Questions tagged [fixed-point-theorems]
A fixed-point theorem is a result saying that a function $F$ will have at least one fixed point (a point $x$ for which $F(x) = x$), under some conditions on $F$ that can be stated in general terms.
233
questions
-5
votes
0
answers
277
views
Is homotopy invariance of the Leray-Schauder fixed point index for compact and compactly fixed maps false?
In Theorem (3.4) page 311 of the book 'Fixed Point Theory' by Granas and Dugundji (see also the paper 'The Leray-Schauder index and the fixed point theory for arbitrary ANRs') Granas defines a Leray-...
- 331
7
votes
0
answers
138
views
Transitive groups with fixed-point free elements of prime power order
A well-known result of Fein, Kantor and Schacher says that if $G$ is a finite group which acts transitively on a set $X$, then $G$ contains an element of prime power order without fixed letters. ...
- 4,503
-4
votes
1
answer
189
views
How to express a quadratic polynomial exactly as a power series [closed]
I claim, for $\operatorname{artanh}(\rho) = \frac{1}{2} \ln\left(\frac{1+\rho}{1-\rho}\right)$, i.e., the inverse hyperbolic tangent function, the following holds approximately under assumptions given ...
- 173
3
votes
1
answer
221
views
Do these polynomials with a complex kind of ‘Vieta jumping’ exist for all $k$?
Inspired by a recent question about sequences defined by $s_{n+1}=s_n^2-s_{n-1}^2$, I started wondering whether non trivial real or complex cycles of any length $k\geqslant3$ fixed by such a sequence ...
- 13.3k
0
votes
0
answers
87
views
Generator of an analytic semigroup
Perhaps I have a naive question. My question is as follows:
When we consider a Cauchy proposition of the following form:
$$
\begin{cases}
x'(t)= -Ax(t)+ F(t,x(t)) &\text{for}\ t> 0 \\
x(0)=...
- 39
4
votes
2
answers
238
views
Teaching suggestions for Kleene fixed point theorem
I will take over two lectures from a colleague in which we discuss fixed point theory in the context of complete partial orders, and culminates in showing the Kleene fixed point theorem (see f.e. ...
- 41
0
votes
1
answer
99
views
Convex sets via fixed point equations
I have an equation of the general form
$$ X = S \cup T X $$
where $S \subset \mathbb R^n$ is a convex polytope (given by its bounding hyperplanes), $T\colon \mathbb R^n \to \mathbb R^n$ is a linear ...
- 759
1
vote
2
answers
240
views
Relationship between fixed points and inversions in permutations
Inversions in a permutation $Y$ are defined as pairs where $Y_a < Y_b$ but $a > b$, while fixed points in $Y$ are defined as elements where $Y_a = a$ (i.e., 1-cycles). Let $S_\alpha$ be the set ...
- 173
2
votes
0
answers
69
views
Is it known whether a homeomorphism close to the identity of a compact manifold with nonzero Euler characteristic necessarily has a fixed point?
I recently saw in Kirby's list of open problems that it isn't known if two commuting homeomorphisms of a compact manifold close to the identity necessarily share a common fixed point, when the ...
- 21
3
votes
0
answers
62
views
Is there a fixed point theorem that applies to $f: \sum_k x_k 10^k \mapsto \sum_k x_k!$?
Let $f:\mathbb{Z}\rightarrow \mathbb{Z}$, $f:x=\sum_k x_k 10^k \mapsto \sum_k x_k!$ where $x_k$ is the $k$-th digit of $x$ in base ten. This function came up in a Project Euler problem. The question ...
- 541
6
votes
0
answers
230
views
Are bounded groups of thin operators on Hilbert space similar to groups of unitaries?
QUESTION. Let $G$ be a group of bounded operators on $\ell^2$, satisfying $\sup_{x\in G} \lVert x\rVert <\infty$, whose elements are all of the form "identity+compact" (sometimes called &...
- 25.6k
1
vote
0
answers
174
views
Is the Poincare Birkhoff theorem valid if we change the volume form of the annulus region?
Is the Poincare-Birkhoff theorem valid if we change the volume form of the annulus region?
Note: A possible approach could be the following: Is it true to say that the answer is affirmative ...
- 452
1
vote
0
answers
53
views
Extension of averaged nonexpansiveness for mappings that are not self maps
Let $\mathcal{H}$ be a Hilbert space and let $\alpha \in (0,1)$. We say that an operator $f:\mathcal{H} \rightarrow \mathcal{H}$ is
Nonexpansive if $\|f(x)-f(y)\|_{\mathcal{H}} \le \|x - y\|_{\...
- 11
0
votes
1
answer
134
views
A contraction mapping theorem
How to use the contraction mapping theorem to prove the following result: Let $X$ and $Y$ be Banach spaces, let $a>0$, and let $$B_a=B_a\left(z_0\right)=\left\{z \in X:\left\|z-z_0\right\| \leq a\...
- 441
1
vote
0
answers
57
views
Convergence of stochastic linear recurrences
Suppose that $\zeta_t$ is a univariate, stationary stochastic process ($t\in\mathbb{N}^+$).
Let $x_0\in\mathbb{R}^n$, and let $f:\mathbb{R}\rightarrow\mathbb{R}^{n\times n}$ be a continuously ...
- 183