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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A technical step in the proof of Atiyah-Bott fixed point formula

From John Roe, Elliptic operators, topology and asymptotic methods, page 135. Here Roe tried to prove that $$ \DeclareMathOperator{\Tr}{Tr} \sum_{q}(-1)^{q} ...
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71 views

Number of limit cycles: Counterexample of the extended Bendixson-Dulac criterion?

The problem concerns the number of limit cycles in the vector field of coupled differential equations (ODEs) in two dimensions, i.e. $$ \ \dot{x} = X(x,y)\\ \dot{y} = Y(x,y) $$ Specifically, let $$ \ ...
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$|f(x)-f(y)|<|x-y|$ on a non-empty closed bounded set of real numbers

Let $A$ be a non-empty closed bounded set of real numbers and $f: A \to A $ be a function such that $|f(x)-f(y)|<|x-y| , \forall x,y\in A$ , then how to show that $f$ has a unique fixed point ?
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Baillon theorem in fixed point theory

Baillon proved that if $X$ is a Banach space with $\rho'_X(0)<\frac{1}{2}$, then $X$ has the fixed point property (by $\rho_X(t)$ we denote the modulus of smoothness). My questions are as ...
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27 views

If $f^N$ is contraction function, show that $f$ has precisely one fixed point.

If $f$ is a mapping of a complete metric space $(X, d)$ into itself and $f^N$(composite $f$ for $N$ times) is a contraction mapping for some positive integer $N$, then $f$ has precisely one ...
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47 views

Finding the fixed points of a recurrence relation (and systems of) analytically?

How would I go about finding the fixed points of the following recurrence? $$X_n = 2X_{n-1}(2- 3X_{n-1}) + X_{n-1}$$ And therein, determining their stability analytically? Also, how does one find ...
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135 views

Fixed point for a continuous function on a compact set?

If $f:X \rightarrow X$ is continuous and X is compact, will $f$ have a fixed point? We know that a contraction will have a fixed point but I have not come across an example of a continuous function ...
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Fixed points of the Gamma Function?

I am interested in complex values of $z$ such that $$ \Gamma (z) =z$$ Clearly, the one trivial value of $z$ is 1. Also, looking at a graph of the gamma function on the real axis, I can tell that there ...
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Reference request: result concerning Leray trace

Let $V$ be a vector space (possibly of infinite dimension). For a linear homomorphism $f\colon V\to V$ define $$N(f)=\bigcup_{n\in\mathbb{N}} \operatorname{ker}(\underbrace{f\circ\ldots\circ ...
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32 views

Homogeneous topological space with the fixed-point property

Let $X$ be a topological space. $X$ is said to be homogeneous if for every $x$ and $y\in X$ there is an self-homeomorphism $f$ of $X$ such that $f(x)=y$. Further, $X$ is said to have the fixed-point ...
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Find the Fixed points

Let $L=\mathcal P(\mathbb N)$ be a complete lattice of subsets of $\mathbb N$. Find the smallest and the greatest Fixed Point: 1) $F(X)=\left\{ x \mid x+1\in X \right\}$ 2) $F(X)=X \setminus \left\{ ...
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Find the Fixed points (Knaster-Tarski Theorem)

Let $L=\mathcal P(\mathbb N)$ be a complete lattice of subsets of $\mathbb N$. a) Justify that the function $F(X)=\mathbb N \setminus X$ does not have a Fixed Point. I don't know how to solve this. ...
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21 views

Firmly nonexpansive mapping with the fixed point set same as for given nonexpansive mapping

I found PAMS publication vol. 113, no. 3, 1991 by Ryszard Smarzewski called "On firmly nonexpansive mappings". It is written that "to each nonexpansive T on set C one can associate a firmly ...
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58 views

Banach Fixed Point Theorem. Measurable version.

The Banach fixed point theorem has the following statement THEOREM ( Banach contraction principle). Let $(Y,d)$ be a complete metric space and $F:Y\to Y$ be contractive . Then $F$ has a uniqe ...
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220 views

Can the proof of fixed point theorems ever be constructive?

Overall, Brouwer fixed point theorem and Kakutani fixed theorem are non-constructive. Is there any established paper that demonstrates that there exists constructive proofs that do exactly what these ...
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Use of closed and convex set in fixed point property

Let $(X, ||.||)$ be a Banach space and C a subset of X. A mapping $T:C{\to}C$ is non-expansive if $||Tx-Ty||\leq||x-y||$ for all $x, y \in C.$ A Banach space is said to satisfy the fixed point (FPP) ...
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Continuity of fixed points of an equation

Say I have a function $f(x,\theta)$ such that for each value of $\theta$ the function $f$ has a unique fixed point $x^*(\theta)$, i.e. $$x^{*}(\theta) = f(x^{*}(\theta),\theta)$$ My question is when ...
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Use a fixed point argument to show there exists a unique solution to the following BVP

Show using a fixed point argument that there exists a unique solution $f\in C[0,1]$ to $$ -f''(x)+\sin(f(x))=\sin(x) , x\in (0,1), y(0)=y'(1)=0 $$ This is what I have so far: We can show ...
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Dense subgroup of an extremely amenble group is extremely amenable??

Recall that a topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space $K$ admit a fixed point. i.e there is a point $\xi\in K$ such that $g.\xi=\xi$ for ...
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28 views

Derivative of the Solution to a Fixed Point Iteration

Let $\theta_s$ is the solution to a fixed point equation $$\theta=f(\theta,\lambda)$$ Let $d(\theta,\lambda)$ be another function of $\theta,\ \lambda$. I know $f$ but I have no explicit expression ...
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41 views

How to compute the Lefschetz number

Given a continuous function $f: X \to X$ how do you compute: $$ \Lambda_f = \sum_{k \geq 0} (-1)^k \mathrm{Tr}(f_*|H_k(X,\mathbb{Q})) $$ which is known as the Lefschetz number. For instance let $X: ...
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How to prove $F$ is a contraction mapping?

Given a $n\times n$ matrix $\mathbf{D}$ in which each entry $\mathbf{D}_{ij} \in [0,1]$. Let $i$-th row vector of $\mathbf{D}$ denote by $\mathbf{D}_{i\ast}$. The mapping $F:[0,1]^n \mapsto [0,1]^n$, ...
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A question on fixed point property

Assume that $0<k<n-1$, Note that $\mathbb{C}P^{k}$ can be considered as a closed subset of $\mathbb{C}P^{n}$, in a natural way. We collapse $\mathbb{C}P^{k}$ to a point. The resulting space is ...
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Why $e^x$ never equal $x$?

Je veux savoir pourquoi $x=e^x$ n'a aucune solution dans $\Bbb R$. Lorsque j'ai essayé de tracer le graphe de la fonction $e^x$, j'ai trouvé en fait qu'elle est une fonction strictement croissante ...
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Converse to Lefschetz fixed point theorem (counterexample)

The celebrated Lefschetz fixed point theorem, in its simplest form, says (following Wikipedia) that if $f\colon X \to X$ is a continuous map of a compact triangulable space $X$ to itself, then $f$ has ...
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completeness of $A=\{(x_1,x_2)\in\mathbb R^2| \max\{ |x_1|,|x_2|\}<1\}$

Let $A=\{(x_1,x_2)\in\mathbb R^2| \max\{ |x_1|,|x_2|\}<1\}$ and $g(x_1,x_2)=\frac14(x_1^2x_2,x_1+1)$. I want to show that $g$ has a fixed point in $A$. So ...
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Fixed point method where the derivative is one - does it converge

I'm trying to see if the iterative method $x_n=g(x_{n-1})$ where $g(x)=2\sqrt{x-1}$ will converge to $2$, if I take $x_0$ that is sufficiently close to $2$. Indeed notice that $g(2)=2$. and we have a ...
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Existence of Galerkin approximations to PDE

Given a weak PDE $$ \langle b(u),v \rangle +\langle a(\nabla u), \nabla v\rangle=\langle f, v \rangle,\qquad v\in H^1_0 $$ where $b\in C(\mathbb{R})$ and $a\in C(\mathbb{R}^n,\mathbb{R}^n)$ grow ...
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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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Trouble understanding the steps to go from $x^x = 1000$ to the fixed point of $f(x) = \frac{log(1000)}{log(x)}$

I'm following through the Structure and Interpretation of Computer Programs, and in an example, they talk about finding fixed points of function (where $f(x) = x$) They then go on to say, if we want ...
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quotient of an amenable group

I have a question about amenable groups. The notion of amenability I am using is: The action of $G$ on $k$ ($k$ locally compact in a topological vector space) is amenable if there exists a point $x ...
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Identification of (Centers of) Cycles in a Discrete Time Dynamical System

I am studying dynamics on nonlinear Discrete Time Dynamical System of the form $$ \vec{X}_{t+1} = D(\vec{X}_t), $$ where D is some nonlinear function. I was looking for a (relatively) quick ...
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Unique fixed point of a contraction defined on a closed ball which maps the boundary back into the ball

Let $X$ be a Banach space, $r > 0$, $A: K_r(X) \rightarrow X$ a contraction (where $K_r(X)$ is the closed ball of radius $r$ and center $0$ in $X$), with contraction constant $0<q<1$, which ...
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An application of Banach fixed point theorem for initial value problem

Find a condition for $\beta>0$ which implies that the differential equation system \begin{align} x'(t)&=x(t)+y(t) ,\\ y'(t)&=t^{2}+tx(t) \end{align} with initial conditions $x(0)=0, ...
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Fixed-Points Limit Set

Let $f : \mathbb{R}^n \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ be compact and convex. For all $p \in \mathbb{R}$, consider $A_p \in \mathbb{R}^{n \times n}$, with $p \mapsto A_p$ ...
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An application of Banach fixed theorem on an integral equation

I'm learning some applications of the Banach Fixed Point Theorem and I have the following question: Consider the integral equation $\displaystyle x(t)=\int_{0}^{\frac{\pi}{2}}\arctan ...
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Is there a proof for the Central Limit Theorem via some fixed point theorem?

This question arose in my mind when I learned that the Gaussian is a fixed point for the Fourier transform. On the other hand, in e.g. the Banach fixed point theorem we have convergence to a fixed ...
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if $f_{n}(x)=f(f_{n-1}(x))$then $f_{10}(x)=x,x\in [0,1]$

Define the function $f:[0,1]\to[0,1]$ by the following. $$f(x)=\begin{cases} x+\dfrac{1}{2},&0\le x\le\dfrac{1}{2}\\ 2(1-x),&\dfrac{1}{2}<x\le 1. \end{cases}$$ Let $f_1(x)=f(x)$ and ...
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Show that $f$ has infinitely many fixed points

Let $f:ℂ→ℂ$ be an entire function. Assume that the equation $f(s)=a$ has infinitely many real solutions for all reals $a$. Show that $f$ has infinitely many fixed points, i.e., there exists infinitely ...
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Continuous function on closed unit ball

Take a continuous mapping $f: \bar{B^{n}} \rightarrow \bar{B^{n}}$, where $\bar{B^{n}}$ is a closed unit ball in $\mathbb{R}^{n}$. Assume that $f(x) \neq x$ for every $x \in \bar{B^{n}}$. Define ...
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A basic question about upper hemicontinuity

Given a correspondence $f:X\rightarrow 2^X$, suppose X is a closed simplex in $\mathbb{R}^n$, and $f$ is compact-valued. We say $f$ is upper hemicontinuous if, $\forall x\in X$ and every open subset ...
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Example of continuous function without fixed point.

I need to find an example of a continuous function without a fixed point, and this is what I've come up with: As {1} is not in the (co)domain, I can evade all $x$ for which $f(x)=x$ up until I ...
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Banach fixed point theorem and Picard-Lindelöf applied to this equation (explanation needed)

Consider the following equation which holds for all $w$ in some space, $$\langle v(t), w \rangle = \langle v(0), w \rangle - \int_0^t \langle F(s,v(s)), w \rangle$$ where $\langle F(s,v),w \rangle$ is ...
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Discrete map: local linearized instability implies nonlinear instability

Let a discrete uni dimensional map $x_{n+1}=f(x_n)=f^{n}(x_0)$. If its linearized version is local instable, that is, $\frac{df}{dx}(x_p)>1$, where $x_p$ is such that $f(x_p)=x_p$ then there is a ...
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Every increasing function from a certain set to itself has at least one fixed point

I need a hint for the following question: Let $S$ be a nonempty ordered set such that every nonempty subset $E\subseteq S$ has both a least upper bound and a greatest lower bound. Suppose $f:S ...
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fixed point iteration $x_{k+1}=f(x_k)$ with multiple fixed points

I have a fixed point iteration problem in my research, as below: $$x_{k+1}=f(x_k)$$ where $$f(x)=\frac{\lambda r}{g}\sum_{i,j}\pi_if_{ij}\left[1- \exp\left\{\displaystyle ...
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Kakutani's FP theorem for $F:X\to 2^Y$

The Kakutani Fixed Point Theorem is postulated for mappings of the form $F:X\to 2^X$, where $2^X$ stands for the powerset of $X$, $X$ is nonempty and $F(x)\neq \varnothing$ for all $x\in X$, $F$ is ...
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On a Variational Inequality

Let $H$ be a Hilbert space with real inner product. Consider $f: C \rightarrow C$, where $C \subset H$ is closed and convex. I am not sure about the variational inequality problem: find $x \in H$ ...
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27 views

What Projections preserve Pseudocontractiveness?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a pseudocontraction, i.e., $$ \left\| f(x) - f(y) \right\|^2 \leq \left\| x-y\right\|^2 + \left\| f(x) - f(y) - (x-y) \right\|^2 $$ for all $x,y \in ...
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42 views

Conditions yielding a unique fixed point of a continuous differentiable function.

Let $f$ be a function defined on $[0,1]$ which is continuous for each point in $[0,1]$ and differentiable for each point in $(0,1)$. Suppose that $f^\prime (x) \neq 1$ for every $x \in (0,1)$. ...