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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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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23 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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71 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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28 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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41 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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18 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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58 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
10 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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2answers
26 views

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
29 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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2answers
28 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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37 views

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
27 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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1answer
39 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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1answer
95 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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50 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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1answer
38 views

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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12 views

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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90 views

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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1answer
26 views

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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18 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
18 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
59 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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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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26 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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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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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
31 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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69 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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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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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
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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) - ...
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1answer
31 views

Does any analytic function from the unit disk to a compact subset of itself have a fixed point?

I was reading a proof by Beardon of the Wolff-Denjoy Theorem in Complex Dynamics. In the proof, a family of maps $$f_\epsilon=(1-\epsilon)f(z)$$ is used, where $f(z)$ is an analytic map from the unit ...
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1answer
37 views

Degree of infinite dimensinal antipodal map

Let $E$ is a infinite dimensional Banach space and $\Omega=\{x\in E: ||x||=1\}$ . $L:\Omega\rightarrow\Omega,L(x)=-x$, how to compute the degree of $L$? In fact ,I just know that the algebra define ...
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26 views

Range of parameter values for a stability of a fixed point for this 2d map

So I am trying to do a linear stability analysis for a very simple 2d discrete system: \begin{equation} \begin{aligned} x_{n+1} &= y_{n}\\ y_{n+1} &= -\frac{x_{n}}{2} + ay_{n} + y_{n}^{3} ...
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3answers
71 views

Prove there is no contraction mapping from compact metric space onto itself

This question is from Foundations of mathematical analysis by Richard Johnsonbaugh The thing with this question is that there is a question that seems to prove the opposite claim Prove the map has a ...
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59 views

Large-time limit of the general solution of an ODE is a fixed-point. Is the fixed-point stable?

This question might well have an obvious affirmative answer (or an obvious counterexample!), which at present I cannot see. Suppose I have a first-order ODE $$ u'(t)=f(u) $$ whose general solution ...
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1answer
24 views

Does a fixed point depend smoothly on the parameters?

Let $(X,d)$ be a complete metric space. A well-known theorem states that, for any map $G: X \to X$ satifying $d(G(x), G(y)) < Ld(x,y)$ for some fixed constant $L < 1$ and arbitrary $x, y \in X$, ...
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4answers
92 views

Fix point of $L:S^2\rightarrow S^2$

Let $L:S^2\rightarrow S^2$ be a bijective continue map. Is there exists $L$ such that $\forall x\in S^2 \Rightarrow Lx\neq x$ and $Lx\ne -x$? I mean that whether $L$ must have fix point? In fact ...
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1answer
30 views

How to find the number of iteration in Fixed point iteration method?

I want to know how to find the number of iterations in fixed point method. The book that i have, gives me 2 ways to find the number of iterations. The first one: $|p_n - p| \leq k^n max ${ $p_0 - ...
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3answers
44 views

Using fixed point iteration find the positive root of $f(x)=e^{-x}-x^2$

Consider $f(x)=e^{-x}-x^2$. I'm suppose to find the positive root using fixed point iteration. after drawing the graph, it's safe to set the interval from [0.25,1]. (I actually want to set it from ...
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0answers
26 views

Computing the set of fixed points of a map

Compute the set of fixed points of the following map: $f(X) := (y,-x)$ when $X=(x,y)$ So for this, do I just have to solve the system of equation such as: $x=y$ $-x=y$? Plugging the first into ...
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1answer
48 views

Is this $x \mapsto k^{k^{\frac{-1}{x}}}$ a contraction?

Given that $k,x \in \mathbb{R}^+$ and $k > 1$, the function $f$ defined by $$f(x) = k^{k^{\frac{-1}{x}}}$$, generates a sequence $x_0,f(x_0), f(f(x_0)), \cdots$ is observed to converge to a fixed ...