangle>0$ ...
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Is $f : [0,\infty]\rightarrow [0,\infty]$ which is continuous and bounded has a fixed point

Question is to check if : $f : [0,\infty]\rightarrow [0,\infty]$ which is continuous and bounded has a fixed point. I have first of all considered boundedness. So, $f(x)$ should not have $x$ as ...
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Is it possible to show that the Gaussian is a fixed point of the Fourier transform using a fixed point theorem?

We know that $\text{exp}(-\alpha |x|^2)$ is a fixed point for the unitary Fourier transform if $\text{Re } \alpha > 0$. Is it possible to show this using a fixed point theorem?
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Homeomorphic to the disk implies existence of fixed point common to all isometries?

A fellow grad student was working on this seemingly simple problem which appears to have us both stuck. (The problem naturally came up in his work so isn't from the literature as far as we know). Let ...
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Does my fixed point proof hold?

I have am looking for existence of a fixed point for an operator that I have. I already looked at some related fixed point theorems such as Schrauder's and Rothe's. But most of them seem to require ...
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Proving $\frac{k-1}{k}$ is an attractor of the logistic map $kx(1-x)$.

Consider the logistic map $f(x) = kx(1-x)$ defined on $\mathbb{R}$. We already know $\frac{k-1}{k}$ is a fixed point of $f$, but my issue is showing it's an attractor when $k \in [1,3]$. There is an ...
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Questions about $\ln(z)$ recurrence and fixed points.

Define property $A_R$ for an analytic function $f(z)$ as $1)$ $f(z)=0$ has exactly one solution being $z=0$ for $|z|<R$ where $R$ is a radius. And $f(z)$ is analytic within the radius $R$ ...
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Question about fixpoints and zero's on the complex plane.

Define property $A$ for an entire function $f(z)$ as $1)$ $f(z)=0$ has exactly one solution being $z=0$ $2)$ $f(z)=z$ has exactly one solution $=>z=0$ (follows from $1)$ ) $3)$ $f(z)$ is not a ...
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In a complete lattice every monotone function has a fixpoint (Knaster–Tarski Theorem)

L is a complete lattice, so every subset has a supremum and infimum. In addition, there exists a function $f:L \rightarrow L$ such that $a \leq b$ implies $f(a) \leq f(b)$. Prove that there exists ...
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Importance of Banach fixed point in mathematics

I know the proof of Banach fixed point theorem in complete metric space and two example of it. It is useful to prove Picard's theorem on existence and uniqueness of ordinary differential equation. In ...
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Question about unique fixed point.

Suppose $f(x,y)\in C^{2}(R^n)$, $g=(g'_1,g'_2)'$, $g_1(y)=\arg\min_xf(x,y)$ and $g_2(x)=\arg\min_yf(x,y)$. $g$ has a fixed point at $(x_0,y_0)$. $h=(h'_1,h'_2)'$, $h_1(y)=x_0+\dot{g}_1(y_0)(y-y_0)$ ...
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72 views

Fixed point of matrix

Suppose that $a$ is a fixed point of matrix $A$, what that means? What is a fixed point of matrix? Thank you!
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37 views

How to decide if these two maps are proper?

We define a mapping $f$ of a topological space $X$ into a topological space $Y$ to be proper if the subspace $f^{-1}(C)$ is compact in $X$ whenever $C$ is a compact subspace of $Y$. Now how to ...
5
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1answer
114 views

Is Poincare-Hopf index theorem connected with Leftschetz fixed point theorem?

Lefschetz Fixed Point Theorem: For a compact triangulable space $X$, and a continuous map $f:X\rightarrow X$, we have ...
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1answer
98 views

Calculating the fixed points of a model

Does anyone know how I would go about calculating the fixed points of this model?
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60 views

Simulating Mixed Nash Equilibria

I have a $N$ person game where each person has a set of $M$ discrete strategies. I know from the theory that at least one mixed strategy Nash Equilibrium exists. Can someone please tell me how do I ...