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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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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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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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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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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Fixed Point in a conditional expectations model

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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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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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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15 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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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19 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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275 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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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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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 ...
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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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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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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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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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$f (x)=\exp(x/2)−25x^2$. 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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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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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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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 ...
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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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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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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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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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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 ...
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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 $\lim_{x \to 1}g'(x) > 1$, ...
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63 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} & ...
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Relationship between the Borsuk-Ulam theorem, Brouwer's fixed point theorem (for the ball) and Tucker's lemma

Which of these - the Borsuk-Ulam theorem, the Brouwer's fixed point theorem (for the ball) and Tucker's lemma implies which? I'm a little confused with this. I suspect they may be equivalent. If that ...
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27 views

Fixed Point Convergence. Finding the interval for which the iteration converges.

I've solved the first part. I think I have something for the second part, but I'm unsure. A) You are given the fixed point problem x=Ax^2 where A>0 is a constant. Compute positive fixed point of the ...
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Fixed point involved ODE

Vector field $v: R^n \to R^n$ is smooth, and $x\cdot v(x)\geq 0$ when $|x|=1$. Then consider the ODE: $$\dot{x}(t)=-v(x(t)) \ \ t\geq 0 $$ $$x(0)=y$$ For $t>0$ fixed, the map $y\mapsto x(t,y)$ ...
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Existence fixed point

Let $f: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^n$ and $g:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and compact valued. Consider the function $F: \mathbb{R}^n \times ...
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19 views

Clarification of a passage in Leray-Schauder theorem's proof

I'm looking at the proof of Leroy-Scauder theorems's. This is the statements: If $X$ is a Banach space, $K \subset X$ a convex, close and bounded set, $F:K \rightarrow K$ compact then $F$ has a ...
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27 views

A sufficient condition for the existence of a fixed point for a continuous function.

How did the author use the intermediate value theorem to prove that period $k$ implies period $1$? Please, see the image which explains every thing. The definitions are in the first paragraph. The ...
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Derivative of a second-iterate map

I have a homework problem I'm working on about the discrete logistic equation: $f(x)=rx(1-x)$ So far, through some experimentation and polynomial division I've dtermined that the fixed poits of ...
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26 views

Complete metric space, fixed point and ?reverse? fixed point theorem.

Let $(X,d)$ be a complete metric space, let $F: X\rightarrow X$ such that $$\exists L > 1, \forall (x,y)\in X^2, d(F(x),F(y))>L\cdot d(x,y).$$ Show that if $F(X)=X$ then there is exactly one ...