The name "functional equation" is used for problems where the goal is to find all functions satisfying the given equation (and maybe some other conditions). So in this case, solving the equation means finding all functions fulfilling the equation. (This is different from the more common use of the ...

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3
votes
2answers
117 views

To find an extremal of a given functional

I have to find extremal of following : $\int_0^1 [(y')^2 + 12 xy] dx$ with $y(0) = 0$ and $y(1) = 1$. I applied the Euler's equation $\frac{\partial F}{\partial y} - \frac{d}{dx}(\frac{\partial ...
4
votes
1answer
434 views

non-continuous function satisfies $f(x+y)=f(x)+f(y)$

As mentioned in this link, it shows that For any $f$ on the real line $\mathbb{R}^1$, $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable. But how to show there exists such ...
3
votes
2answers
674 views

Finding all continuous functions such that $f^2(x + y) − f^2(x − y) = 4f(x)f(y)$

I've been working on the following homework problem: Find all continuous functions $f : \mathbb{R} → [0,∞)$ such that $f^2(x + y) − f^2(x − y) = 4f(x)f(y)$ for all real numbers $x, y$. The first ...
0
votes
0answers
39 views

Prove $\lim\limits_{x \to +\infty} \frac{f_1(x)}{f_2(x)} = \text{Constant}$

Let $g(x)$ be a real-analytic strictly rising function for $x>0$. Define for $x>1$ two real analytic functions $f_1,f_2$ such that : $$f_1(x) - f_1(x-1) = f_1(g(x))$$ $$f_2'(x) = f_2(g(x))$$ ...
3
votes
1answer
228 views

What is known about functions of type $f(n+1) = f(n)^ {f(n)}$?

Update : having looked at Knut's Double Arrow Notation ( Thank you DJC), it seems that this question is nothing more than a frivourless wondering that should be undertaken by whom ever wonders it, it ...
7
votes
2answers
131 views

For all $x,y\in G$ we have: $f(xf(y))=f(x)y$. Prove that $f$ is an isomorphism?

Let $G$ be a group and $f : G \to G$ a function such that for all $x,y\in G$: $$f(x f(y)) = f(x) y.$$ Prove that $f$ is an isomorphism. There are two problems here: we don't know that $f$ is a ...
2
votes
4answers
342 views

Problem in solving functional equation.

To find all functions $f$ which is a real function from $\Bbb R \to \Bbb R$ satisfying the relation $$f(x^2 + yf(x)) = xf(x+y)$$ It can be easily seen that the identity function $i.e.$ $f(x)=x$ and ...
9
votes
4answers
168 views

Solve $f\left(\frac{x-3}{x+1}\right)+f\left(\frac{x+3}{x+1}\right)=x$ $\forall x\neq -1$

Given function $y=f(x)$ such that $$f\left(\frac{x-3}{x+1}\right)+f\left(\frac{x+3}{x+1}\right)=x \quad\forall x\neq -1$$ find $f(x)$ and $f(2007)$.
4
votes
2answers
70 views

Solve the functional equation $2f(x)=f(ax)$ for some $a$.

I am trying to solve the following functional equation, and could use some help.$$ 2f(x)=f(ax)$$ For some $a\in\mathbb{R}$. By repeated adding $2f(x)$ together we notice that $$2nf(x)=f(a^nx).$$ ...
0
votes
2answers
91 views

$f(xy)=\frac{f(x)+f(y)}{x+y}$ Prove that $f$ is identically equal to $0$

For all $x,y\in\mathbb{R}$. also $f : \mathbb{R} → \mathbb{R}$ and $x+y\not=0$. My attempt: I restated it as $a[x^2 y^2 (\frac{x}{y}+\frac{y}{x}-\frac{1}{y^2}-\frac{1}{x^2})] + ...
0
votes
1answer
125 views

Solutions to functional equation $ \gamma(s,t)=f(t \cdot g(s))+h(t) $

Let $$ \gamma(s,t)=f(t \cdot g(s))+h(t) $$ where $\gamma$ is a known function of $s \in \mathbb{R}$ and $t \in \mathbb{R}$ while $f$, $g$, and $h$ are unknown functions. Assume ...
4
votes
0answers
45 views

Solution techniques for f'(x)=f(g(x))

I stumbled over this seemingly natural question and was surprised, that I couldn't find a satisfying answer. Differential equations of the type $f'(x)=g(f(x))$ are studied for all kind of classes of ...
1
vote
1answer
99 views

Are all functions on vectors in $GF(2^n)$ representable as polynomial functions?

In $GF(2)$, any function from an n-dimensional vector to a number is equal to a polynomial function of n variables. (See proof below.) Question: Is this true for other $GF$, especially $GF(2^8)$? ...
3
votes
3answers
156 views

Solving functional equation $f(x)f(y) = f(x+y)$

I'm having some trouble solving the following equation for $f: A \rightarrow B$ where $A \subseteq \mathbb{R}$ and $B \subseteq \mathbb{C}$ such as: $$f(x)f(y) = f(x+y) \quad \forall x,y \in A$$ ...
0
votes
1answer
22 views

Calculation of Sigma (Multiple Sigmas)

When I was studying Game Theory, I came across this equation: $$F_i(q_1, \ldots, q_n) = \sum_{s_1 \in S_1} \: \ldots \: \sum_{s_n \in S_N} \big\{ \prod_{j=1}^n q_j(s_j) \big\} \: \: f_i(s_1, ...
1
vote
1answer
34 views

A functional equation over integers

I was working in a problem in number theory and I blocked over the problem : Given functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$, $g:\mathbb{Z^2}\rightarrow \mathbb{Z}$ and ...
0
votes
1answer
35 views

Fixed Point Iteration $x = g(x)$ method for $y_1 = e ^{-x}$ and $y_2= \cos x$

The question reads as follows: Find the x and y coordinates of the intersection points by means of the $x = g(x)$ method. ( I believe they are referring to the Fixed Point Iteration method) The ...
18
votes
2answers
392 views
6
votes
4answers
550 views

Find a solution for $f\left(\frac{1}{x}\right)+f(x+1)=x$

Title says all. If $f$ is an analytic function on the real line, and $f\left(\dfrac{1}{x}\right)+f(x+1)=x$, what, if any, is a possible solution for $f(x)$? Additionally, what are any solutions for ...
3
votes
1answer
46 views

Difficulty with Jensen's Equation.

Its easy to find all continuous function $f: \Bbb R \to \Bbb R $ satisfing the Jensen equation $$f \left( \frac{x+y}{2}\right )=\frac{f(x)+f(y)}{2}$$ But I am finding difficulty in finding all ...
0
votes
0answers
7 views

find the distribution of 100 heart transplant patients at a low volume and high volume hospital using boxplot graph 0-40 mortality

Using a boxplot graph find the distribution show mortality rates within one year of 100 patients having heart transplants at various hospitals. The low volume hospital perform between 5 and 9 ...
-3
votes
2answers
87 views

Functional equation $f(ax)=bf(x)$ [duplicate]

What are all the solutions to the functional equation $f(ax)=bf(x)$, where $a,b>0$, and $f$ is continuous, strictly monotone and increasing, and $x$ ranges over the reals? references? proof? ...
2
votes
3answers
273 views

Solve the following functional equation

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ so that: a) $[f(x)+f(y)][f(x+2y)+f(y)]=[f(x+y)]^2+f(2y)f(y)$ b) for every real $a>b\ge 0$ we have $f(a)>f(b)$ As much as I know: ...
10
votes
4answers
130 views

Polynomial equation $f(x)f(2x^2)=f(2x^3+x)$

Find all polynomials $f(x)$, for which $f(x)f(2x^2)=f(2x^3+x)$. I have no idea how to do it.
2
votes
1answer
40 views

Functional equation extended solution

The question is Find all functions $f:R \to R$ such that $$f(x+y)f(x-y)=(f(x)+f(y))^2-4x^2f(y)$$ Taking $x=y=0$, we get $f(0)^2=4f(0)^2 \implies f(0)=0$. Now take $x=y$ which immediately gives ...
4
votes
0answers
104 views

Symmetry and the zeta function

It is an incontestable fact that symmetry is one of, if not the, most powerful tools a mathematician can have at his or her disposal. What I want to know, is if there are any good reasons as to why it ...
0
votes
1answer
32 views

Functional equation, probably involving discrete differentiating?

I want to know how to solve this problem on functions. Question: Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ satisfying $$f(x+1)-f(x)=nf(x)$$ where $\mathbb{R}$ is set of real ...
2
votes
0answers
45 views

Is the product rule for logarithms an if-and-only-if statement?

If a function $f(x)$ is proportional to $\ln x$, then we know $$ f(xy) = f(x) + f(y). $$ My question is, Is the converse true? If we know that, for an unknown function f, $$ f(xy) = f(x) + f(y), $$ ...
3
votes
2answers
124 views

Solutions of the functional equation $f(2x) = \frac{f(x)+x}{2}$

How can I solve the following functional equation? $$f(2x) = \frac{f(x)+x}{2},$$ for $x \in \mathbb{R}$ with $f$ being a continuous function.
0
votes
1answer
48 views

Solutions of the functional equation $f(x) + f(qx) = 0$

How can I find the solutions of $$f(x) + f(qx) = 0,$$ where $q \in \mathbb{Q}, q\neq1, x \in \mathbb{R}$, with $f$ being a continuous function?
3
votes
1answer
55 views

Solve functional equation $f(a)f(b)=\frac{1}{2}f(a+b)+\frac{1}{2} f(max(|b-a|,a))$

Solve functional equation $$f(a)f(b)=\frac{1}{2}f(a+b)+\frac{1}{2} f(max(|b-a|,a)),(*)$$ where the following conditions are satisfied: 1)$f:\mathbb{R}_{+}\cup \{0\}\rightarrow [0,1] ,$ 2)$f(0)=1,$ ...
0
votes
1answer
11 views

Criteria for elevating a rational FE solution to reals

I have a functional equation which I have solved for rational numbers. Now to 'elevate' my solution to reals, I use the given fact that f is continuous. I could have done the same process, if I knew ...
2
votes
3answers
50 views

Solve the funtional equation $f(xf(y)+x)=xy+f(x)$

Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ so that $f(xf(y)+x)=xy+f(x)$. If you put $x=1$ it's easy to prove that f is injective. Now putting $y=0$ you can get that $f(0)=0$. $y=\frac{-f(x)}{x}$ ...
-2
votes
1answer
102 views

Find $f(x)$ such that $f(x+y)+f(x)=2f(x-y)+2f(y)$ [closed]

Problem: Let $f(x): \mathbb{R} \to \mathbb{R}$ such that: $$f(x+y)+f(x)=2f(x-y)+2f(y) \ \ \ \forall x,y \in \mathbb{R}$$ This is a problem in my analytics exam, I can't find it if $f$ is not a ...
5
votes
1answer
44 views

Can $f(x+1) = f(x)^{\ln(x)}$ be expressed as integral transform $\int g(x,t) dt $?

Let $x$ be a real number. Can some real-analytic function $f$ that satisfies for $x>3$ :$f(x+1) = f(x)^{\ln(x)}$ be expressed by standard functions as an integral transform : $$f(x) = ...
1
vote
1answer
24 views

Is $Tf=x_0+\int_0^tf(t)(1-f)\,dt$ a Lipschitz function?

Consider the Banach space $C[0,1]$ with the uniform norm and the operator given by $(T(f))x=x_0+\int_0^xf(t)(1-f(t))\,dt$ for $x\in[0,1]$ and $x_0\geq0$. I want to show that there is a number ...
2
votes
1answer
36 views

Existence solution pendulum equation using the contraction principle

How can I prove that the differential equation $\ddot{x}(t)=-(g/\ell)\sin (x(t))$ must have a solution by using the contraction principle (by Banach). The numbers $g$ and $\ell$ are fixed constants ...
0
votes
1answer
10 views

Proving that $(Sf)(x)=1+\int_0^x t^2f(t)\,dt$ is $K$-Lipschitz

I want to show that $S:C[0,1]\to C[0,1]$ where $$(Sf)(x) = 1+\int_{0}^{x} t^2\cdot f(t) \,dt$$ is a Lipschitz map for $x\in[0,1]$. There I use the Euclidean norm on $\mathbb{R}$ and the uniform norm ...
9
votes
3answers
319 views

Is Zorn's lemma necessary to show discontinuous $f\colon {\mathbb R} \to {\mathbb R}$ satisfying $f(x+y) = f(x) + f(y)$?

A UC Berkeley prelim exam problem asked whether an additive function $f\colon {\mathbb R} \to {\mathbb R}$, i.e. satisfying $f(x + y) = f(x) + f(y)$ must be continuous. The counterexample involved ...
1
vote
1answer
49 views

Is there a $f\in C[0,1]$ such that $f(x)=\frac12 \sin (f(1-x))$

Is there a $f\in (C[0,1],\lVert\cdot\rVert_{\infty})$ such that $f(x)=\frac12 \sin (f(1-x))$? I feel like you need to apply Banach´s fixed point theorem, which means it suffices to prove that ...
2
votes
2answers
54 views

Is $T:C[0,1]\to C[0,1]: Tf(x)=\frac12 x f(x^2)+1$ a contraction?

I am a non-mathematician (a physicist) who wants to find a continuous function on $[0,1]$ that satisfies the relation $f(x)=\frac12 x f(x^2)+1$ with $0\le x\le 1$. I need it for one of my models. ...
19
votes
2answers
787 views

Which $f$ satisfy the equation $\,\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y\,$?

Find all continuous functions $f$ which satisfy the functional equation $$ f(x)\,f(y)-f(x+y)=\sin x\,\sin y, $$ for all $x,y\in\mathbb R$. I can prove that $f(n\pi)=\cos\left(n\pi\right)$ for all ...
2
votes
1answer
39 views

General Solution to Almost Riccati Like Equation

Consider the differential equation $$ y' = a_0(x) + a_1(x)y + a_2(x)\frac{1}{y}$$ I am attempting to find the general solution to this. One thing I can note is that the entire equation can be ...
3
votes
0answers
49 views

Problem in Putnum competition? [closed]

Suppose $f:\mathbb{R}\longrightarrow \mathbb{R}$ is a continuous function and $f(2x^2 -1)=2xf(x)$ for all $x\in \mathbb{R}$. Prove $f(x)=0\,\,\text{for all} \, x\in [-1,1].$
4
votes
1answer
61 views

Is distributivity sufficient to define composition?

Function Composition has the property of distributivity: $$(f\star g)\circ h = (f\circ h)\star(g\circ h)\;\forall f,g,\star \in\{+,-,\times,\div\}$$ I was wondering if these properties uniquely ...
2
votes
1answer
36 views

Is this a valid equivalence between the classes of Differential Equations?

Consider the general first order Linear Ordinary Differential Equation: $$ \frac{dy}{dx} = A(x,y) = \frac{F(x,y)}{G(x,y)}$$ This equation is characteristic equation of the Partial Differential ...
1
vote
1answer
43 views

Entire $f,g$ such that $f(f(z)) = p(g(z))$

Let $p(z)$ be a given polynomial of degree $3$. How to find nonconstant entire functions $f,g$ such that $f(f(z)) = p(g(z))$ or prove they do not exist ? I considered the related equation $f(f(z)) ...
1
vote
2answers
87 views

Continuous function with differentiation

Let $f: \mathbb{R} \to \mathbb{R} $ be a continuous function with $f(x+y) = f(x) +f(y),\forall x,y\in \mathbb{R}.$ Find $\frac {df} {dx} $, if it is exist?
2
votes
0answers
45 views

Functions that hasn't any root

we say that a function like $f:X \to X$ has root if exists a function like $g:X\to X$ that for every $x \in X$: $$f(x) = g(g(x))$$ what is a necessary and sufficient condition for $f$ that it has a ...
30
votes
7answers
1k views

How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$?

How to get $f(x)$, if we know that $f(f(x))=x^2+x$? Is there an elementary function $f(x)$ that satisfies the equation?