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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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}\\\ ...
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Trapping Region of a Map [on hold]

For the following map: find the interval A such that for α ∈ A, interval D = [0, 1] constitutes a trapping region. Locate the fixed points of this map and determine their stability: ...
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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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1answer
19 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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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: ...
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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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1answer
72 views

Statement of Markov-Kakutani fixed-point theorem

Markov-Kakutani fixed-point theorem is usually stated as follows: "Let $E$ be a locally convex topological vector space. Let $C$ be a compact convex subset of $E$. Let $S$ be a commuting family of ...
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50 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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34 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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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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27 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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72 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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46 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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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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43 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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20 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) ...
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60 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
13 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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35 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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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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32 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} = ...
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29 views

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
28 views

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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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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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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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 ...
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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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21 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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2answers
19 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
61 views

Quotient Spaces Defined By Bijection

I was working with a question in topology and came to the following statement that I can't seem to figure out: Let $f:\mathbb R^2\rightarrow\mathbb R^2$ be a homeomorphism with no fixed points. ...
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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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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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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 ...
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42 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 ...
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1answer
32 views

Whether there exists some compact manifold whose continuous map must have a fixed point?

Whether there is some compact manifold $M$ such that $f:M\rightarrow M$ is continuous $\Rightarrow f$ has a fix point. I mean that any continuous map from $M$ to $M$ must have a fixed point. I ...
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On the in-comparability of the Gauss-Seidel and Jacobi iteration schemes.

Given the system $x_1 + x_2 = 2$ $-x_1 + x_2=0$ $x_1 + 2x_2 - 3x_3=0$ the Jacobi iteration converges and Gauss-Seidel iteration diverges. Is there a way we can derive these two facts using the ...
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1answer
23 views

Every nonempty, compact convex set $M$ in a locally convex space has fixed point property

I need to prove that "Every nonempty, compact convex set $M$ in a locally convex space has fixed point property". In the book the reference has been given to "Eisenack & Frenske, 1944, page 44". ...
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71 views

Generalized Fixed Point Theorem

Suppose that $T: M \to M$ is a self map of a nonempty closed set $M$ in a complete metric Space $(X,d)$. Suppose further that $$d(Tx,Ty) \le k(a,b)d(x,y)$$ for all $x,y \in M$ with $0 \lt a \le d(x,y) ...
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Contractive Operators on Compact Spaces

Suppose that $T: M \to M$ is a compact contractive Operator on a nonempty compact subset $M$ of a complete metric space $X$. Show that $T$ has a unique fixed point. Further show that the sequence ...
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36 views

Finding an interval of convergence for a given $g(x)$

I am trying to do a fixed point iteration on the function: $f(x) = x^2 -3x+2 $, analyzing different forms of $g(x)$. I solved for the actual roots and they equate to $x=1$ and $x=2$. I am currently ...
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25 views

Establish the sufficient condition $|g'(x)| < 1$ for convergence of an iteration using the Banach fixed point theorem?

If $x_n = g(x_{n-1})$ is an iteration, it converges if $g$ is continuously differentiable and $|g'(x)| < 1$. The Banach FPT says that if $T$ is a contraction on a complete metric space $X$ then it ...
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1answer
11 views

Need help finding the smallest contraction constant.

I have to show $ T : X \to X$ given by $x \mapsto x/2 + 1/x$ is a contraction map, where $X = \{x \in R : x \ge 1 \}$ and find the smallest contraction constant. I have worked out that $|T(x) - ...