The fixed-point-theorems tag has no wiki summary.

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### Brouwer fixed points via flow

Let $B$ be a closed ball in $n$-dimensional space, let $f$ be a "sufficiently smooth" map from $B$ to $B$, and let $p$ be a point on the boundary of $B$.
It seems to me that something like the ...

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### Fixed point relation $\ Fix\ $ for pairs of manifolds

First the classical definition: a topological space $X$ has the fixed point property (fpp) $\ \Leftarrow:\Rightarrow\ $ for every continuous $\ f : X\rightarrow X\ $ there exists $\ p\in X\ $ such ...

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### Relation between modulus of smoothness and reflexivity

Baillon proved that if $X$ is a Banach space with $\rho'_X(0)<\frac{1}{2}$, then $X$ has the fixed point property (by $\rho_X(t)$
we denote the modulus of smoothness). My questions are as ...

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**1**answer

328 views

### A question on fixed point theory

I asked this question in MSE, but I did not received any answer, so I repeat it here:
http://math.stackexchange.com/questions/858238/a-question-on-fixed-point-property
Assume that $0<k<n-1$, ...

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### Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof)

Let $m \geq 2$ and let $m'$ be its conjugate. Let $w_j$ for $j=1, ..., k$ be a basis of $H_1 \cap L^{m'}$. The task is to show that there is a $u(t) \in \text{span}(w_1, ..., w_k)=:A$ such that
...

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### Is Schauder's Conjecture Resolved?

Schauder's Conjecture: "Every continuous function, from a nonempty
compact and convex set in a (Hausdorff) topological vector space into
itself, has a fixed point." [Problem 54 in The Scottish ...

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258 views

### Fixed point problem with a monotone vector as a fixed point?

Suppose $F : [0,1]^n \to [0,1]^n$ is continuously differentiable and $0 < \frac{\partial F_1}{\partial x_i} \leq \dots \leq \frac{\partial F_n}{\partial x_i} < \beta < 1$ for all $i ...

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### Application of Schauder fixed point theorem

Let $\Omega$ an open bounded in $\mathbb{R}^n$, and let $A(x,u)$ an Caratheodory function, bounded and coercive, i.e.,
$$|A(x,u)|\leq \beta$$
$$A(x,u) \geq \alpha I,\quad \alpha > 0$$
and let
...

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### A variation of the Banach fixed-point theorem

Let $(X, d)$ be complete metric space, $q \in [0, 1)$ be a real number, and $f$ be a map that satisfies $$d(f(x), f(y)) \leq q \cdot d(x, y)$$ for all $x, y \in X$. Then, Banach fixed-point theorem ...

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### Fixed points and their continuity (2)

Yesterday I asked a question about fixed point. Here is the link.
In summary, the question was,
Let $f : I^2 \to I$ be a continuous map, where $I := [0,1]$ is the unit interval. It is a basic ...

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**1**answer

166 views

### Fixed points and their continuity

Let $f : I^2 \to I$ be a continuous map, where $I := [0,1]$ is the unit interval. It is a basic fact that for each $y\in I$, the function $x \mapsto f(x,y)$ admits a fixed point. I want to ask whether ...

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**1**answer

324 views

### What is the relation between Lefschetz fixed point theorem and Poincare-Hopf theorem on vector fields?

In Dubrovin/Fomenko/Novikov Modern geometry--Methods and applications, Part II, the (Poincare-)Hopf theorem is treated in section 15.2 (see theorem 15.2.7 on page 129), while the Lefschetz theorem on ...

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178 views

### Seemingly ill-founded recursion and the recursion theorem

The following line well-defines a family of subsets $\{S_i\}_{i\in\mathbb N}$ of $\mathbb N$:
$n\in S_i$ iff $n=2i$ or $\exists j<i$ such that $n\in S_j$.
The following line does not:
...

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703 views

### Periodic Orbit property

A topological space $X$ satisfies "Periodic orbit property", briefly POP, if for every continuous map
$f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that ...

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### A fixed point problem in infinite dimensions for monotonic algebras

Let $A$ be an algebra over $[0,1]$, whose operations are all unary monotone (increasing or decreasing) bijections, except that $A$ also includes the infimum operation over finite or countably many ...

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### Fixed point for a self-mapping on subset of C[0,1]

Let $f_1$ and $f_2$ be arbitrary self-mappings on $C([0,1])$ with $f_2 > f_1$. Define set $F = \{f \in (C[0,1])| f_1 \leq f \leq f_2 \mbox{ and } f \mbox{ is increasing}\}$. Is it true that every ...

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**1**answer

295 views

### A weak fixed point property for Grassmannian

Let $f$ be a continuous function on complex Grassmannian $G(k, 2n+1)$. Is it true to say that there is a $k$-plane $Y$ such that $Y$ has nontrivial intersection with $f(Y)$?
A motivation for ...

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**1**answer

221 views

### Infinite-dimensional hex

Suppose $n$ players take turns selecting vertices of the grid $[k]^n = \left\{0, 1, 2, \ldots, k-1\right\}^n$. Each player is assigned a pair of opposite faces of the grid, and wins the game if they ...

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203 views

### An approximate infinite-dimensional fixed point theorem

Given $\epsilon > 0$ and $f : [0, 1]^{\omega} \rightarrow [0, 1]^{\omega}$, can we find $x$ such that $x \in \textrm{Conv}\left( \left\{f(y) : ||y - x||_{\infty} < \epsilon\right\}\right)$?
In ...

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### Fixed point theorm that does not require the hemi-continuity of the set valued map?

All of the fixed point theorem I have seen (like Kakutani and Brower, Browder ) required the set valued map to be hemi-continuous (lower). Is any fixed point theorem that can assure the existence of ...

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### can we say fixed point existance of a set valued map over a compact set is homotopy invariant?

Consider two set valued maps over different compact sets as $F(\mathbf{x}):D\rightarrow\rightarrow D$, $G(\mathbf{x}):E\rightarrow\rightarrow E$ where $D,R\subset Y$. Assume there is a homotopy pair ...

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### A Fixed point Theorem that does not need the convexity of set valued map?

I am looking for a fixed point theorem for set valued maps that does not assume the set valued map should be convex valued.
Something like contractiblity or other properties can be replaced with ...

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**1**answer

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### Browder's fixed point theorem in non-Hausdorff topological vector spaces

Browder proved the following fixed point theorem in his 1968 Mathematische Annelen paper (Theorem 1):
Theorem. Let $K$ be a non-empty compact convex subset of a topological vector space $E$ (where we ...

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### fixed point property for maps of compacts

Definition. A topological space $X$ has the Fixed Point Property (FPP) if every continuous self-map $X\to X$ has a fixed point.
Question. If $X$ and $Y$ are homotopy-equivalent compact metrizable ...

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### Applications of Lawvere's fixed point theorem

Lawvere's fixed point theorem states that in a cartesian closed category, if there is a morphism $A \to X^A$ which is point-surjective (meaning that $\hom(1,A) \to \hom(1,X^A)$ is surjective), then ...

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### Fundamental groups of reduced subgroup lattices

Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a ...

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### A fixed point problem

Let $A = \lbrace (tr,1-t)\; | \; t \in [0,1], r \in \Bbb{Q}\rbrace$. Is it true that any continuous function from $A$ into $A$ has a fixed point?

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### Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem?

The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective ...

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### Does a self map from the wedge sum of two spheres have either a fixed point or a point of period 2?

Let $X$ be the wedge sum of two $2$-dimensional spheres and $f$ a continuous function from $X$ into $X$. Does $f$ have either a fixed point or a periodic point of order 2?
Thanks

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### Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...

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### Has the Fundamental Theorem of Algebra been proved using just fixed point theory?

Question:
Is there already in the literature a proof of the fundamental theorem of algebra as a consequence of Brouwer's fixed point theorem?
N.B. The original post contained superfluous ...

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### What structure has been found for functions with this relationship.

Given $f$ and $g$
$\forall x y. f(x) = f(y) \Longrightarrow f(g(x)) = f(g(y))$
Or equivalently
$ker\ f \subseteq ker\ (f \circ g)$.
Note: if $f$ is injective then this holds for any $g$.
...

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### Uniqueness of fixed points for rational transformations

Background
Let $a,b,c,d$ be nonnegative constants and consider the map $T\colon [0,1]\times[0,1] \rightarrow [0,1]\times[0,1]$ defined by
$$
T(x,y) := \left( \frac{1}{1 + ax + by}, \frac{1}{1 + cx + ...

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### Connection between codata and greatest fixed points

This is, I'm afraid, another question that MSE couldn't answer.
It's easy to see how inductively-defined data types correspond to least fixed points. Let's take the natural numbers as an example, ...

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### Fixed point theorems

It is surprising that fixed point theorems (FPTs) appear in so many different contexts throughout Mathematics: Applying Kakutani's FPT earned Nash a Nobel prize; I am aware of some uses in logic; and ...

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### Is the Binomial Expectation of Convex Function Convex in p?

Suppose $X$ has a binomial distribution with success probability $p$ and $n$ trials and let $h(\cdot)$ be a positive convex real-valued function.
Is the function $g(p)=\mathbb{E}[h(X)\ |\ p]$ convex ...

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565 views

### Fixed point theorem on graphs?

Originally posted here: http://math.stackexchange.com/questions/276167/fixed-point-theorem-on-graphs
--
I have a graph $G=(V,E)$ where to each vertex $v$ I have associated a value, $\hat{v}$ (ie I ...

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### Binomial Expectation of Convex Function

Suppose $x$ has a binomial distribution with chance $\alpha$ drawn $k$ times, and let $f(x)$ be a positive convex real valued function. I would like to evaluate
$$\frac{\partial}{\partial \alpha} ...

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### Lefschetz Fixpoint theorem for non-orientable manifolds

The classical lefschetz fixpoint theorem is stated for oriented compact manifolds $M$ and a smooth map $f:M\to M$ as follows:
the intersection number $I(\Delta, \mathrm{graph}(f))$ is equal to the ...

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### fixed point of a particular vector valued function

Hi
I have a function $F:\mathbb{R} ^ n\rightarrow \mathbb{R}^n$ for which I know there exist a unique fixed point $x ^ *$ (say). I also know that the Jacobian of $F$ at each point $x$ in $\mathbb{R} ...

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### Length of intersection of intervals

Can anyone prove this statement? It seems true, but I'm finding it tricky to give a concise proof.
Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. Define $B(c,r)\equiv[c-r,c+r]$, where $[\cdot, ...

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### fixed points of system of quadratic equations

Let $\Phi: R^n \to R^n$ satisfy
$\Phi(x)=u+Ax+Q(x)$, with $x=(x_1, x_2,\ldots, x_n) \in R^n$. $u$ is a given positive vector, $A$ non negative matrix, and $Q(x)$ quadratic mapping with
...

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### Reference request: an elementary proof of Brouwer fixed-point theorem.

One of the elementary way to prove of the Brouwer fixed-point theorem is, making it follow from the (smooth) Non-Retraction theorem. The latter is then proven by contradiction by means of a simple ...

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### Can we actually find any fixed points with Brouwer's theorem?

Background
At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is ...

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### Fixed point indices of simplicial maps

Let $K$ be a simplicial complex and let $f:K \to K$ be a simplicial map.
Assume the existence of an embedding $\iota: K \hookrightarrow \mathbb{R}^n$ of this complex into Euclidean space and note ...

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### Sperner Lemma Applications

I was always fascinated with this result. Sperner's lemma is a combinatorial result which can prove some pretty strong facts, as Brouwer fixed point theorem. I know at least another application of ...

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### Contraction mapping with no fixed point

I am interested in constructing the following "counter-example" to the Banach's fixed point theorem.
Let $K=$ {$ g\in L_1: \|g\|=1, g(\cdot)\ge0 $}.
Clearly, $K$ is not a compact and $K$ is not ...

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285 views

### a question about the isotropy subgroup of circle action on manifolds with isolated fixed point

Given a circle action on a closed, oriented smooth manifold $M^{2n}$ with isolated fixed points. My question is, does there always exist a point $p\in M$ such that the isotropy subgroup of $p$ is ...

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### Knaster Tarski theorem, example needed

http://en.wikipedia.org/wiki/Knaster%E2%80%93Tarski_theorem
Let $L$ be a complete lattice and let $f : L \to L$ be an order-preserving function. Then the set of fixed points of $f$ in $L$ is also a ...

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### algebraic proof of Atiyah-Bott fixed point formula?

Hi,
Atiyah and Bott apparently proved the following theorem:
Let $X$ be a smooth projective complex variety and $L$ a line bundle on $X$.
Let $f:X\to X$ be an automorphism of $(X,L)$ with finitely ...