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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Complete a proof that $F(x,y)$ is contracting.

Can anyone fill in the dots in this proof? Let $D := [0,\frac{1}{2}]^2$. Show there is exactly one $(x,y)=(x^*,y^*)\in D$ such that \begin{align*} x &= \frac{x^3}{2} + y^4 + \frac{1}{4} \,, \...
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Prove that a function is contractive

I'm stuck with the following. I need to prove that in $D:=[0,1]\times[0,1]$ the function $F$ is contractive, where $F:\mathbb{R}^2\rightarrow\mathbb{R}^2$ is defined as: \begin{align} F(x,y):=(\frac{...
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PDEs - prove continuity of operator

Consider the following nonlinear problem $$ \begin{cases} -div(a(u)\nabla u ) =0 & \text{in $\Omega$} \\ u=0, & \text{on $\partial \Omega$ } \end{cases} $$ We can assume $\Omega$ to be a ...
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34 views

Show there exists $C\in\Bbb{R}^n$ such that $|C-A_i|=|B-A_i|+u_i$, with $A_i,B\in \Bbb{R}^n$ and $u_i$ close enough to $0$

Let $A_1,...,A_n,B$ be vectors in the $n$-dimensional Euclidean Space, such that they are never on the same affine $(n-1)$-dimensional subspace. (What? Is that a way to say they span $\Bbb{R}^n$?). ...
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54 views

Proving a function isn't homotopic to a map to the boundary

Let $X$ be a compact Hausdorff space and $S$ a finite dimensional, convex, Hausdorff space. Moreover, let $f:X \to S$ be a closed, surjective map with $\tilde{H}^q(f^{-1}(s))=0$ for all $q\ge 0$ and $...
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47 views

Can there be a limit cycle without a fixed point in 3D space?

I am working with a population dynamics model. Basically, I have a nonlinear ODE in $R^3$ space, (X,Y,Z), and I know that if I start in the an open region ($0<X<1,0<Y<1,0<Z<1$, ...
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98 views

Example of contraction mapping theorem failing for strict metric map

Is there an example of $f: [0,1] \to [0,1]$ s.t. $|f(x)-f(y)|<|x-y|$ but a sequence $x_0,f(x_0),f^2(x_0)...$ doesn't converge to its fixed point? where $f^n$ denotes repeated application. Also, ...
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71 views

Open ball does not have fixed point

How we can prove that the open ball in $R^n$ does not have fixed point property (by algebraic topology concepts)? I know $D^n$ -closed ball in $R^n$- has fixed point property by Brouwer's theorem, but ...
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22 views

Fixed Point in the Space of Rational Functions

Let $\mathcal R$ be the space of rational functions and $F: \mathcal R \to \mathcal R $ be a function that transforms a rational function into another rational function. Is there a fixed point ...
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45 views

Fixed Point and Contraction Mapping

Consider $Tf(x) = \int_0^x e^{-f(s)^2} \; ds$ for $x \in [0,\infty)$. I want to use the contraction mapping theorem to show that $T:C^1([0,\infty)) \to C^1([0,\infty))$ has a unique fixed point. From ...
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45 views

Banach fixed-point theorem for a recursive functional equation

I was asked to prove that the functional equation $$ x(t) = \begin{cases} \frac{1}{2} x(3t) + \frac{1}{2},& 0 \leq t \leq \frac{1}{3}\\\ f(t), & \frac{1}{3} < t\leq \frac{2}{3}\\\ \frac{1}{...
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1answer
27 views

Is this an isolated equilibrium point?

I've just been learning the definition of an isolated equilibrium point. From my understanding of this definition, I would expect (as an example) the point $x=1$ to be an isolated fixed point for the ...
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21 views

Where should I begin the study of fixed point theory, especially of multi-valued maps?

How should one begin one's study of fixed point theory, especially of multi-valued maps? What background --- in topology, analysis, functional analysis, algebra, and set theory --- should one have? ...
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31 views

How can I see that we have $\Lambda_a=s\Lambda_a\mathbin{\dot\cup} \big(s\Lambda_a+(1+s)\big)\mathbin{\dot\cup}s\Lambda_b$ (Silver mean substitution)

Consider the (Silver mean) substitution $$\varrho:\begin{aligned}&a\mapsto aba\\&b\mapsto a\end{aligned}.$$ If we take $w^{(1)}=a|a$ and $w^{i+1}=\varrho(w^{(i)})$, then we get: $$a|a\...
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24 views

Topology: every continuous function on $\mathbb{R}^2$ scales a point

The question is simple: Suppose $f : \mathbb{R}^ 2 \to \mathbb{R}^ 2$ is continuous. Show that there exist $\lambda > 0$ and $x \in \mathbb{R}^2$ such that $f(x) = \lambda x$. So basically, we ...
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51 views

What's the meaning this DOT notation?

I'm reading a chapter in a Model Checking book. I came across this chapter "Symbolic Model Checking", in which the author mentions Fixed Point representation. I don't know how to explain the context, ...
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35 views

What can we say about the convergence of these fixed-point iterations for $\phi:\mathbb{R}\to \mathbb{R}$

Let $\phi: \mathbb{R}\to \mathbb{R} \in C^2(\mathbb{R})$ and let $x^{*}$ be a fixed-point of this function. Further assume that $|\phi'(x^{*})| \neq 1$. We define two sequences $\begin{align} &...
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Holomorphic map from closed convex domain in hilbert ball into itself has fixed point

Let $B=\{x \in \ell_2, ||x|| \leqslant 1\}$ - Hilbert ball $X \subset B$ - open convex connected set in Hilbert ball, $\bar{X}$ - closure of $X$. $F: \bar{X} \to \bar{X}$ - continuous map that ...
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65 views

Continuous map in $\mathbb{R}^2$ has a (scaled) fixed point

Let $\phi:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a continuous map. How do I prove that there exist $a>0$ and $x\in\mathbb{R}^2$ such that $\phi(x)=ax$? What I know: I thought maybe this can ...
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33 views

Transform system to polar and sketch phase portrait. Show that $(0,0)$ is an unstable focus.

Transform the system $$x' = y - x(x^2+y^2-1)$$ $$y' = -x - y(x^2+y^2-1)$$ to polar coordinates, and sketch the phase portrait. Show that it has a unique limit cycle and that all ...
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29 views

Problem with proof of the upper hemicontinuity of correspondence

I have a problem with a proof I found here of the upper hemicontinuity of the best-reply correspondence in the Nash Theorem. Below there is the proof, and here my problems: Problems: Is here ...
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74 views

Find the fixed points of the system, and sketch the trajectories of the system

I am given the following system: $$x' = [(x-1)^2 + y^2]y$$ $$y' = -[(x-1)^2 + y^2]x \tag{*}$$ where $x = x(t), y = y(t)$. I am supposed to Find the fixed points of the system, and ...
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49 views

Show Existence of Fixed Point in $\mathbb{R}^n$ with Euclidean Metric

Consider the closed unit ball in $\mathbb{R}^n$, $B = \{ x \in \mathbb{R}^n : \|x\| \leq 1 \}$ with the Euclidean metric. Then let $g : B \rightarrow B$ be a function such that $$\|g(x) - g(y)\| \leq \...
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30 views

If $X$ has the FPP then does $X\times I$ have the FPP.

If $X$ is compact subset of $\mathbb{R}^2$ and all continuous maps $f:X\rightarrow X$ have a fixed point, do all continuous maps $f:X\times I\rightarrow X\times I$ have a fixed point?
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22 views

Does this functional equation have a (unique) solution?

Does the following equation have a (unique) solution for $b_1$? \begin{equation} b_1 = \dfrac{1}{2b_1} \left\lbrace -Var(p) + Cov \left(\left[ (p + b_1 + b_0 + \alpha)^2 -4b_1(p + b_0) \right]^{0.5},...
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78 views

closed unit ball without a fixed point.

What I've done so far: Let $B=\{x\in\mathbb{R}^n : \|x\| \leq 1\}$ be the closed unit ball in $\mathbb{R}^n$ equipped with the standard Euclidean metric. Let $f \colon B \to B$ be a function such ...
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1answer
14 views

Find 3 fixed points of function with 2 arguments

I'm looking for a way to determine the fixed points of a function with 2 arguments. The specific function is the following: $F:\bigl( \begin{smallmatrix} x \\ y \end{smallmatrix} \bigr) \mapsto \...
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44 views

Uniqueness of solution to integral equation with “endogenous” kernel

Let $x,y\in C$ and consider the functional equation $T:y\mapsto x$ implicitly defined by the following integral equation: $$ x(t) = \int_{-\infty}^\infty K(x(t),t,s) y(s) \, ds, $$ with $K$ given. ...
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Understanding fully the proof of the uniqueness of a fixed-point

I was reading the proof of the uniqueness of the fixed-point. The uniqueness is stated as follows: ... If, in addition (to the fact that we know that $\exists$ a fixed-point in a range $[a, ...
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1answer
33 views

How many iterations are required to reduce the convergence error by a factor of 10?

I've the following function $$g(x) = x^2 + \frac{3}{16}$$ for which I found the two fixed points $x_1 = \frac{1}{4}$ and $x_2 = \frac{3}{4}$. I noticed that the fixed-point iteration $$x_{k+1} = g(...
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Find if a fixed-point iteration converges for a certain root

I'm asked to find if the fixed-point iteration $$x_{k+1} = g(x_k)$$ converges for the fixed points of the function $$g(x) = x^2 + \frac{3}{16}$$ which I found to be $\frac{1}{4}$ and $\frac{3}{4}$. ...
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Extension of Kakutani's fixed point theorem.

Can the Kakutani's fixed point theorem's be extended to say that there exists a fixed point inside the set (not on boundary)(I am not sure how to formally state this). For a $n$-dimensional compact, ...
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1answer
101 views

How many fixed points must $f$ have in the disk? [closed]

Let $\Omega$ be an open subset of $\mathbb{C}$. Assume $f \in H(\Omega)$, $\Omega$ contains the closed unit disk, and $|f(z)| < 1$ if $|z| = 1$. How many fixed points must $f$ have in the disk?
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Provinf Orbits are eventually fixed or eventually prime-2-periodic

Please I need help with this question: Let $S = \{a,b,c,d\}$ be a finite set and suppose that $f \colon S \to S$ has the property that: $$f(a) = f(b) = f(c) = d$$ Prove that each of the orbits of ...
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48 views

Suppose $f(z)$ is analytic in the closed unit disc…

Suppose $f(z)$ is analytic in the closed unit disc and $$|f(z)|<1 \quad \text{for} \quad |z|=1$$ Show that $f(z)$ has one and only one fixed point; that is, there exist a unique point $z_0$ in the ...
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58 views

How many fixed points are there for $f:[0,4]\to [1,3]$

Let , $f:[0,4]\to [1,3]$ be a differentiable function such that $f'(x)\not=1$ for all $x\in [0,4]$. Then which is correct ? (A) $f$ has at most one fixed point. (B) $f$ has unique fixed point. (C) $...
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How to confirm that if a function $f\circ f$ is a strong contraction, then $f$ has a fixed point or not?

Suppose that $(X,d)$ is a complete metric space and $f:X \rightarrow X$ is such that $f\circ f$ is a strong contraction. Must $f$ have a fixed point? So, it is given that $f\circ f$ is a strong ...
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Is there any difference between fixed point and decimal point?

Source: Introduction to Computers' 1999 Ed.1999 Edition Fixed point number 774.3675 is just a decimal number with a decimal point to show a fractional part 3675/10000. I see no difference in the fixed ...
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Does this iterative sequence converge?

Let $f:[0,1] \rightarrow [0,1]$be a continuous function. Choose any point $x_0 \in [0,1]$ and define a sequence recursively by $x_{n+1}=f(x_n)$. Suppose $\lim_{n \rightarrow \infty}x_{n+1}-x_n =0$, ...
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How to prove convergence of a sequence maximizing a sum of exponential distances?

I want to find the argument $x$ that maximizes $f(x)=\sum_i e^{-(x-d_i)^2/c}$ for some data values $d_i$ and an arbitrary positive constant $c$. I assume that $f(x)$ has only a single maximum (most ...
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$f:A\rightarrow B$ and $a_n=f(a_{n-1})$ $L=[a_1,a_2,…a_n,…]$ For which $f$ and $a_0$ is $\bar L=B$?

This is a generalization of a question I posted a few days ago regarding a particular recursive sequence . I am trying to find general conditions (if they do exist) on which the problem has a ...
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22 views

Convergence criterion of vector fixed point iteration

As we know, the convergence criterion of scalar fixed point iteration, e.g. $x^{k+1}=f(x^k)$, is that $|f^{\prime}(x^*)|<1$. For the vector variable version, it has been proved in the case when $...
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36 views

Proof that fixed points of a null field are zero

In this context, a null field means a field constructed of planar waves $e^{I k_{\mu} x^{\mu}}$ that satisfy the null condition $k_{\mu} k^{\mu} = 0 \implies c^2 k^2 = \omega^2 $ Suppose we have a ...
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22 views

Show that there is an equilibrium point

Show that if $x_{k+1}=f(x_{k})$, where $f$ is continuous, has two stable equilibrium points, then there's a third equilibrium point between then. I'm trying to approach this problem using the ...
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1answer
26 views

connection between Newton’s method and fixed point iteration

This is from my lecture slide I can understand Newton’s method, but I don't understand the context in red which requires rewriting th equation $x=g(x)$ as the Newton’s method require the right ...
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32 views

Number of fixed points of a meromorphic function

I would like to know whether a meromorphic function on the whole complex plane with at most one pole can have infinitely many fixed points or not. Many thanks in advance.
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37 views

Is every Boyd-Wong mapping also a contraction?

I know every contraction is a BW map, so I suspect the BW class of mappings is strictly larger. Can someone help me construct an example of a mapping that is not a contraction but is a BW map? Edit: ...
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Prove that if $X=[0,1]$ and $T:X \to X$ is defined as below then $d(Tx, Ty) \le \alpha (d(x, Tx) + d(y,Ty))$.

$Tx = x/4$ if $x \in [0, 1/2)$ and $Tx =x/5$ if $x \in [1/2, 1]$ and $\alpha$ is a constant such that $0 \le \alpha < 1/2.$ What I have tried so far: if $x,y \in [0, 1/2)$ then $d(Tx, Ty) = 1/4 ...
2
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43 views

An inverse problem for a parabolic equation and fixed-point theorem

I'm reading the paper "The determination of a parabolic equation from initial and final data" http://www.ams.org/journals/proc/1987-099-04/S0002-9939-1987-0877031-4/S0002-9939-1987-0877031-4.pdf. A ...