### Archive

Posts Tagged ‘cauchy’

## IMC 2016 – Day 1 – Problem 3

Problem 3. Let ${n}$ be a positive integer. Also let ${a_1,a_2,...,a_n}$ and ${b_1,b_2,...,b_n}$ be reap numbers such that ${a_i+b_i >0}$ for ${i = 1,2,...,n}$. Prove that

$\displaystyle \sum_{i=1}^n \frac{a_ib_i -b_i^2}{a_i+b_i} \leq \frac{\sum_{i=1}^n a_i \cdot \sum_{i=1}^n b_i - \left(\sum_{i=1}^n b_i \right)^2 }{\sum_{i=1}^n (a_i+b_i)}.$

## Sierpinski’s Theorem for Additive Functions – Simplification

1. If ${f}$ is a solution of the Cauchy functional equation which is surjective, but not injective, then ${f}$ has the Darboux property.

2. For every solution ${f}$ of the Cauchy functional equation there exist two non-trivial solutions ${f_1,f_2}$ of the same equation, such that ${f_1}$ and ${f_2}$ have the Darboux property and ${f=f_1+f_2}$.

These two results were proven in this post. The version presented here is a simplified one, identifying exactly what we need in order to obtain the desired results.

## Miklos Schweitzer 2013 Problem 7

November 22, 2013 1 comment

Problem 7. Suppose that ${f: \Bbb{R} \rightarrow \Bbb{R}}$ is an additive function (that is ${f(x+y) = f(x)+f(y)}$ for all ${x, y \in \Bbb{R}}$) for which ${x \mapsto f(x)f(\sqrt{1-x^2})}$ is bounded of some nonempty subinterval of ${(0,1)}$. Prove that ${f}$ is continuous.

Miklos Schweitzer 2013

Categories: Algebra, Analysis, Geometry

## Cauchy Problem with two solutions

Suppose $f:\mathbb{R}^2 \to \Bbb{R}$ is continuous and $t_0, x_0 \in \Bbb{R}$. Prove that if the Cauchy Problem $\begin{cases} \dot{x}=f(t,x) \\ x(t_0)=x_0 \end{cases}$ has two distinct solutions then it has infinitely many solutions.

## Cauchy Functional Equation

We say that a function $f: \mathbb{R} \to \mathbb{R}$ satisfies the Cauchy functional equation if $f(x+y)=f(x)+f(y),\ \forall x,y \in \mathbb{R}$.
1. Prove that $f(qx)=qf(x),\ \forall q \in \mathbb{Q},\ \forall x \in \mathbb{R}$. ( There fore, $f$ is a linear application, of we consider $\mathbb{R}$ considered a vector space over $\mathbb{Q}$.)
2. The following statements are equivalent:
i) $\exists a \in \mathbb{R},\ such \ that \ f(x)=ax,\ \forall x \in \mathbb{R}$.
ii) $f$ is continuous.
iii) there exists a point $x_0$ such that $f$ is continuous in $x_0$.
iv) $f$ is non-decreasing/non-increasing.
v) $f$ is bounded on some interval.
vi) $f(x)$ is positive/negative when $x$ is positive/negative.
Solutions of the type above are called “trivial” solutions of the Cauchy functional equation.
3. Prove that there exist non-trivial solutions of the Cauchy functional equation.
4. Prove that if $f$ is a non-trivial solution of the Cauchy functional equation then $\overline{f([a,b])}=\mathbb{R}$, for any $a,b \in \mathbb{R},\ a.
5. Prove that there exist solutions of the Cauchy functional equation which have the Darboux property, which means that $f(I)$ is an interval for any interval $I\subset \mathbb{R}$.