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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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
12 views

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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12 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
14 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
25 views

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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1answer
33 views

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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1answer
76 views

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 ...
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2answers
52 views

Solve for y in sin(y) = cos(y) using a fixed point procedure

I'm reading an programming book that uses a lot of math equations and formulas as coding examples. In one lesson, it demonstrates finding the fixed point for $\sin(x) + \cos(x)$ by repeatedly calling ...
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1answer
26 views

Metric Fixed Point Theory

I am learning Metric Fixed Point Theory by Mohammed A Khamsi and William A Kirk. I need help in understanding a step in the proof of the following theorem(Chapter 3, Theorem 3.2, Page No. 43): ...
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1answer
30 views

Newton Iteration Function

So I'm having trouble figuring this problem out so if someone can help me out that'd be great. Find all the fixed points for the associated Newton iteration function for $$ f(x) = \frac{x}{(x-1)^n} ...
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23 views

Show that the fix points of a function couldn't be in the interior

I want to solve the following problem: Show that the fix points of a function $f:\mathbb B^n\rightarrow \mathbb B^n$ could possibly not be in the interior. By this, Show that the Brouwer fixed-point ...
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2answers
53 views

Is there a metric proof for the Caristi theorem?

I'm writing a paper about hyperconvexity in metric spaces and came across Caristi's theorem: Let $(X, d)$ be a complete metric space and $\phi\colon X \to \mathbb{R}^+$ a continuous function. An ...
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22 views

Fixed Point involving limits of Integrals

Assume $\omega$ is a random variable with a p.d.f $f(\omega)$. There is a function $\lambda(\omega):[0,1]\rightarrow[0,1]$ such that $\int_0^1\lambda(\omega)f(\omega)d\omega=\bar{\lambda}$ with ...
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1answer
39 views

Function on a Power Set

Let $f\colon \mathcal{P}(A)\mapsto \mathcal{P}(A)$ be a function such that $U \subseteq V$ implies $f(U) \subseteq f(V)$ for every $U, V \in \mathcal{P}(A)$. Show there exists a $W \in \mathcal{P}(A)$ ...
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18 views

Fixed point theorem for multivalued functions on b-metric space

Hy guys! I'm studying some fixed point results for multi-valued function on a b-metric space $(X,d,s)$. I'm looking for the proof of a theorem wich is just a generalization of Nadler's results for ...
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1answer
16 views

Fixed point implication question

Suppose $f:[a,b] \to [a,b]$ is continuous and $f''>0$. Use the fundamental theorem of calculus to argue that if $f(x^*) = x^*$ and $f'(x^*) \geq 1$, then $f(x) > x$ for all $ x > x^*$. My ...
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1answer
36 views

Fixed point property of “3-star”

Let $X = (I_1\sqcup I_2 \sqcup I_3)/(0_1 \sim 0_2\sim0_3),(I_i=[0,1]_i)$. I spend much time to trying to prove that any continuous map $X\to X$ have fixed point, but with no results..
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19 views

Problems with the hypothesis in a fixed point theorem

Leray-Schauder fixed point theorem : If $D$ is a non-empty , convex , bounded and closed subset of Banach space $B$ and $T:D \to D$ a compact map , then $T$ has a fixed point in $D$. I have ...
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1answer
52 views

System of equations and the Brouwer's Fixed-Point Theorem.

Let's consider the following system of equations: \begin{eqnarray}{ e^{xyz} = \frac{x}{\sqrt{e^{2xyz}+1}}\\ \cos(x+y+z) = \frac{y}{\sqrt{e^{2xyz}+1}}\\ \sin(x+y+z) = \frac{z}{\sqrt{e^{2xyz}+1}} ...
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21 views

A fixed point theorem revisited paper.

Let $X$ be a metric space. A function $G:X \to [0,\infty)$ is said to be $T$-orbitally lower semicontinuous at $x \in X$ if every sequence $\{x_{n}\}$ in $O(x,\infty)$ which $x_{n} \to x$ , then ...
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1answer
18 views

If $\frac{dy}{dx}=A(x)y+B(x)$ and if $A(x)$, $B(x)$ are bounded and integrable, show the fixed point theorem solves the initial value problem

Given the linear differential equation $\frac{dy}{dx}=A(x)y+B(x)$, show that if $A(x)$ and $B(x)$ are bounded and integrable on $I=\{x|a \leq x \leq b\}$, then the fixed point theorem yields a ...
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1answer
21 views

If $f$ maps a complete metric space $S$ onto $S$ and $f^2=f \circ f$ is a contraction mapping, show $f$ has a unique fixed point.

If $f$ maps a complete metric space $S$ onto $S$ and $f^2=f \circ f$ satisfies the fixed point theorem given below, show $f$ has a unique fixed point. The following is the fixed point theorem: If f ...
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2answers
62 views

Given $\frac{dy}{dx}=x^2+y^2$ and initial condition $\varphi (0)=1$, find the first 6 terms in the Taylor expansion solution $y=\varphi (x)$

Given $\frac{dy}{dx}=x^2+y^2$ and initial condition $\varphi (0)=1$, use the method of reduction to an integral equation and successive approximation to find the first 6 terms in the Taylor expansion ...
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23 views

Using a fixed point theorem.

Let $x,y \in[0,1] $, consider the following system of equations: $$ ((x+y)/2)^n-x=0 $$ $$ {x^n \over x^n+y^n+1}-y=0 $$ where $ n \in N $ a) Transform the system of equations into equivalent fixed ...
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2answers
32 views

Show using the fixed point theorem that if $f(x)=\frac{1}{4}[1-x-\frac{1}{10}x^5]$ is defined on $I=\{x|0 \leq x \leq 1\}$ then it has a zero in $I$

The main idea here is to apply the fixed point theorem to $g(x)=f(x)+x$, in order to show that f has a zero in $I$. If $g$ has a fixed point (i.e. $g(x_0)=x_0$), then $f(x_0)=0$. I just don't see how ...
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1answer
21 views

Question about function compositions

Let us have $m,n$ positive integers, and suppose, that $ f o f ... f(m$ times$)$ and $f o f... f(n$ times$)$ have an $x$ fix point. For what $(m,n)$ positive integers will it be true, that $x$ is a ...
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1answer
66 views

What is the role of fixed point theorems in modern mathematics?

About Fixed Point Theorems, Wikipedia says: Results of this kind are amongst the most generally useful in mathematics. This seems an accurate statement: indeed, there are many journals ...
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36 views

Proof of Banach's homeomorphism theorem without the contraction map principle.

Let $E$ a Banach's space and $X\subset E$ open. The Banach's homeomorphism theorem tells us that if a function $F:X\to E$ is a contraction on $X$ then $(I+F):X\to E$ is a homeomorphism of $X$ onto ...
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45 views

How to get the proper fixed point iteration function?

When we find the approximated root of a function $f(x)$ in an interval $[a,b]$ from the fixed point iteration method, we derive a new function $g(x)$ which has a fixed point as a root of $f(x)$. Is ...
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28 views

So i read that kannan maps are caristi maps, how do I prove it?

Let $(X,d)$ be a complete metric space and $f:X \rightarrow X$ be a function such that $d(f(x),f(y)) \leq k(d(x,f(x))+d(y,f(y)))$ for $k \in[0,\frac{1}{2})$ I have tried to prove the following ...
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0answers
56 views

Condition for maximizer of convex combination to be expansion mapping

I have $\Pi_n:\mathbb R^{n+1}\rightarrow \mathbb R$ and $F_n:\mathbb R^2\rightarrow \mathbb R$ with $$F_n(x,a)=\Pi_n(x,...,x,a)$$ $$f_n(x)=\operatorname{ArgMax}_{a\in\mathbb R}\{F_n(x,a)\} $$ such ...
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1answer
20 views

Fixed point location for functions

How are fixed points calculated? Are intersections of $ y = f(x) ,y = f^{-1} (x) $ graphs give real fixed points for all $f$ ?
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5answers
277 views

Can't there be more than one fixed points in a contraction? or none?

I was going through the contraction mapping theorem in my book where it says, that if $\phi: G\to G$ is a contraction, then $\phi$ has a unique fixed point $\alpha$ on $G$. Sequence {$x_n$}, $x_{n+1} ...
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0answers
30 views

Deriving the fixed point for $\omega$ (i.e. $\lambda x.xx$) and proving it to be so

I am studying the simply typed $\lambda$-calculus, and I am struggling a bit with really understanding fixed-points and the $\mathbf Y$ combinator. I have read or skimmed all the questions on here ...
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2answers
92 views

Example of a function, so that $g(x)\neq x$

I'm trying to find an example of a function $g:\mathbb{R}\to \mathbb{R}$ (or $g:[1,\infty) \to \mathbb{R}$), so that $$|g(x_1)-g(x_2)|<|x_1-x_2|$$ for all $x_1, x_2\in \mathbb{R}$ ( or $x_1,x_2\in ...
0
votes
1answer
16 views

Unclear passage of a theorem concerning compact operators (Schauder fixed point theorem)

I'm looking at this proof of Schauder theorem and I am struggling with a passage. This is my problem: Let $X$ be a Banach space, $K \subset X$ a convex, close and bounded set and $F:K \rightarrow ...
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1answer
59 views

Which of the followings have a fixed point?

Consider the following sets : $$S=\left\{(x,y)\in \mathbb R^2:x^2+y^2=1\right\}.$$ $$D=\left\{(x,y)\in \mathbb R^2:x^2+y^2\le 1\right\}.$$ $$E=\left\{(x,y)\in \mathbb R^2:2x^2+3y^2\le 1\right\}.$$ ...
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1answer
26 views

Atiyah-Bott fixed point formula; signs

In classical paper by Atiyah-Singer on page 16 (or 560) stated formula $(3.1)$. It should give classical Lefschetz fixed-point formula if the operator is $d + d^* : \Omega^{even} \rightarrow ...
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27 views

Noisy contraction mapping

I am trying to describe the behavior of the iterates of a contraction mapping when noise is added. Given a real valued random variable $X_{0}$ a sequence $\{Z_{n}\}_{n=0}^{\infty}$ of i.i.d real ...
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1answer
37 views

$f (x)=\exp(x/2)−25x^2$. Show that f on $ (4\log(20), \infty)$ has exactly one root.

Let $$f (x)=\exp(x/2)−25x^2$$ Show that $f$ on $ (4\log(20), \infty)$ has exactly one root $x^*$. (Note that log the natural logarithm) I'm struggling with this question, we were given a hint, ...
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1answer
20 views

Verify (without Banach Contraction Priciple), that the function g(x) = 1 + x - (1/8)x^3 has a unique fixed point

I know how to show that there exist a solution by the intermediate value theorem but I'm not sure how to show that the root is unique?
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0answers
26 views

How do I show that infinite application of this function gives a constant?

I want to show that g(x) returns the same value independent of x and hence is a constant. $$g(x) = \lim_{n \rightarrow \infty}(\underbrace{f \circ f \circ \cdots \circ f}_{n\text{ times}})(x)$$ ...
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2answers
37 views

Example of contraction mapping

Please give me some examples of contraction mapping on $(C[0,1]), \lvert \lvert \cdot \rvert \rvert_\infty)$ and $(C[0,1],\lvert \lvert \cdot \rvert \rvert_1) $. Note that : 1. $\lvert \lvert f ...
2
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1answer
34 views

If $f$ is a homeomorphism then any periodic point have period less or equal 2

How can one prove the followiong statment? Let $f:[0,1]\to [0,1]$ be a homeomorphism. If $x\in\operatorname{Per}(f)$ then the period of $x$ can't be greater than $2$, i.e, $f(x)=x$ or $f^2(x)=x$.
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42 views

Proof of a fixed point theorem on the disk

There is a very nice fixed point theorem which I'd have liked to give to my students : Let $n$, $m$ be two integers larger or equal to one. Let $B_n$ be the open unit ball in $\mathbb{R}^n$, and ...
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17 views

Simple closed curve with non-zero index implies fixed point.

I'm searching for a proof(sketch) of something similar to the following: Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ be an orientation-preserving homeomorphism and let $\gamma: S^1 \rightarrow ...
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0answers
43 views

Find the fixed point of $\cos(x)$ (equivalently of $\cos(\cos(x))$) restricted to $[0,\frac \pi 2]$.

I've proved that $\cos(\cos(x))$ restricted to $[0,\frac \pi 2]$ is a contraction, which imply by Banach's fixed point theorem that it has a unique fixed point on this interval. I've also proved ...
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0answers
19 views

Suppose $S^m$ is a contraction on a complete metric space $(X,d)$. I want to show that this implies $S$ has a unique fix-point. [duplicate]

Let $(X,d)$ be a complete metric space and let $S: X \rightarrow X$ be a mapping. Suppose there exist $m \ge 1$ such that $\underbrace {S^m = S \circ S \circ \dots \circ S}_{\text {m times}}$ is a ...
0
votes
1answer
24 views

A problem about fixed points

We got a real valued continuous function $g$ which is defined on $[0,1]$, $g(0)>0$, $g(1)=1$ and $g'(x) > 0$, $g''(x) > 0$ on $(0,1)$. We need to prove that if $\lim_{x \to 1}g'(x) > 1$, ...
1
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2answers
67 views

Lotka-Volterra First Integral and Fixed Point

I have the following problem that I am dealing with, quite a long time, I must say. Let us assume that we have a predator-prey, Lotka-Volterra system given to us by: \begin{align} & ...