# Questions tagged [fixed-point-theorems]

Fixed-point theorem is a result about existence of fixed points, i.e. points fulfilling $F(x)=x$, under some conditions on the function $F$. Results of this type appear in many areas of mathematics, e.g. functional analysis (Banach), algebraic topology (Brouwer), lattice theory (Knaster-Tarski, Kleene) etc.

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### Using Banach's Fixed Point Theorem on an Integral Equation

I have been studying the solutions $f(x)$, $x \in [0,1]$ to integral equations of the form $$f(x) = \int_{0}^{1}K(x,y)\frac{f(y)}{\sqrt{h^{2}+(f(y))^{2}}}dy$$ where the ...
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### Prove that in a given coloring of a square, there exists a sub-square with a certain coloring.

I was given the following question: There is a square which is divided to sub-squares by edges which are parallel to the edges of the big square (vertices are connected by the edges). The vertices of ...
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### Bivariate fixed points

Let $f(x):\mathbb{R}\rightarrow\mathbb{R}$, be a strictly decreasing, convex and continuous function in $x$ with $f(x)>0, \forall x>0$, with $\underset{x\rightarrow\infty}{\textrm{Lim}} f(x)=0$. ...
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### Showing that $\mathbf{X}^{2} + \mathbf{X} = \mathbf{A}$ has a solution

Show that there exists some $\epsilon >0$ s.t. for all $\mathbf{A}\in \mathbb{R}^{2\times 2}$ with $|( \mathbf{A})_{i, j}| < \epsilon$ for all $i, j$ (let the space of all such matrices be $E$)...
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### Does there exist a continuous function which satisfies Kannan contraction but not Banach contraction?

If possible, give an example of a continuous function defined on a convex subset of a Banach space $X$ satisfies Kannan contraction but does not satisfy Banach contraction. Definitions. Let $C$ be ...
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### How can I prove that the next sequence converges? [closed]

Let $f:[a,b]\to[a,b]$ such that is continuously differentiable, $f^{-1}$ exists and $$\underbrace{\min\text{ }x}_{x\in[a,b]} |f'(x)|>1.$$ How can I prove that the sequence $x_{k+1}=f^{-1}(x_k)$ ...
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### Fixed-Point iteration method fails on converging on equation.

I'm looking to solve the following equation in $x$. $$\frac{Wa}{b} = \left((\frac{a}{b}+x)\Phi(\frac{a}{b}+x)+\phi(\frac{a}{b}+x)\right)-\left(x\Phi(x)+\phi(x)\right)$$ , where $W, a$ and $b$ are ...
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### Is there a constructive proof of Brouwer's fixed-point theorem that does not rely on triangulation?

I'm aware of the constructive proof of Brouwer's fixed-point theorem via Sperner's Lemma, and I love it for its simplicity, directness, and constructiveness. However, I still have a lingering ...
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### Quadratically convergent procedure so that the associated procedure function is without division

Hey I'm having trouble solving the following exercise. Do you have a solution? Let $a>0$ and $1/a > δ > 0$. Give a locally quadratic convergent procedure for the determination of $1/a$ on the ...
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### Goedel's Fixed Point Theorem

Let's consider an arithmetic theory such as Peano Arithmetic. Suppose the Gödel codes of all the true sentences are even and all the false sentences are odd. Then, how can there be a fixed point for ...
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### Can you disprove this counterexample to the diagonal lemma?

I was looking at the Diagonal Lemma or Fix point theorem which states in every Theory $T$ every formula with one variable $B(n)$ has a fix point: $T \vdash G \leftrightarrow B(\# G)$. Where $\#F$ ...
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I have the following formula: $$\lambda = \lambda(1-F(S-2)) + \frac{r}{c}p$$, where $S, p, r$ and $c$ are constants and $F(.)$ is the CDF function of a random (Poisson distributed) variable, so \$F(S-2)...