0
votes
0answers
15 views

Convex Feasibility problem on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
0
votes
0answers
17 views

Projection and Pseudocontraction on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
4
votes
2answers
114 views

Prove the Contraction Mapping Theorem.

Prove the Contraction Mapping Theorem. Let $(X,d)$ be a complete metric space and $g : X \rightarrow X$ be a map such that $\forall x,y \in X, d(g(x), g(y)) \le \lambda d(x,y)$ for some ...
0
votes
0answers
17 views

The number of periodic orbits

Let $\{a_n\}_{n=2}^\infty$ be a sequence of nonnegative integers. How to construct a continuous map $f$ on the 2-dimensional closed disk such that the number of $n$- periodic orbits is $a_n$? So far, ...
6
votes
0answers
107 views

Fixed point of a shrinking map. Proof.

Could you please verify my proof? Let $f:X \to X$ be a shrinking map on a compact metric space $(X,d)$. In other words: $d(f(x),f(y)) < d(x,y)$ for all $x,y \in X$. Prove: $f$ has a unique ...
0
votes
0answers
73 views

Application of fixed point theorem for an integral equation.

Consider the following non-linear integral equation: $f(x)=\lambda \int_a^b K(x,y;f(y))dy + \phi(x)$ with $K$ and $\phi$ continuous functions , such that $K$ satisfies the Lipschitz condition on the ...
4
votes
0answers
97 views

Showing a sequence $x_{n+1} = Tx_n$ forms a contraction mapping

I want to show that the sequence given by $$x_{n+1} = Tx_n = x_n-\frac{(x_n^2-2)}{x_n+x_{n-1}}$$ forms a contraction mapping. That is $$|Tx_1-Tx_2|\leq c|x_1-x_2|.$$ Where $c$ is to be determined. I ...
1
vote
0answers
69 views

Importance of Banach fixed point in mathematics

I know the proof of Banach fixed point theorem in complete metric space and two example of it. It is useful to prove Picard's theorem on existence and uniqueness of ordinary differential equation. In ...
0
votes
1answer
37 views

How to decide if these two maps are proper?

We define a mapping $f$ of a topological space $X$ into a topological space $Y$ to be proper if the subspace $f^{-1}(C)$ is compact in $X$ whenever $C$ is a compact subspace of $Y$. Now how to ...
2
votes
2answers
509 views

Contradict the Contraction Mapping Theorem

I am trying to show that the function $f(x) = 2\pi+x-\tan^{-1}x$ is contractive but has no fixed points. Finally I wish to conclude that it does not contradict the contraction mapping theorem. $f$ is ...
1
vote
1answer
93 views

upper semi-continuity of a multi-valued function $T$ and lower semi-continuity of $d(x,T(x))$

Let $(X,d)$ be a complete metric space, $CB(X)$ the set of closed and bounded subsets of $X$, and $T:X\rightarrow C(X)$ be a multi-valued function. How can you prove this: If $T$ is upper ...
1
vote
2answers
48 views

Brouwer fixed poin in 1-D

In $\mathbb R_{+}$ (non-negative) I realize that any nonnegative valued continuous function $f\colon X \rightarrow Y$ where $X, Y \subseteq \mathbb R_{+}$ with a negative finite gradient has a fixed ...
3
votes
1answer
91 views

Show that sequence approaches fixed point of a function

Problem Let $f(x)$ be a differentiable function on $\Bbb R$ with $\left|\,f ' (x)\right| \leq r < 1$, where $r$ is constant. Then consider the sequence $\{x_n\}$ such that $x_1 = 0$, $x_{n+1} = ...
1
vote
1answer
31 views

Is there any space with normal structure but not uniform normal structure?

It is clear that uniformly normal structure implies normal structure. But, is there any space that has normal structure but it doesnt have uniformly normal structure?? Would you mind to give me some ...
1
vote
2answers
131 views

Showing a contraction without a fixed point

Suppose $f: [1, \infty) \to [1, \infty]$ defined by $f(x) = x + \frac{1}{x}$ for all $x \geq 1$. I want to prove that: \begin{equation} |f(x)-f(y)| < |x-y| \end{equation} except when $x=y$, but ...
3
votes
1answer
147 views

Understanding the Banach fixed point theorem

The Banach fixed point theorem is stated in my book (Applied Asymptotic Analysis by Miller) as Let $\mathcal B$ be a Banach space with norm $\|\cdot\|$. Let $X$ be a nonempty bounded subset of ...
1
vote
1answer
132 views

The wedge sum of two circles has fixed point property?

The wedge sum of two circles has fixed point property? I'm trying to find a continuous map from the wedge sum to itself, that this property fails, I couldn't find it, I need help. Thanks
1
vote
1answer
97 views

about fixed point set?

let $K$ be a closed convex subset of a normed space $V$. For any $f: K \to K$ define the fixed-point set of $f$ as follows: $fix(f)=\{x$ belongs to $K$ $|f(x)=x \}$. I have to show that a nonempty ...
0
votes
1answer
49 views

IVP- Has at most one solution

Suppose $f(t, x)$ is nonincreasing with respect to $x$ for all $t \geq 0$ and $x \in \mathbb{R}$. Prove that the IVP problem $$ \left\{ \begin{array}{l} x'(t)=f(t,x) \\ x(t_0)=x_0 \end{array} \right. ...
7
votes
2answers
316 views

Continuous Brouwer's fixed point theorem via Stokes's theorem?

Let $B$ denote the closed unit ball in $\mathbf{R}^n$. Brouwer's fixed point theorem states that every continuous map $f:B\to B$ has a fixed point. There is a simple proof using Stokes's theorem, at ...
1
vote
0answers
59 views

Fixed point of continuous function on compact metric space [duplicate]

Possible Duplicate: Prove the map has a fixed point Let $X$ be a compact metric space and $f:X\rightarrow X$ be a continuous function such that $d(f(x),f(y))<d(x,y)$ for every $x,y\in X$ ...
5
votes
3answers
641 views

Continuous function on unit circle has fixed point

The question I have is: Let $f: S^1 \rightarrow S^1$ be a continuous function, where $S^1$ is the unit circle. Prove that if $f$ is not onto, then $f$ must have a fixed point.
2
votes
2answers
433 views

Analogue to Fixed Point Theorem for Compact metric spaces

If $\omega:H \rightarrow H$ (into) is continuous and $H$ is compact, does $\omega$ fix any of its subsets other than the empty set?
7
votes
1answer
273 views

Minimax Theorems V.S. Fixed Point Theorems.

Is there any relationship between the minimax theorems $$ \mbox{Regularly hypothesis on } f:U\times V\to\mathbb{R} \implies \min_{u}\max_{v}f(u,v)=\max_{v}\min_{u}f(u,v) $$ and fixed point ...
1
vote
0answers
62 views

Fixed Point Theorem for Set-to-Set Mappings

Is there a fixed-point theorem regarding mappings of the form $T:2^S\to 2^S$, i.e. mappings that map subsets of $S$ to other subsets in $S$: $$ T:S\supseteq A \mapsto T(A)\subseteq S $$ Where $S$ ...
1
vote
1answer
225 views

A corollary of Banach's fixed-point theorem

Can someone give me an idea how to generalize Banach's fixed-point theorem for complete metric spaces such that the constant contraction coefficient $c$ (as in $d(Tx,Ty)\leq c \ d(x,y)$ ) may be ...
3
votes
1answer
178 views

Brouwer FPT and solutions to a system of equations

I am trying to solve the following problem: Let f, g be continuous positive functions $\mathbb{R}^2 \to \mathbb{R}$: show that the system of equations $$(1-x^2)f^2(x,y) = x^2 g^2(x,y)$$ ...
4
votes
1answer
494 views

Continuous bijections from the open unit disc to itself - existence of fixed points

I'm wondering about the following: Let $f:D \mapsto D$ be a continuous real-valued bijection from the open unit disc $\{(x,y): x^2 + y^2 <1\}$ to itself. Does f necessarily have a fixed point? I ...