0
votes
2answers
23 views

Fixed Point of a Function

I am trying to answer the following question: Prove the function $f(x)=1-x^2$ has a fixed point on $[0,1]$. Find the value of this fixed point explicitly. I know how to prove that it has a fixed ...
2
votes
1answer
21 views

Strictly increasing on R

Choose correct options , more than one may be correct Let f be the function defined by $$h(x)=e^x (x-1)+x^2$$ we've : $h$ is positive on $(0,\infty)$ $h$ is negative on $(0,1)$ $h$ is ...
0
votes
1answer
39 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 ...
1
vote
1answer
62 views

Root of a function? (Proof with Banach theorem)

Given is a function $f$: $\left [ -1,1 \right ]\rightarrow \mathbb{R}$, which is continuous and differentiable. The function $g$: $\left [ -1,1 \right ]\rightarrow \left [ -2,2 \right ]$ is a ...
2
votes
1answer
44 views

Limit of a monotone function

Let $f\colon [a,b] \to[a,b]$ be a non-decreasing function in a sense that $f(x)\leq f(y)$ whenever $x\leq y$. Although there may be several fixpoints of $f$, at least one does always exist and there ...
3
votes
1answer
69 views

Fixed point in a metric space with distance at most 1

The question is: Suppose that $X$ is a complete metric space such that the distance function is at most 1, and $f:X\rightarrow X$ is such that $d(f(x),f(y))\le d(x,y)−1/2(d(f(x),f(y)))^2$. Prove that ...
4
votes
2answers
115 views

Prove the Contraction Mapping Theorem.

Prove the Contraction Mapping Theorem. Let $(X,d)$ be a complete metric space and $g : X \rightarrow X$ be a map such that $\forall x,y \in X, d(g(x), g(y)) \le \lambda d(x,y)$ for some ...
1
vote
4answers
132 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
154 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
3answers
217 views

Fixed-Point Theorem Proof

Merry Christmas everybody. Let $a,b\in\mathbb{R}$ and $a<b$. Prove that: If $f: [a,b] \rightarrow [a,b]$ is continuous, then there is a fixed-point in $f$. So basically, if f is continous I ...
3
votes
1answer
129 views

Brouwer Fixed Point Theorem via the Jordan Curve Theorem

There is a proof of the Brouwer Fixed Point Theorem via the Jordan Curve Theorem ? The Brouwer Fixed Point Theorem. Let $B=\{x\in \mathbb R^2 :∥x∥≤1\}$ be the closed unit ball in $\mathbb R^2$ . ...
1
vote
1answer
74 views

Brouwer Fixed Point Theorem $f(S^1)\subset B$

I have a question about the Brouwer Fixed Point Theorem: Theorem 1.(Brouwer Fixed Point Theorem) Let $B=\{x\in \mathbb R^2 :∥x∥≤1\}$ be the closed unit ball in $\mathbb R^2$ . Any continuous ...
1
vote
1answer
114 views

Function iteration and intervals of attraction for fixed points

I am currently studying iteration sequences and I am a bit hung up on one specific bit which involves determining intervals of attraction of fixed points. I've been given a graphical method to ...
0
votes
2answers
53 views

Number of Fixed points of an odd degree polynomial

Let $p(x)$ be a polynomial of degree $2n+1$ with real coefficients. then $p(x)$ has (I) exactly $2n+1$ fixed points (II) at least one fixed point (III) at most one fixed point (Iv) $n$ fixed ...
0
votes
1answer
71 views

Is $f : [0,\infty]\rightarrow [0,\infty]$ which is continuous and bounded has a fixed point

Question is to check if : $f : [0,\infty]\rightarrow [0,\infty]$ which is continuous and bounded has a fixed point. I have first of all considered boundedness. So, $f(x)$ should not have $x$ as ...
1
vote
0answers
49 views

Proving $\frac{k-1}{k}$ is an attractor of the logistic map $kx(1-x)$.

Consider the logistic map $f(x) = kx(1-x)$ defined on $\mathbb{R}$. We already know $\frac{k-1}{k}$ is a fixed point of $f$, but my issue is showing it's an attractor when $k \in [1,3]$. There is an ...
0
votes
1answer
37 views

How to decide if these two maps are proper?

We define a mapping $f$ of a topological space $X$ into a topological space $Y$ to be proper if the subspace $f^{-1}(C)$ is compact in $X$ whenever $C$ is a compact subspace of $Y$. Now how to ...
5
votes
1answer
142 views

No fixed points imply no periodic points

Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a smooth injective function with $\operatorname{det}[f'(x)]\not=0 $ for all $x\in\mathbb{R}^n$. Moreover assume that $f$ has no fixed points. Can $f$ have a ...
1
vote
1answer
93 views

upper semi-continuity of a multi-valued function $T$ and lower semi-continuity of $d(x,T(x))$

Let $(X,d)$ be a complete metric space, $CB(X)$ the set of closed and bounded subsets of $X$, and $T:X\rightarrow C(X)$ be a multi-valued function. How can you prove this: If $T$ is upper ...
0
votes
1answer
61 views

Fixed point and non-fixed point function

For constructing another proof I need two functions explicitly and therefore I was wondering whether there exists a function that has nowhere a fixed point and a function that (maybe depending on the ...
6
votes
0answers
148 views

A fixed point theorem [duplicate]

Let $\emptyset \not = X\subseteq \Bbb{R}^n$ be convex and compact and let $\cal{A}$ be a family of affine maps from $\Bbb{R}^n$ into $\Bbb{R}^n$ such that $X$ is invariant under each element of $\cal ...
0
votes
0answers
37 views

correctness of functional iteration and contraction proof

I need to prove: If $F$ is contractive from $[a, b]$ to $[a, b]$ and $x_{n+1} = F(x_n)$ with $x_0\in[a,b]$ then $|x_n -s| \leq C\lambda^n$ for an appropriate $C$ where $s$ is a fixed point of $F(x)$. ...
1
vote
2answers
48 views

Brouwer fixed poin in 1-D

In $\mathbb R_{+}$ (non-negative) I realize that any nonnegative valued continuous function $f\colon X \rightarrow Y$ where $X, Y \subseteq \mathbb R_{+}$ with a negative finite gradient has a fixed ...
0
votes
1answer
80 views

A smooth or analytic function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer

In A continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer, I asked about continuous function. This time, I would like to ask the ...
0
votes
2answers
58 views

A continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer

Can we find a continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ for every integer $k$ and every prime number $n$? And other points other than these are not fixed ...
1
vote
1answer
221 views

How many fixed points can a differentiable function have?

Let $f$ be a differentiable function on $\mathbb{R}$. Then could any one tell me which of the following statements are necessarily true? If $f'(x)\le r<1$ for all $x$ then $f$ has at least one ...
4
votes
1answer
84 views

Unique solution to $x^4 + 7x -1 = 0 $ on $[0,1]$ (Banach's fixed point theorem)

I want to show that $x^4 + 7x -1 = 0 $ has a unique solution on $[0,1]$. The idea is to use Banach's fixed point theorem. However, I see a problem with this as the statement of the theorem says that ...
3
votes
1answer
288 views

Application of Banach fixed-point theorem

I am looking for a function $f:[0,1]\rightarrow\mathbb R$ which satifies $\int_{0}^{1}\frac{\sin(f(t)-y)}{2}\,\mathrm dy=f(t)$ for $t\in[0,1]$. The first thing I do is to define a function ...
1
vote
2answers
116 views

Limit of a sequence of fixed points also a fixed point?

Suppose I have a continuous function $f : [0,1]^n \rightarrow [0,1]^n$ (maybe $n$ is infinite). Suppose I have a sequence $\{a_n\}_{n=1}^\infty$ of points in $[0,1]^n$ where each $a_n$ is a fixed ...
3
votes
1answer
78 views

Brouwer's fixed point theorem for infinite dimensional real space in subsystems of second order arithmetic

$\text{WKL}_0$ proves Brouwer's fixed point theorem for continuous functions on $\lbrack 0,1 \rbrack^n$, when $n$ is finite. What subsystem of second order arithmetic is needed to prove Brouwer's ...
3
votes
1answer
83 views

$x_{n+1}=\frac{2x_n+3f(x_n)}{5}$ showing $f$ has a fixed point

Let $f: \Bbb{R} \rightarrow \Bbb{R}$ be a differentiable function , and suppose that there is a constant $A<1$ such that $|f'(t)|\le A$ for all real $t$. Define a sequence $\{x_n\}$ by $ ...
2
votes
2answers
134 views

does it have unique fixed point?

$p:C[0,1]\rightarrow C[0,1]$ defined by $p(f(x))=\int_{0}^{x} (x-t)f(t)dt$, well, I am getting all constant functions are fixed points, but the answer says that it has unique fixed point. I got ...
1
vote
3answers
113 views

Let $0 \leq \alpha < 1$ and let f be a function from $\mathbb{R} $ into $\mathbb{R}$ which satisfies…

Let $0 \leq \alpha < 1$ and let f be a function from $\mathbb{R} $ into $\mathbb{R}$ which satisfies $$ | f(x) - f(y)| \leq \alpha|x-y| \; \forall x,y \in \mathbb{R}.$$ Let $a_{1} \in \mathbb{R}$ ...
0
votes
2answers
109 views

Question on fixed point

I had some trouble to approach the question above. Especially (2) and (3). I appreciate if you can help!
3
votes
1answer
282 views

Convergence of fixed-point iteration for convex function

Let $f:[0,1]\to[0,1]$ be a smooth, convex (downward) function satisfying $$ f(0)=f(1)=1,\quad \lim_{x\to 0}f'(x)=-\infty,\quad \lim_{x\to 1}f'(x)=+\infty. $$ I am confident to be able to argue that ...
1
vote
0answers
84 views

Conjecture on continuous selection of fixed points of a correspondence

I have the following conjecture to show some sort of a continuous selection of fixed points in a correspondence: Let $S$ and $\Theta $ be non-empty, compact and convex subsets of some Euclidean space ...
3
votes
2answers
251 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| ...
2
votes
2answers
91 views

Distance between fixed points of functions

I'm trying to bound the distance of fixed points of two functions assuming there's some bound on the distance between the functions. Specifically, assume $f_1, f_2:[0,1] \rightarrow [0,1]$ are two ...
1
vote
1answer
164 views

Implicit Function theorem and Bifurcation points

So let us say we have a function $\dot{x} = f(x,r)$ that has some critical point at $(x_0,r_0)$ such that $f(x_0,r_0)=0$. The question now is: when is this a bifurcation point? I understand that ...
0
votes
1answer
88 views

a fixed point theorem for rectangle in $\mathbb{R}^2$

Well, Could any one tell me how to prove this one or any reference? Let $f$ be a continuos map on $\mathbb{R}^2$, and $S$ be a rectangular region such that as the boundary of $S$ is traversed, the ...
1
vote
0answers
59 views

Fixed point of continuous function on compact metric space [duplicate]

Possible Duplicate: Prove the map has a fixed point Let $X$ be a compact metric space and $f:X\rightarrow X$ be a continuous function such that $d(f(x),f(y))<d(x,y)$ for every $x,y\in X$ ...
1
vote
0answers
123 views

Fixpoint of monotone operators

Let $X$ be some set and let $F$ be the set of all functions with a domain $X$ and a range $[0,1]$. We consider $F$ to be a partially ordered set with $f\leq g$ if and only if $f(x)\leq g(x)$ for all ...
0
votes
2answers
56 views

Applying a contraction to balls' centers increases the size of the balls' intersection?

The following statement seems clearly true, but I'm having a hard time proving it: Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. For $r\ge 0$, let $B(c,r)\equiv[c-r,c+r]$. Fix ...
4
votes
2answers
396 views

banach fixed point theorem

Let $T:X \to X$ be a map on a complete non-empty metric space. Assume that for all $x$ and $y$ in $X$, $\sum_n d(T^n(x),T^n(y))<\infty$. Then $T$ has a unique fixed point. guess: I assume that the ...
4
votes
1answer
452 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 ...
8
votes
3answers
397 views

In this case, does $\{x_n\}$ converge given that $\{x_{2m}\}$ and $\{x_{2m+1}\}$ converge?

I'm playing around with a sequence $\{x_n\}$ defined by $$ x_{n+1}=\frac{\alpha+x_n}{1+x_n}=x_n+\frac{\alpha-x_n^2}{1+x_n}. $$ Here $\alpha\gt 1$, and $x_1\gt\sqrt{\alpha}$. I'm trying to compute ...
5
votes
2answers
330 views

Banach theorem example

By Banach fixed point theorem, if a metric on a metric space $X$ is such that $d(f(x),f(y))\leq K d(x,y)$ for $K\in (0,1)$ then $f$ has one unique fixed point. Is there an example where ...
5
votes
2answers
593 views

Opposite of a contraction mapping

I am taking Real Analysis and we recently went over the Banach Fixed-point Theorem, also commonly known as the Contraction Mapping Theorem which states: If $(X,d)$ is a complete metric space, and ...
3
votes
2answers
857 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
178 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 $$(1-x^2)f^2(x,y) = x^2 g^2(x,y)$$ ...