# 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.

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### On the existence of fixed points of a matrix iteration

Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e. all eigenvalues of $A$ have negative real part). Let $\succeq$ denote the standard partial order in the cone of positive semidefinite ...

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### Convergence of a fixed-point algorithm for a concave objective function

Let's suppose we have an objective function $\max_\limits{x} \sum_\limits{i} f_i(x_i)$ with the constraint that $\ x_i \geq 0, \sum_\limits{i} x_i = 1$.
Each function $f_i$ is continuous and ...

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55 views

### If $X$ has F.P.P, Does $(X\oplus X , \lVert .\rVert )$ have F.P.P?

If X is a Banach space with F.P.P, Does the space $(X\oplus X , \lVert .\rVert )$ have F.P.P? (F.P.P : For every closed convex bounded subset $C$ of $X$ and every nonexpansive map $T:C\to C$ there is ...

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266 views

### Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P?

The space $( \ell^2 ,\lVert \cdot \rVert _2 )$ is a Hilbert space. The space
$X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert_\infty )$ is a Banach space. Does X have fixed point property? (For any ...

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62 views

### Takahashi minimization theorem for lower pseudo-continuous functions on complete metric spaces

Takahashi minimization theorem says : Let $(X,d)$ is a complete metric space, $f:X\to \mathbb{R}\cup\{+\infty\}$ is a proper(not constantly +$\infty$) lower semi continuous function, which is bounded ...

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### Property of Fixed Point Function

Given an operator $\mathcal{T}$ that maps from a function $f: \mathbb{R}^d\rightarrow \mathbb{R}$ to another function $f': \mathbb{R}^d\rightarrow \mathbb{R}$, we are interested in the fixed point $f^*...

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113 views

### On the notion of multiplicity of a fixed point [closed]

I am trying to understand the notion of multiplicity of a fixed point of a map $f: M \to M$, say, $M$ being a smooth closed manifold, and $f$ being a smooth diffeomorphism.
There is a notion of ...

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59 views

### Betti sum, cup-length, Lusternik-Schnirelmann category, and critical points

I am trying to make some order in the notions of cup-length, sum of Betti numbers, LS category, critical points of functions.
Let $M$ be smooth closed compact manifold. We denote by Crit($M$) the ...

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139 views

### On Lefschetz theorem and sum of Betti numbers as lower bounds for fixed points

Let $M$ be a closed manifold with holomorphic cell decomposition (if it is complex), or at least with only even cohomology. In particular, its Euler characteristic is equal to the sum of its Betti ...

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### Integrand in equivariant Atiyah-Patodi-Singer index theorem

I am trying to determine the integrand in the equivariant Atiyah-Patodi-Singer index theorem by Donnelli ("Eta invariants for G-spaces") for spin manifolds if one only has isolated fixed points. In "...

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89 views

### Fixed-point properties for affine actions of topological groups

T. Mitchell [Illinois J. Math. 14 (1970) 630--641] defined four properties of a topological semigroup, and in particular of a topological group $G$.
Two of them are:
(F2) Every jointly continuous ...

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372 views

### Existence of Solution, System of Equations

Suppose $P(\lambda, i)$ is the probability that a Poisson random variable with average $\lambda$ is equal to $i$, i.e. $\frac{\lambda^i}{e^{\lambda}i!}$
I think the following system of equations ...

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239 views

### Is it possible to continuously select a probability distribution over fixed points in Brouwer's fixed point theorem?

According to Brouwer's fixed point theorem, for compact convex $K\subset\mathbb{R}^n$, every continuous map $K\rightarrow K$ has a fixed point.
However, these fixed points cannot be chosen ...

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186 views

### Question on K.Gobel's paper 1969

Let $X$ be uniformly convex Banach space. $f:K\rightarrow K$, such that $\parallel fx-fy\parallel \leq\parallel x-y\parallel\,\,\forall x,y\in K $, with $K$ a nonempty, closed, convex, bounded subset ...

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97 views

### prove the singularity of a matrix as solution of a non-linear equation

Let $B$ ($n \times n$) and $R$ ($m \times m$) be two square matrix with $n>m>0$ who satisfie:
$B=(I-KH)B(I-KH)^T+K RK^T$
with $K=BH^T(HBH^T+R)^{-1}$ and $rank(H)=m$
I would like to prove $...

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339 views

### Solution of an equation with Jacobi theta function

I have been struggling with this equation for some time and I do not seem to find any conclusive answer (it's from my research, not a homework).
It has to do with the real solutions $x$ to the ...

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189 views

### Brouwer fixed-point for open ball and bijective uniformly continuous function?

Let $B^-$ be the open unit ball in $\mathbb{R}^n$ for $n\in\mathbb{N}$, and let $f$ be a uniformly continuous function from $B^-$ to itself, with a uniformly continuous inverse $f^{-1}$.
Under these ...

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144 views

### A particular measure of noncompactness?

I am working on an article based mainly on the notion of Measure of non-compactness, to study a particular type of fixed point theorems.
Let $\mathcal M $ to be the family of all nonempty bounded
...

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125 views

### Banach spaces with unconditional basis have w-FPP

A Banach space $X$ is said to have w-FPP (weak fixed point property) if for every non-empty, weakly compact and convex subset $K\subseteq X$; every non-expansing mapping $T:K\longmapsto K$ i.e.
$$\|...

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347 views

### Base of topology for metric-like space

Let $X$ be a nonempty set and $p:X\times X\rightarrow\mathbb{R}^+ $ be a function satisfying the following conditions for all $x,y,z\in X$: \begin{align} &1)\enspace p(x,y)=0\implies x=y \\ &2)...

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### Can we internalize topological fixed point theorems in an effective topos?

Reflective oracles are a kind of Turing oracle that give stochastic answers about the outputs of Turing machines. This works in a self-referential way, where they can answer queries about Turing ...

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78 views

### Automorphisms of rationally connected varieties

Let $X$ be a smooth, rationally connected variety over an algebraically closed field of characteristic zero. Denote by $\mathrm{Aut}(X)$ the space of automorphisms of $X$ and for a given $\phi \in \...

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### Existence and uniqueness of fixed point in generalized condition of triangular norm

Definition 1) A Menger space is defined as a triple $\left( S,F,T \right)$ where $S$ is a set , $F$ is a collection of distribution functions and $T$ is a triangular norm function
$T:[ 0,1 ]\...

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63 views

### $f$ strictly decreases with a unique fixed point, under what condition $f^2$ has more than one fixed point?

Consider a strictly decreasing function $f$ from $[0,1]$ to $\mathbb{R}$. We know $f$ has a unique fixed point in $(0,1)$, denoted by $z$. Apparently, $z$ is also the fixed point of $f^2$. Does there ...

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133 views

### On fixed point probability in discrete logarithm?

Given integer $n>2$ what is the probability that for a given $h\in\Bbb Z_n$ there is no $x\in[0,\varphi(n)-1]\cap\Bbb Z$ such that $h^{x\bmod\varphi(n)}\equiv x\bmod n$?

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414 views

### Fixed point of a group action

Let $\mathbb{R}^\infty$ be the product of countably many real lines.
Assume that a finitely generated group $\Gamma$ acts on $\mathbb{R}^\infty$ (linearly and continuously) and there is a nonempty ...

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301 views

### Continuity of mapping sending a function to its (brouwer) fixed point

Let $f:[0,1]^n \rightarrow [0,1]^n$ be a continuous mapping. Brouwer's fixed point theorem says that $f$ has a fixed point, i.e., some $x$ such that $f(x) = x$.
Suppose we have a continuous family, i....

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79 views

### Formal way to prove existence and continuity in an integral equation

In the paper "A Boundary Value Problem Associated with the Second Painlevé Transcendent and the Korteweg-de-Vries Equation" by Hastings and McLeod, the authors study the ODE
$$\frac{\mathrm{d}^2 y}{\...

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353 views

### Equivariant Riemann-Hurwitz

The Riemann-Hurwitz formula starts with a genus $g$ algebraic curve $Y$ and a ramified cover $\pi\colon X\to Y$ of degree $N$, with ramification indices $e_P$ and computes invariants of $X$, such as ...

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268 views

### Is it possible to find a $\mathbb{C}$-algebra endomorphism of $\mathbb{C}[x,y]$ that fixes a given $w \in \mathbb{C}[x,y]-\mathbb{C}$?

Let $w \in \mathbb{C}[x,y]-\mathbb{C}$ and let $u \in \mathbb{C}[x,y]-\mathbb{C}[w]$.
Is it possible to find a $\mathbb{C}$-algebra endomorphism $f$ of
$\mathbb{C}[x,y]$ such that $f(w)=w$ and $...

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125 views

### Random iteration of a set of monotone maps until fixed point

Let $P$ be a poset with a least element $\bot$ ($\forall x \in P.\ \bot \le x$).
Let $M$ be a set of monotone maps $P \to P$.
Call $x \in P$ reachable if $x = f_1(f_2(...f_n(\bot)...))$ for some ...

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### Closed manifolds with the fixed point property

The real projective plane ${\bf P}^2({\bf R})$ is the only closed surface with the fixed point property: all continuous maps from the plane to itself has a fixed point. The projective spaces ${\bf P}^{...

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179 views

### How to compute the joint spectrum?

Let $(A_{1},A_{2}, \ldots,A_{k})$ be $k$ matrices in $M_{n}(\mathbb{R})$.
Is there an algebraic formula, as a generalization of "Determinant" for $k=1$, to compute the joint spectrum of ...

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354 views

### Is $\{x_n\}$ a Cauchy sequence?

Let $(X,d)$ be a complete metric space and $f$ a mapping of $X$ into itself. Let $\{f^n(x)\}=\{x_n\}$ be the sequence of iterated transforms.
Suppose $f$ satisfies that for each $\varepsilon >0$,...

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349 views

### Does a certain contractive mapping have a fixed point?

Let $f:X\rightarrow X$ be a contractive mapping of a complete metric space satisfying
$$d(f(x),f(y))\leq\alpha(d(x,y))d(x,y)$$
where $\alpha:\mathbf{R}^+\rightarrow [0,1)$, and $\alpha(t_n)\rightarrow ...

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### What are the major differences between real and complex Banach space?

Most theorems under real Banach space settings have their twin brothers for complex ones, say, the Hahn-Banach theorem. However, some theorems are not valid in complex Banach spaces, and vice versa.
...

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435 views

### Reference request: an example of Bott residue formula's usage

Could you give me an example of a clear and beautiful application of Bott residue formula in torus-equivariant cohomology (see below)?
I found an example calculating a product of Chern classes on ...

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### spiral forward orbits of analytic functions near repelling fixed points

An anonymous referee informs me that forward orbits near fixed points of analytic functions, such that the members of the forward orbits lie on spirals, are well-known. His citation for this (p. 31 of ...

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### Fixed-point iteration depending on a parameter

Let $f\colon X\times \mathbb{R}\to X, (x,\varepsilon)\mapsto y$, with $X$ open, be a continuous function in both arguments. Consider the following fixed-point iteration
\begin{align}
x_{k+1} = f(x_k,\...

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208 views

### Fixed point theorem for a nonconvex set in a Banach space

Generally speaking, I am looking for a generalization of the Schauder fixed point theorem, which applies to the situation described briefly below.
All references I read (e.g. E. Zeider 'Nonlinear ...

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305 views

### Number of fixed points in Zagier's involution (Fermat's Theorem) [closed]

Zagier's has found a famous one sentence proof for Fermat's theorem on sums of two squares. It centers on the following involution of the set $S= \lbrace (x,y,z) \in N^3: x^2+4yz=p \rbrace $ having ...

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152 views

### characterization of normality by selection theorem

The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...

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### Fixed point shape property

Question: Provide (or prove that it's not possible) a metric compact space which has the fixed point property but not the fixed point shape property.
Here is the definition of f.p.s.p.("map" means ...

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586 views

### leray schauder fixed point and schauder fixed point

I have seen these 2 fixed point theorem and I think the condition of Leray Schauder fixed point theorem is very strong and we require to consider the fixed point of $u=\sigma Tu$ $\forall \sigma \in[...

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### Proof of Lefschetz-Hopf Fixpoint Theorem with de Rham cohomology?

Looking for a proof of the Lefschetz-Hopf Fixpoint Theorem with the de Rham Cohomology.
(I´m more interestet in the Formula then just the simple statement that if the Lefschetz number is not zero ...

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### Can the following system of equations be solved analytically/in a closed form?

From a constrained non-linear maximization problem I obtained the following system of equations:
$a_1=\frac{1+a_3-\sqrt{a_2a_3}\sqrt{v_1}}{1+\sqrt{\frac{a_3}{a_2}}\sqrt{v_1}}$
$a_2=\frac{2-a_3-\sqrt{...

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265 views

### On a matrix algorithm involving rank-one projections

Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors spanning $\mathbb{R}^n$. Let $p\in [0,1]$ be a rational number and consider the iteration
\begin{equation}
X_{k+1}=\frac{1}{N}\sum_{i=1}^...

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198 views

### Fixed point of quantum operations

A quantum operation is defined as
\begin{equation}
\varepsilon(\rho)=\sum_{k}M_k\rho M_k^{\dagger}
\end{equation}
where $\varepsilon(\rho)$ takes an initial state $\rho$ to some final state $\rho'$ ...

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607 views

### Does every automorphism of a separably rationally connected variety have a fixed point?

Let $k$ be an algebraically closed field. Let $X$ be a smooth, projective variety over $k$ that is separably rationally connected, i.e., there exists a $k$-morphism $u:\mathbb{P}^1_k \to X$ such that ...

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383 views

### A property stronger than the fixed point property

Assume that $X$ is a topological space. We say that $X$ satisfies the strong fixed point property if the graph of every surjective continuous self-map intersect the graph of every continuous self-...