# 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
Filter by
Sorted by
Tagged with
200 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-...
• 422
133 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. ...
• 4,443
188 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
213 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
84 views

• 441
1 vote
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
332 views

### Does a contractible locally connected continuum have an fixed point property?

I'm surprised that I can't find any research on this topic. Maybe it's too obvious? Kinoshita proved that contractible continuum do not have FPP, but his example is not locally connected. Maybe if we ...
191 views

• 121
316 views

### Converse of Knaster-Tarski's theorem as choice principle

Knaster-Tarski's theorem States that if $(A,\le)$ is a complete lattice, then every monotone function $f :A \to A$ has a fixed point. The proof is carried out in $\mathsf{ZF}$. By $\mathsf{KTC}$ we ...
• 231
207 views

• 1,252
240 views

### Measure of non-commutativity of two invertible functions

I need to estimate $|x - f^{-1}(g^{-1}(f(g(x))))|$ for various values of $x$ for two smooth invertible functions $f$ and $g$ on $\mathbb{R}$ (actually some other spaces, but $\mathbb{R}$ will do.) Are ...
• 21
324 views

### The Tarski-Lindenbaum theorem of the middle value

In Ettore Casari, La Matematica della Verita' (2006), the following theorem (attributed to Tarski and Lindenbaum 1926), is stated, which Casari calls the theorem of the middle value of Tarski-...
• 629
104 views

• 808
1 vote
160 views

### Classical fixed-point argument and invertible function

Let $n\in\mathbb{N}$ and $W^{1,\infty}(\mathbb{R}^n)=\lbrace f:\mathbb{R}^n\rightarrow \mathbb{R}^n : \text{ f is bounded and Lipschitz continuous } \rbrace$. Suppose $f\in W^{1,\infty}(\mathbb{R}^n)$ ...
• 237
I am trying to prove a convergence result on an iterative scheme which has the initial point defined as $$x_1 = \frac{1 - s(x_0)}{s(x_0)}$$ where s(x) is some unknown function. Here is my theorem and ...