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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If $f$ is Lipschitz continuous on a closed interval $[a,b]$ such that $f([a,b])\subseteq [a,b]$ then it has a unique fixed-point

I am stucked at this problem: Prove or give a counter-example for the following sentence: If $f:[a,b]\to\Bbb{R}$ is Lipschitz continuous on a closed interval $[a,b]$ and $f([a,b])\subseteq [a,b]$ ...
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Notion of fixed point for the sequence $x_{n+1}=f_n(x_n)$

Let $(X,d)$ be a complete metric space and let $f_n:X\to X,\ n\in \mathbb{N}$ be a sequence of contractions. Is there any notion of fixed point for the following sequence? $$x_{n+1}=f_n(x_n),\ ...
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A question on the Banach fixed point theorem.

Suppose $f:(X,\tilde{d})\rightarrow(X,d)$ be a continuous function satisfying \begin{eqnarray}d(f(x),f(y))\leq \lambda d(x,y),\end{eqnarray} $\lambda > 1$. Let $\tilde{d}(x,y)=\lambda d(x,y)$. I ...
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+50

Reference request: fixed point and first-order logics

I'm looking for materials on the relationship between first-order and fixed-point logics, specifically on the condition for a formula in a fixed-point logic to have an equivalent first-order formula. ...
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Question about assumptions for Picard-Lindelöf Theorem in Zeidler's functional analysis text

In Zeidler's text on functional analysis pg.24 he wrote... The Picard Lindelöf Theorem: Assume the following: (a) the function $F: S \to \mathbb{R}$ is continuous and the partial derivative ...
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43 views

Show that the sequence does not converge

My Try: $|f'(a)|>1$. Assume that the sequence converges to a limit $b$. Then $f(b)=b$. Since $a$ is the only fixed point it implies that $b=a$. Hence, given any $\frac{1}{m}$ where $m\in ...
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Convergence of sequence as $O(1/n)$ using damped iterations

I am trying to understand the argument for convergence here (page 16) for damped iterations of non-expansive maps. Say we have a sequence $\{x^n\}_{n=0}^{\infty}$ that is generated as $x^n = \theta ...
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34 views

Convergence of continuation scheme for fixed-point via homotopy

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a non-expansive map, i.e. $$\|f(x) - f(y)\| \leq \|x - y\|$$ for all $x,y\in\mathbb{R}^n$. Further, assume $f$ has at least one fixed-point $x^\star$. ...
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33 views

Fixed-point analysis similar to Banach Fixed Point Thm

I have a fixed-point question similar to the Banach fixed-point theorem. Let $f\colon \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a given function and let $x^\star \in \mathbb{R}^n$ be a known ...
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43 views

Application of fixed point theory in Physics

Is there any application of fixed point theory in Physics?
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Find all real solutions to the following system of equations (involving fixed point iteration)

From the 1996 Canada National Olympiad. I have emphasised the real point of the question. Find all real solutions to the following system of equations. Carefully justify your answer. ...
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Proof Attempt of Brouwer (via Separating Hyperplane Theorem)

In part motivated by the discussion here, I have been playing with trying to prove Brouwer's theorem appealing as minimally as possible to topology. In the 1-dimensional case I believe one can ...
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1answer
22 views

Banach contraction principle: closed sets mapped to itself

I'm shoring up my understanding of basic real analysis and encountered this problem. Consider the operator $$K(x)(t) = \int_0^2 B(t,s)x(s) ds + g(t)$$ where $B$ and $g$ are continuous and $|B(t,s)| ...
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What kinds of functions have fixed points?

Among continuous functions, can we characterize those which have fixed points and those which do not? Geometrically, these are the functions that intersect the line $f(x) = x$. Is that the most ...
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36 views

Find the Lipschitz constant of a multi-variate Gaussian density function

I would like to find the Lipschitz constant of a multi-variate Gaussian density function: $$f_{\mathbf x}(x_1,\ldots,x_k) = \frac{1}{\sqrt{(2\pi)^{k}|\boldsymbol\Sigma|}} ...
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1answer
28 views

How do i show stability of fixed points of this two.dimensioanl system:$f(x,y)=(y,y²−x²)$? [on hold]

Is there someone show me how do i show stability of fixed points of this two.dimensioanl system:$$f(x,y)=(y,y²−x²)$$ and what is it relation to henon map ? Note :thank you for any help
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39 views

Finding a metric such that $\Phi$ becomes a contraction

Let $$ f:\:[0,1]\rightarrow\mathbb{R};\\ \Phi: \mathcal{F}([0,1], \mathbb{R}) \rightarrow \mathcal{F}([0,1], \mathbb{R}),\\ f\mapsto\Phi(f):=Φf := \left(x\mapsto\left\{ ...
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Contraction on a metric.

Let $M=\{x\in \mathbb{R}|x\geq 1\}$ with the absolute metric, being a metric space. Show that: a) The mapping $f:M\to M$ with $f:M\to M,~f(x)=\frac{1}{x}+\frac{x}{2}$ is a ...
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53 views

Can we find an example?

I have this theorem, and i want to know if i can find an expression of the operator $A$ which satisfy this theorem:
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Counterexamples of Brouwer fixed point theorem applied on the close unit ball

Brouwer fixed point theorem states that for any compact convex set $X$, a continuous mapping from $X$ to $X$ has at least one fixed point. Brouwer fixed point theorem applies in particular on the ...
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multidemensional fixpoint iteration

We want to solve the following system of equation, which shows an intersection between a circle and an ellipse . $x^2+y^2=5$ $\frac{x^2}{16}+y^2=\frac{5}{4}$ We can expres this system as a fixpoint ...
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Leray-Schauder fixed point theorem and remark

Leray-Schauder fixed point theorem from Gilbarg and Trudinger book is quoted below. I do not understand remark below this theorem. Could you explain? Theorem 11.2 from this text is Schauder fixed ...
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Convergence of a particular fixed point iteration scheme

Setup I have the following non-linear system of equations: $$ \mathbf{x} P(\mathbf{x}) = 0 $$ where $\mathbf{x} \in \mathbb{R}_{>0}^n$ is a probability distribution, i.e., $\sum_i x_i = 1$, and ...
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Every continuous map of a closed interval into itself has a fixed point

The Question: Please show this theorem: Let $f: I=[a,b] \rightarrow \mathbb{R}$ be a continuous map such that $f(I) \supset I $. Then $f$ has a fixed point on I. My Attempt: Suppose there is a ...
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26 views

Show that a zero of $f$ is a fixed point of $g$

I want to show that a solution of the equation $x^2+cos(x)-10x=0$ is a fixed point of $g(x)=(x^2+cos(x))/10$. I tried using the quadratic equation but my solution doesn't simplify nicely in $g$. I'm ...
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How to do fixed point iteration with matrices?

I am trying to follow solution to solve $$\min[\mathbf{z},\mathbf{q+Mz}]=0$$ by fixed point iteration. If $\mathbf{M=C+B}$ then a recursive algorithm with $k$ showing the iteration can be written as ...
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1answer
48 views

Prove that the function in $[0,\pi]$ defined by $f(x)=\sin(x)/x$ and $f(0)=1$ is a contraction

Let $f$ be a function $f:[0,\pi]\to\mathbb{R}$ such that: $$f(x)=\left\{ \begin{array}{ll} \frac{\sin(x)}{x} & \mbox{if } x \neq 0 \\ 1 & \mbox{if } x = 0 \end{array} \right.$$ I want ...
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1answer
37 views

A fixed-point theorem by Zamfirescu

I am having a trouble with understanding the proof of a fixed-point theorem by Zamfirescu. Could somebody please explain how the inequality in the inner, pink rectangle is obtained from the previous ...
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Fixed points and infinite series

Consider the formula $1 + \frac{y}2$. This has a fixed point at $y = 2$. And if we use the equation $y = 1 + \frac{y}2$ to substitute for $y$ in our formula, we get $1 + \frac{1 + \frac{y}2}2$, or $1 ...
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On (known) applications of fixed point theorems to some conjectures in elementary number theory

Let $\sigma$ be the classical sum-of-divisors function. Call an integer $n$ almost perfect if $\sigma(n)=2n-1$. The only known examples are $n=2^k$ for $k \geq 0$. Let $I(n)=\sigma(n)/n$ be the ...
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Numerically solving a transport equation

I would like to solve this transport PDE numerically : $$ \partial_t f + v(f) \partial_x f = 0 $$ What I would want to do is "freeze" the velocity $v$ and solve a classical transport equation by ...
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1answer
36 views

A generalization of Poincare-Birkhoff theorem

What could be the statment of a possible generalization of Poincare Birkhoff theorem for $M\times [0,\; 1]$ where $M$ is a compact orientable manifold?
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Finite groups whose non-trivial elements have no fixed points

(I first asked this question on MathOverflow, but was recommended to ask here at Mathstackexchange instead.) I am interested in finite groups $G$ acting on a finite set $X$ with the following ...
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Finite groups whose non-trivial elements have no fixed points [duplicate]

I am interested in finite groups $G$ acting on a finite set $X$ with the following property: (*) fix(g)=$\emptyset$ for all $g\in G\setminus\{1\}$, where fix(g):=$\{x\in X|gx=x\}$ denotes the set ...
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Brouwer fixed-point theorem on non-convex sets [duplicate]

I read about the Brouwer fixed-point theorem in Wikipedia, and got confused about whether it holds when the domain of the function is non convex. On one hand, convexity is explicitly mentioned as a ...
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Theorem about number of crossing-points between a function and a line

Assume $f(x)$, with $x \in [a,b]$. Take $u$ so that $f(a)<u<f(b)$. By the Intermediate value theorem, we know that $f(x)$ crosses $u$ at least once. My question is, given some extra information ...
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1answer
31 views

Can somebody explain the notation $f \in C^4$

To give some context the full question is: Suppose $f \in C^4$ in a interval containing the root $\alpha$ and that Newton’s method gives a sequence of iterates $\{x_k\}$, $k = 0, 1, 2, \dots$ which ...
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1answer
56 views

Brouwer's fixed-point theorem and the intermediate value theorem?

In one dimension, Brouwer's fixed-point theorem (BPFT) can be proved easily based on the Intermediate Value Theorem (IVT). Is the inverse also true? I.e, is it possible to prove the IVT directly from ...
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1answer
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Show that the comb space has fixed-point property

By "comb space", I mean the space $X=([0,1] \times \{0\}) \cup (K \times [0,1])$, where $K=\{ \frac{1}{n} : n \in \mathbb{N}^+ \}$, without the leftmost vertical line segment. How to prove that this ...
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1answer
46 views

Application of the Banach fixed point theorem

Let $a > 0$. We consider the function: $f: (0, \infty) \to (0, \infty)$, defined by $f(x) = \frac{1}{2}(x + \frac{a}{x})$. Let $(x_n)_{n \in \mathbb{N}_0}$ be defined by: $x_0 \in (0, \infty)$, ...
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Translate this proof from German to English

I need your help to translate some exercises from German to English. I will attach like images. Thanks :) Satz 3. Es sei $(X,d)$ ein ultrametrischer Raum. $X$ ist genau dann transvollständig, wenn ...
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Show that F can have at most two fixed points

Consider a function $F:\mathbb{R}^n\to\mathbb{R}^n$, where $F=(F_1,...,F_n)$. Suppose that $F$ is strictly quasi-concave, and that for all $i=1,...,n$ the function $F_i:\mathbb{R}^n\to\mathbb{R}$ is ...
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Brouwer's fixed point continuous function

Can anyone point me out the continuous functions without brouwer fixed point's for the following sets $$A = \{x \in \mathbb{R}^2 | x_1,x_2 \geq 0 \text{ and }x_1^2+x_2^2 = 1 \}$$ $$B = \{x \in ...
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What are conditions for Tarski's fixed point theorem so that greatest and least fixed points are not the same?

Tarski's fixed point theorem claims that for any monotone function on a complete lattice there are least and greatest fixed points (or more generally, that the set of fixed points is a complete ...
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1answer
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What is the difference between a period $n$ point, and a point of least period $n$?

What is the difference between a period $n$ point, and a point of least period $n$? Simply what is the definition of the two of them, and how do they differ. I think I have a rough idea of one ...
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14 views

Function whose fixed points are a convergent sequence with derivatives at each term $\neq 1$ and derivative at limit point $=1$

The question asks to find an example of a function where the fixed points of the function are a sequence that converges to a fixed point, where the derivative of the function at each of the fixed ...
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1answer
30 views

Spectral radius and convergence of fixed point iteration

Let $F: \mathbb{R}^n \to \mathbb{R}^n$ be a differentiable map. -editted- Let $x^\star$ be a fixed point of $F$. Then, is it true that the fixed point iteration $x_{n+1} = F(x_n)$ converges locally ...
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1answer
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How to find periodic points non-algebraically

For example, $f(x) = x - x^2$ by observation has a fixed point at $x = 0$, and an eventually periodic point at $x = 1$ (that goes to $0$). For the second iteration, to see if there's a periodic point ...
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Prove there exists $a \in E$ such that $a = f(a)$, assuming $d(f(x), f(y)) \le Kd(x,y)$ with $K<1$

Let $f: E \rightarrow E$, $E$ a complete metric space. Assume that there exists $K$ such that $0 < K < 1$ and $d(f(x), f(y)) \le Kd(x,y)$ for all $x,y \in E$. Prove that there exists $a \in E$ ...
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No clear analytic method to prove unique maximum? ($2^{-x}+2^{-1/x}$)

Prove that $f(x) = 2^{-x}+2^{-1/x}$ has the unique local maximum $(1,1)$ for $x>0$. Do not use computer software. Proving that $(1,1)$ is a maximum is easy, but I'm having trouble with the ...