0
votes
0answers
22 views

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\{ ...
1
vote
2answers
38 views

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. ...
0
votes
1answer
41 views

Proving Banach's fixed point theorem

The hint tells me how to proceed but I am stuck. I define the sequence ${z_n}$ as is stated in the hint, First off, I want to prove that $|z_{n+k} - z_n| < \epsilon$ I add $z_{n+k-1}$ and ...
2
votes
1answer
42 views

Show an iterative fixed point method does not converge

I was asked the following question: let $g(x)=\frac{30}{1+x}$. Notice that $g(5)=5$. Is there an $\epsilon >0$ such that the series $\{x_k\}_{k=0}^{\infty}$ defined by $x_{k+1}=g(x_k)$ and ...
0
votes
1answer
87 views

Prove that $\exists\ C$ such that $T: C \to C$ with $C=\overline{\text{conv}}T(C)$

I have a problem: Let $K$ be a nonempty, closed, bounded and convex subset of reflective Banach space $X$. Suppose that $$T:K \to K$$ is nonexpansive. Prove that $\exists\ C$ such that ...
0
votes
1answer
26 views

Problems finding fixed point.

$x_n=f(x_{n-1})={ax_{n-1} +e^{-x_{n-1}}\over 1+a}$ for $a>0$ Setting $f(x)=x$. The problem is while trying to find the fixed point of $x={ax +e^{-x}\over 1+a}$ I only get $x=e^{-x}$. What's the ...
1
vote
2answers
95 views

Applying the Banach's Contraction Principle

I have a problem: For a system of linear equations: $$x_i=\sum_{j=1}^{n}a_{ij}x_j+b_i,\ \ i=1,2, \ldots , n \tag 1$$ Prove that, if $$\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^2 \le q<1$$ then ...
1
vote
4answers
145 views

Fix-Point Theorem Proof.

Firstly, the assignment: Let $a,b \in\mathbb{R}$ and $a < b$. Furthermore let $f: [a,b] \rightarrow [a,b]$ be monotone increasing. Show that if $x:= \mathbf {sup}\{y \in [a,b] \| \ y ≤ f(y)\}$ ...
4
votes
1answer
180 views

Proof that the solution to cosx = x, is the limit of a recursive sequence.

So I've got this question. Exists a sequence $a_n$ such that: $$a_0 = \frac \pi4, a_n=\cos\left(a_{n-1}\right)$$ Prove that $\lim_{n\rightarrow\infty} a_n = \alpha$ Where $\alpha$ is the solution to ...
1
vote
2answers
50 views

Prove that a real variable function that satisfies certain conditions has a fixed point.

Let $f: \mathbb R \to \mathbb R$ be a differentiable function such that $f(0)=1$ and $|f'(x)|<\dfrac{1}{2}$. $i)$ Prove that there exists $x_0 \in [0,2] :f(x_0)=x_0$. $ii)$ Let ...
2
votes
1answer
69 views

Application of fixed point theorem in $R^n$

Let $A=(a_{ij}) \in \mathbb R^{n \times n}$ a matrix such that $|a_{ij}|<\frac{1}{n}$ for every $i,j$. Prove that $I-A$ is invertible. My attempt at a solution: $I-A$ is invertible $\iff$ ...
3
votes
2answers
243 views

Using the Banach Fixed Point Theorem to prove convergence of a sequence

Use the Banach fixed point theorem to show that the following sequence converges. What is the limit of this sequence? $$\left(\frac{1}{3}, \frac{1}{3+\frac{1}{3}}, ...
1
vote
1answer
99 views

Stokes' and Green's Theorem Integral Setup

a) For the vector : $$v= 4y\hat{x}+x\hat{y}+2x\hat{z}$$ evaluate, $$\int(\nabla \times v) \cdot da$$ over the hemisphere represented by the upper half plane of $$x^2 + y^2 + z^2 = a^2$$ (this is ...
5
votes
2answers
225 views

How to show something is a contraction?

If we let $X$ be a complete metric space, and let $S:X\to X$ be a map, such that $S^m$ is a contraction. We now want to show, that $S$ has a unique fixed point This is what I've thought so far: Due ...
1
vote
2answers
184 views

Apply Banach's fixed point theorem

Let $$T:f\mapsto (x\mapsto \frac{2}{5}\int_0^1 (x^2+t^5)f(t) dt + \sin(x))$$ for any $x\in[0,1]$, $f\in C([0,1])$. I want to show that that there is a uniqu $\tilde{f}$ that solves that equation ...
3
votes
2answers
271 views

Some homework questions about a Lipschitz function (cauchy sequence)

Do you want to help me with my homework? The exercise is as follows: Consider a Lipschitz function, $h:\mathbb{R}\rightarrow\mathbb{R}$, satisfying for every $x, y$: $$\left| h(x)-h(y) \right| ...
1
vote
1answer
102 views

Topology Fixed Point Theorem

suppose the wind is blowing on the surface of the earth in a constant and continuous fashion. Suppose also that at every point on the equator, the wind is blowing directly east, so the wind doesnt ...
6
votes
2answers
506 views

Proof $\lim\limits_{n \rightarrow \infty} {\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}=2$ using Banach's Fixed Point

I'd like to prove $\lim\limits_{n \rightarrow \infty} \underbrace{\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}_{n\textrm{ square roots}}=2$ using Banach's Fixed Point theorem. I think I should use the ...
2
votes
2answers
266 views

Fixed point iteration

Assume that $g$ is a continuously differentiable function and that the Fixed-Point Iteration $g(x)$ has exactly three fixed points, $-3, 1$ and $2$. Assume that $g '(-3) = 2.4$ and that FPI ...
4
votes
1answer
491 views

If $f: \mathbb R^n \to \mathbb R^n$ is a contraction, then $x-f(x)$ is a homeomorphism

I am stuck in following homework question. Let $f : \mathbb R^n \to \mathbb R^n$ be a uniform contraction and $g(x) = x - f(x)$. Investigate whether $g : \mathbb R^n \to \mathbb R^n$ is a ...
3
votes
2answers
887 views

Contraction mapping does not hold in metric space

Let $X=\mathbb{Q}\cap [1,2]$, i.e $X$ is the set of rational number between 1 and 2 inclusive. We can consider $X$ to be a metric space by endowing it with the usual distance function, i.e for $x,y ...
3
votes
1answer
181 views

Brouwer FPT and solutions to a system of equations

I am trying to solve the following problem: Let f, g be continuous positive functions $\mathbb{R}^2 \to \mathbb{R}$: show that the system of equations $$(1-x^2)f^2(x,y) = x^2 g^2(x,y)$$ ...
7
votes
1answer
232 views

Do there exist sets $A\subseteq X$ and $B\subseteq Y$ such that $f(A)=B$ and $g(Y-B)=X-A$?

This is a little exercise I've been fiddling with for a while now. Let $f\colon X\to Y$ and $g\colon Y\to X$ be functions. I want to show that there are subsets $A\subseteq X$ and $B\subseteq Y$ ...
4
votes
1answer
518 views

Continuous bijections from the open unit disc to itself - existence of fixed points

I'm wondering about the following: Let $f:D \mapsto D$ be a continuous real-valued bijection from the open unit disc $\{(x,y): x^2 + y^2 <1\}$ to itself. Does f necessarily have a fixed point? I ...