Let $X$ be a generic Banach space over $\mathbb R$ (also infinite dimensional), and let $B$ be the unit ball in $X$. I want to prove that the following proposition $B$ is a fixed-point space if and only if $\partial B$ is not a retraction of $B$ proof. $(\Rightarrow)$ If $r: B\rightarrow \partial B$ is […]

In the answering of another question in MSE I’ve dealt with the iteration of $x=b^x$ where the base $b=i$. If I reversed that iteration $x=log(x)/log(i)$ then I run into a cycle of 3 periodic fixpoints. Another complex base on the unit circle however, $b = \sqrt{0.5}(1+i)$ gives only one fixpoint. I asked myself, whether there […]

Use the Contraction Mapping Principle to show that $x=\frac19\sin\left(3x\right) + \sqrt{x}$ has exactly one solution $x\geqslant\frac{8}{9}$. I have literally no idea if this is right, please could someone check my answer? Let $T$ be defined by $Tx=\frac19\sin\left(3x\right) + \sqrt{x}$. To prove that $T$ is a contraction: $|Tx-Ty| = \left|\frac{\sin(3x)}{9} + \sqrt{x} – \frac{\sin(3y)}{9} – \sqrt{y}\right| […]

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 am aware that without the bijective property, it is not necessarily true – indeed, I have constructed a counterexample without any […]

I am familiar with the Banach Fixed Point Theorem and I have used it to prove existence and uniqueness of functional equations in Banach spaces like $C(X)$, the space of bounded continuous function $f:X\rightarrow \mathbb R$ with the sup norm $||f||=\sup_{x\in X}|f(x)|$, $X\subset \mathbb R^n$. I am trying now to show existence and, particularly, uniqueness […]

I ask my question in two parts: though the topic is similar, I would like to distinguish linear and general cases since methods may be too different while my questions are broad. Consider a space $X$ which we assume to be Banach. We define a linear operator $A:X\to X$ which is bounded: $$ \|\mathcal A\| […]

Let $n\in \mathbb N^*$, $f$ a map from $\mathbb N \cap [0,n]$ to $\mathbb N \cap [0,n]$, with : $$\forall k\in\mathbb N \cap [0,n], \max(f(k),f(k+1))\leq k \text{ or } \min(f(k),f(k+1))\geq k$$ Is it true that $\exists a \in \mathbb N\cap [0,n],f(a)=a$ ? I think I have an answer, and I allow myself to submit it […]

I was asked to prove that the functional equation $$ x(t) = \begin{cases} \frac{1}{2} x(3t) + \frac{1}{2},& 0 \leq t \leq \frac{1}{3}\\\ f(t), & \frac{1}{3} < t\leq \frac{2}{3}\\\ \frac{1}{2}x(3t – 2) – \frac{1}{2},&\frac{2}{3} < t \leq 1 \end{cases} $$ (where $f(t) $ is the equation of the straight line that passes through $(\frac{1}{3}, \frac{1}{2}x(1) + […]

Consider $X = \{ x \in \mathbb{R} \mid 1 \le x \lt \infty \}$, taken with the usual metric of the real line, and $T \colon X \to X$ defined by $x \mapsto x +x^{-1}$. Show that $|T(x) – T(y)| \lt |x-y|$ when $x \neq y$ but the mapping has no fixed points. Any help […]

For the following functions (i) find the fixed points and (ii) use the graphical analysis to determine the dynamics for all points on R: $f(x)=2sin(x)$. I have no problem finding the fixed points, but I don’t even know what the second part of the problem is asking for or even what to do. Can someone […]

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