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0
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0answers
76 views

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 ...
0
votes
1answer
209 views

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 ...
6
votes
2answers
159 views

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 ...
6
votes
1answer
158 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 ...
6
votes
1answer
245 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 ...
2
votes
4answers
167 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: ...
13
votes
1answer
594 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 ...
2
votes
0answers
44 views

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 ...
1
vote
2answers
214 views

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 ...
3
votes
1answer
286 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 ...
7
votes
1answer
214 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 ...
4
votes
2answers
192 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 ...
1
vote
0answers
57 views

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 ...
0
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0answers
49 views

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 ...
3
votes
2answers
190 views

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 ...
2
votes
1answer
195 views

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 ...
14
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3answers
543 views

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 ...
14
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0answers
330 views

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 ...
11
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0answers
141 views

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 ...
2
votes
1answer
221 views

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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2answers
789 views

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 ...
11
votes
1answer
272 views

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
10
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0answers
335 views

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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2answers
2k views

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 ...
2
votes
2answers
359 views

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 $g$. ...
3
votes
2answers
212 views

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 + ...
3
votes
2answers
292 views

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, ...
26
votes
33answers
4k views

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 ...
5
votes
2answers
381 views

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 ...
4
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2answers
526 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 ...
4
votes
1answer
182 views

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} ...
1
vote
1answer
207 views

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 ...
6
votes
1answer
174 views

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} ...
0
votes
1answer
159 views

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, ...
0
votes
2answers
293 views

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 ...
10
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6answers
1k views

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 ...
27
votes
6answers
2k views

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 ...
2
votes
0answers
109 views

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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3answers
541 views

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 ...
5
votes
1answer
1k views

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 ...
2
votes
2answers
281 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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2answers
610 views

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 ...
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3answers
645 views

Do finite groups acting on a ball have a fixed point?

Suppose that $G$ is a finite group, acting via homeomorphisms on $B^n$, the closed $n$-dimensional ball. Does $G$ have a fixed point? A fixed point for $G$ is a point $p \in B^n$ where for all ...
4
votes
2answers
568 views

A Brouwer fixed point theorem on finite sets

I have casually almost (i.e. up to details that shoud work) proved the following discrete version of Brouwer's fixed point theorem. I should have obtained this result as a corollary of quite ...
1
vote
2answers
520 views

Does convex continuous mapping have a unique fixed point?

(Note: I have a very little knowledge in the area related to this question. (My research is more related to Combinatorics.) So, I apologize in advance if I say something wrong or trivial.) One ...
2
votes
2answers
191 views

Maximization of a matrix product by iterative methods

This might not be very difficult, but I think I may have gotten a little confused. Suppose we are given a matrix A, and would like to find the vector x of modulus 1 which maximises the product xt A x ...
4
votes
2answers
382 views

In what sense do the categorical trace and coend count fixed points?

According to the nlab, the categorical trace of a 1-endomorphism $F:C\to C$ in a 2-category is the set hom$(1_C, F)$ of global elements of $F$. If $F$ is a functor in the 2-category Cat, the ...
2
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1answer
280 views

Equilibria Exist in Compact Convex Forward-Invariant Sets

Theorem. Consider a continuous map $f : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ and suppose that the autonomous dynamical system $\dot{x} = f(x)$ has a semiflow $\varphi : ...
3
votes
2answers
663 views

Sets invariant under sections

Let $X$ be a compact in the Polish space (metric, complete, separable) and $G\subseteq X\times X$ is open. For $x\in X$ we define the section of $G$: $$ s(x) = (y\in X|\langle x,y \rangle \in ...
11
votes
1answer
576 views

Converse to Banach’s fixed point theorem for ordered fields?

Suppose $R$ is an ordered field. Call a continuous map $f: R \rightarrow R$ a contraction if there exists $r < 1$ (in $R$) such that $|f(x)-f(y)| \leq r |x-y|$ for all $x,y \in R$ (where $|x| := ...