# Tagged Questions

1answer
37 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 ...
2answers
43 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 ...
1answer
52 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 ...
1answer
58 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 ...
1answer
73 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 ...
2answers
54 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 ...
1answer
68 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 ...
1answer
63 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 \$ ...
2answers
73 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 ...
3answers
99 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 $a_{1} \in \mathbb{R}$ ...
2answers
77 views

### Question on fixed point

I had some trouble to approach the question above. Especially (2) and (3). I appreciate if you can help!
1answer
138 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 ...
0answers
44 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 ...
2answers
154 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| ...
2answers
64 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 ...
1answer
101 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 ...
1answer
74 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 ...
0answers
44 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 $x,y\in X$ ...
0answers
98 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 ...
2answers
44 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 ...
2answers
308 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 ...
1answer
329 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 ...
3answers
329 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 ...
2answers
200 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 ...
2answers
293 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 ...
2answers
452 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 ...
1answer
160 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 $$(1-x^2)f^2(x,y) = x^2 g^2(x,y)$$ ...
1answer
383 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 ...
2answers
907 views

### Prove Fixed Point Theorem using the Mean Value Theorem

Assume $f$ has a finite derivative and $|f'(x)| \leq y < 1$ for all $x \in (a,b)$ $f$ is continuous and $a \leq f(x) \leq b$ for all $x \in [a,b]$. Prove $f$ has a unique fixed point in ...
3answers
303 views

### Finding the fixed points of a contraction

Banach's fixed point theorem gives us a sufficient condition for a function in a complete metric space to have a fixed point, namely it needs be a contraction. I'm interested in how to calculate the ...
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