Questions tagged [fixed-point-theorems]
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, Kleene) etc.
1,834
questions
3
votes
0
answers
29
views
Navier-Stokes, admit space of divergence free or no?
Consider the problem of Navier-Stokes
$$\frac{du}{dt}-\mu \Delta u+u \cdot \nabla u+\nabla p=f$$
such that $u(0)(x)=u_0(x)$ for $x \in \mathbb{R}^N$ in a adequate space (for example $L^p(\mathbb{R}^N)$...
0
votes
1
answer
16
views
Least fixed point and monotone mappings in complete lattice
Let $(E_1,\leq_1)$ and $(E_2,\leq_2)$ be two complete lattices. also let $f_1 : E_1 \times E_2 \rightarrow E_1$ and $f_2 : E_1 \times E_2 \rightarrow E_2$ be mappings monotonic with respect to their ...
- 3
0
votes
1
answer
59
views
Solving the functional equation $f^2(x) = 1 + x^2$ for iterated functions
I am currently conducting independent research on iterated functions and have successfully solved equations of the form $f^a + f^b + ... + f^n = c$, but I am struggling with the specific equation $$f^...
1
vote
1
answer
18
views
Given bound on difference btw fixed points of two contractions, is there a bound on difference after $k$ iterations?
Let $T, T' \in \mathbb{R}^{n \times n}$ and contractions. Assume that $J, J' \in \mathbb{R}^n$ are the respective fixed points $J' = T'J'$ and $J = TJ$.
Further assuming that the respective fixed ...
- 209
1
vote
0
answers
28
views
CLT as a special case of the Banach fixed-point theorem?
I'm wondering if the Lindeberg–Lévy CLT can be seen as a special case of the Banach fixed-point theorem. I have in mind some map $T:P\rightarrow P$ for some complete metric space of CDFs $P$ with ...
- 23
2
votes
0
answers
41
views
$(\beta_n)_{n \geq 0}$ converges uniformly to the solution of $x' = F(t,x)$, variation of Picard iteration?
This is exercise 2.7. from Differential Equations: A Dynamical Systems Approach to Theory and Practice by Marcelo Viana and José Espinar.
Let $F \colon \mathcal{U} \to \mathbb{R}^n$ be continuous and ...
- 1,066
0
votes
0
answers
36
views
Prove that mapping has a fixed point
Let $x_0 \in \mathbb{R^n}$ and $y_0 \in \mathbb{R^m}$. Given
$F:B_r(x_0)\times B_r(y_0) \mapsto \mathbb{R^m}$ a continuously
differentiable function and $x \in B_r(x_0)$ fixed. Show that the
mapping $...
- 1
0
votes
0
answers
27
views
Are median filters nonexpansive operators?
Let $m:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be the median filter associated with the family of windows defined by the rule
$$
W(i) = \{ j \in \{1,2, \dots, n\} : \max \{ 1,i-k\} \le j \le \min\{i+k,...
- 111
6
votes
1
answer
76
views
Let $g(y) := \lim\limits_{n\to \infty}x_{n}(y)$. Find $\int_{0}^{3}g(y)dy$.
Let $f(x)$ be the function on $\mathbb{R}$ defined by $f(x)\!:=\sin(\pi x/2)$. For $y$ in $\mathbb{R}$, consider the sequence $\{x_{n}(y)\}_{n\geqslant0}$ defined by $$
x_{0}(y) := y\;\;\text{ and }\;\...
- 471
0
votes
0
answers
39
views
Estimate overlap of trial state $x$ with the (unknown) eigenvector $x_0$ of a real symmetric, sparse matrix H
Suppose we have a very large, sparse, real symmetric matrix $H$ of dimension $n$ ($n > 10^7$). Being diagonalizable, it has a spectrum of eigenpairs $(\mathbf{x}_i, \lambda_i)$. The problem is the ...
- 13
2
votes
1
answer
30
views
Is the Bellman-Ford algorithm a contraction mapping?
I'm not sure this is the right community to ask this question, so feel free to move or close the discussion and point me to the right community.
I was wondering whether the Bellman-Ford (BF) algorithm ...
- 85
1
vote
0
answers
39
views
Existence and uniqueness of fixed point for increasing and strictly concave continuous functions $f:]0,+\infty[^n\to]0,+\infty[^n$
Let $I=\{x\in\mathrm{I\!R}\,|\, x>0\}$ and $n\geq 1$. Suppose $f:\,I^n\to I^n$ is continuous, strictly concave and increasing. (Herein, concavity and monotonicity are meant componentwise.)
Suppose ...
- 11
2
votes
1
answer
71
views
Topological conjugacy of dynamical systems knowing all there periodic orbits
Let's assume we have a discrete dynamical system with $M \subset \mathbb{R}^n$ and
$$x_t=f(x_{t-1}), \text{ where } f: M \rightarrow M \text{ continuous} $$
Let's further assume we know all its cyclic ...
- 417
0
votes
0
answers
15
views
Proof of Fixed Point for nonlinear systems in Pazy's Semigroups of linear operators and applications to PDEs book
In Theorem 1.2 of Pazy's book called Semigroups of linear operators and applications to PDEs, a fixed point method is used to show that the nonlinear problem has a solution. The map $F:C([t_0,T];X) \...
- 57
0
votes
2
answers
58
views
How to prove the contraction of $f: [0,1] \to \mathbb{R}, x \mapsto \frac{1}{1+x^2}$?
I am currently studying for my analysis exam by working through older exams.
One question that I am stuck on however, is the following:
"Using the Banach fixed-point theorem, prove that for $f(x) ...
- 61
1
vote
0
answers
23
views
Second order fixed-point problem
Let's say we have the following fixed-point problem: find $y\in\mathbb{R}^n$ that satisfies
$$
y = p + f(y)
$$
where $p\in\mathbb{R}^n$ is a constant and $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$. In ...
- 825
0
votes
0
answers
23
views
Example of a continuous game without a Nash equilibrium
It is well-known that pure Nash equilibria need not exist in continuous games: for e.g. consider two players both playing over $[0,1]$ with payoff functions $u_1(x,y) = -u_2(x,y) = (x-y)^2$.
However, ...
- 11
6
votes
3
answers
244
views
If $\lim_{n \rightarrow \infty}\sum_{k=0}^np_{k}=\infty$ show that $f$ has a fixed point
We have $(p_{n})_{n\geq0}$ a sequence of strictly positive real numbers and $a$ a real number and the continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the sequence $$\left(\frac{\...
- 1,011
3
votes
1
answer
100
views
How to prove this problem related to fixed point theorem?
$M$ is a bounded,convex and closed subset of Banach space $X$, $A:M\rightarrow M$ satisfies:
$$||Ax-Ay||\le ||x-y||$$
for all $x,y\in M$
Show that $\forall \varepsilon>0$,there exist $x\in M$ such ...
- 85
1
vote
0
answers
28
views
What can we know about the fixed points of the action of a strongly continuos one-parameter subgroup of unitary operators?
My motivation is physical, but my question in mathematical. In quantum mechanics, time evolution of states is the action of a one-parameter subgroup of projective-unitary transformations of a ...
- 1,798
2
votes
3
answers
95
views
Will this function always have a fixed point?
$f:[0,\infty) \to [0,\infty)$ where $|f(x)-f(y)| \le \frac{1}{2}|x-y|$
Does this function always have a fixed point?
My attempt:
The function is continuous. If it becomes differentiable then it will ...
- 1,970
0
votes
0
answers
38
views
Are the fixed points of a continuous map from a compact, convex subset of the Euclidean space to itself always isolated? If not, is there a condition?
Let $C \subset \mathbb{R}^n$ be a compact, convex subset of $\mathbb{R}^n$. Let $f: C\to C$ be a continuous map that's not the identity map (see Qiaochu Yuan's comment below). If necessary here, ...
- 2,028
0
votes
0
answers
35
views
Fixed points of the equation $x' = rx \ln(K/x)$
$$x'\ =\ r\cdot x\cdot \ln\left(\frac{ {K} }{x}\right)$$
Analyze fixed points and their stability depending on the parameters.
I found one fixed point $x = K$ and the stability of this point. If $r &...
Matys
0
votes
1
answer
30
views
If a contractive map has a fixed point, is it necessarily unique?
Is it true that if $T:X\to X$ is contractive and has a fixed point, then this fixed point is unique (even if $X$ is not complete)?
If $(X,d)$ is a complete metric space and $T:X\to X$ is a contractive ...
- 93
0
votes
0
answers
31
views
Speed of convergence of an iterative method
Problem: Let $f: \mathbb{R} \to \mathbb{R}$ be an analytical function with a simple root $\alpha$. Given an iteration
$$x_{r + 1} = x_r - \frac{f(x_r)^2}{f(x_r + f(x_r)) - f(x_r)},$$
prove that the ...
- 321
1
vote
1
answer
39
views
Find position knowing 2 points and distances to those points
I have a problem that I am trying to solve and I don't know how to approach it.
I am trying to position myself in 2d space using Bluetooth Beacons.
Basically i have my 2 beacons:
BEACON 1 at ...
- 13
0
votes
0
answers
44
views
Is the Brouwer's fixed point theorem applicable for convave functions?
In my book it says that Brouwer's fixed point theorem requires the set to be compact and convex. Yet, in one of the exercises (Let f: [0,1] -> [0,1] be continuous and concave. Show that if f(0)>...
4
votes
0
answers
39
views
Application of Kannan Fixed Point theorem to the integral equations
Let $(X,d)$ be a complete metric space. The map $T:X\longrightarrow X$ is called a Kannan type contraction if
$$d(T x, T y) ≤ α[d(x, T x) + d(y, T y)], ∀x, y ∈ X,\alpha\in[0,\frac{1}{2}).$$
Kannan's ...
- 1,250
1
vote
0
answers
37
views
Characterization of normal structure of a Banach space
I'm reading the Kirk-Goebel's book, "Topics in metric fixed point theory" and I don't get one implication of an equivalence proof. I'm talking about the Lemma 4.1. It sais as follow:
A ...
- 360
0
votes
1
answer
41
views
Choosing initial approximation and the function in Fixed point iteration method
In Numerical analysis, to solve an equation of the form $f(x)=0$ in $[a,b]$, fixed point iteration method is useful.
To this end, we can write $f(x)=0$ in the form $g(x)=x$ and try to find a fixed ...
- 23
0
votes
3
answers
42
views
Fixed points for increasing function
Let $f(x):[a,b] \to \mathbb{R}$ a real-valued function that is strictly increasing in $x$. Further, $f(b)>b$. I would like to show that the function has at most two fixed points; is that true?
...
2
votes
0
answers
46
views
Can you solve $y'(x) = y^2(x)\;\land\;y(x_0) = y_0 $ on $]-1, 1[$ using the Banach contraction theorem?
I would like to solve the following Cauchy problem:
$$
\begin{cases}
y'(x) = y^2(x)\\
y(x_0) = y_0
\end{cases}\tag 1
$$
In my opinion, using the Banach contraction theorem it can only be solved in $I:=...
user1107523
2
votes
0
answers
23
views
Comparative statics for a fixed point or functional equation
I was wondering if I could get some references regarding the following problem. I have a functional equation: $$f(x;\alpha) = (T \circ f)(x;\alpha)$$ where $x \in [0,1]$. This equation involves a ...
1
vote
0
answers
23
views
What is the fixed point (in sense of a 2D function) of Radon transformation?
Looks that if I repetitively apply radon transform on image (use the output again as the input), I always get something like enter image description here
I'm curious what kind of function is that.
- 11
2
votes
0
answers
39
views
Proving surjectivity of the exponential using Lefschetz's fixed point theorem
I've read in a few places (e.g., here), that one can prove Cartan's result on the surjectivity of the exponential map on compact Lie (throughout, assumed to be connected) groups using Lefschetz's ...
- 280
2
votes
2
answers
91
views
Does a function whose derivatives never attains values $\pm 1$ have a fixed point?
I started wondering if the following is true.
Consider a differentiable function $f:\mathbb R \to \mathbb R$. If $f'(\mathbb R) \cap \{-1,1\} = \emptyset$, then $f$ has a fixed point.
From Darboux ...
- 1,351
2
votes
3
answers
211
views
Lower bound of integration such that integral equals that bound [closed]
While doing some math I came up with the following task and I don't know how to solve it and whether it's solvable at all. Maybe anybody knows how to solve it or at least shows me the right direction. ...
- 31
1
vote
1
answer
51
views
Contractive mappings and fixed points
Hy friends,
In a metric space, we now that $d(f(x),f(y))<d(x,y)$ is not sufficient for the existence of a fixed point for $f$.
However, the results of Rakotch (A Note on Contractive Mappings, 1962) ...
- 301
0
votes
1
answer
69
views
Does $f'(x_*)=0\Rightarrow |e_{n+1}|<C|e_n|^2$ in the fixed-point iteration method?
(Multiple options could be correct!)
Q. Let $f:[0,1]\to[0,1]$ be a twice continuously differentiable function with a unique fixed-point $f(x_*)=x_*$. For a given $x_0\in [0,1],$ consider the ...
- 415
1
vote
1
answer
33
views
Proof of non-convergence of fixed point iteration
Suppose we have a function $f$ which has a fixed point $\alpha$. Moreover, $|f'(\alpha)|>1$.
I aim to prove that the fixed point iteration does not converge locally to $\alpha$ (only sequences ...
- 1,134
-2
votes
1
answer
63
views
Fixed point theorem
Define $f : \mathbb R\rightarrow \mathbb R$ by $f(x) = (3x^2+1)/(x^2+3).$ Let $f^{\circ1} = f,$ and let $f^{\circ n} = f^{\circ(n-1)} \circ f$ for all integers $n \geq 2.$ Which of the following ...
- 881
0
votes
1
answer
37
views
Fixed point of 1-Lipschitz functions on compact sets
I'd like to show that for $f:I\to I$, where $I$ is compact, with $|f(x)-f(y)|<|x-y|\,\,\forall x\neq y\in I$ there exists a unique fixed point.
My first idea was using the mean value theorem and $|...
2
votes
1
answer
45
views
Finding f(x) for Fixed Point Iteration
The question is simple, find the $f(x)$ that was used for this fixed point iteration knowing it was created using Newton's Method:
$$x_{k+1} = 2x_k - x_k^2y$$
So we just rearrange to get it into $x - ...
- 21
3
votes
0
answers
101
views
Two contractions with the same fixed point
We have $f,g: \mathbb{R} \to \mathbb{R}$ two contractions with the same fixed point and $(x_{n})_{n\geq1}$ a sequence with real numbers with the propriety that $x_{n+1}\in \{f(x_{n}),g(x_{n})\}$, for ...
- 1,011
0
votes
1
answer
39
views
Order of a numerical iteration method
Suppose that we have the definition of order $p$ of a numerical method as in the first snippet below. Now I want to prove that for a one-point iterative method this order $p$ is a positive integer. To ...
- 3,911
3
votes
1
answer
55
views
Prove the existence of a fixed point for the sum of two mappings
Let $X$ be a real valued Banach-space and $A \subseteq X$ which is bounded, closed and convex. Furthermore let $f,g,h : A \mapsto X$ be continuous with $f=g+h$ and $g(A)+h(A) \subseteq A$. Lastly $g$ ...
- 129
0
votes
0
answers
27
views
Iterative way to "exactly" compute smallest eigenvalue of AA^T such that it is positive
I have a matrix $A$ and I'm interested in its smallest singular value, i.e. the smallest eigenvalue of $B = A A^T$ (which is clearly positive).
Using common algorithms on $B$ to solve the eigenvalue ...
- 155
1
vote
1
answer
77
views
alternate proof of fixed point theorem
There is a famous application of fixed point theorem as follows :
If $f:\mathbb{R}^n\to\mathbb{R}^n$ be a continuous function such that
$\langle f(x),x\rangle\geq0$ for all $|x|=R>0$ , then $f(x_0)...
- 1,894
1
vote
0
answers
97
views
$|f(x) - f(y)| \leq c|x - y|$, with $0 < c < 1$, implies $(f^n(x_0))_n$ converges to a fixed point
Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $|f(x) - f(y)| \leq c|x - y|$, with $0 < c < 1$. Prove that for all $x_0, (f^n(x_0))_n$ converges to a unique fixed point.
Note: This ...
- 2,161
1
vote
1
answer
69
views
show that finite orbit has a periodic point [closed]
i am trying to prove that a finit orbit has a periodic point. The true statement of the exercise is:
"prove or disprove that if a point x has a finit orbit, ...
- 41