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 ...

learn more… | top users | synonyms

7
votes
0answers
62 views

Find all pairs of functions $(f,g)$, $\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$

Find all pairs of functions $(f,g)$ : $\mathbb{R} \to \mathbb{R}, g : \mathbb{R} \to \mathbb{R}$ satisfying : $$\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$$ I am really ...
7
votes
0answers
52 views

What are the densities of branches of the euclidean tree?

The Euclidean algorithm shows how all coprime pairs of positive integers can be uniquely obtained from the pair $(1,1)$ by applying the two operations $(a,b) \to (a+b,b)$ and $(a,b) \to (a,a+b)$. (or ...
6
votes
0answers
86 views

Nonlinear inhomogeneous recurrence $f(x)^2=f(x+1)+S(x)$ to find nested radical

I want to solve the recurrence relation $$f(x)^2=f(x+1)+S(x)$$ where $S(x)$ is a given polynomial. The background is to find nested radicals expressions of the form ...
6
votes
0answers
86 views

How to prove subadditive function?

Let $f: [0, \infty) \to \Bbb R$ be a function satisfying the following conditions: (1) For any $x,y \geq 0, f(x+y) \geq f(x) + f(y)$. (2) For any $x \in [0,2], f(x) \geq x^2 - x$. Prove that, for ...
6
votes
0answers
209 views

Differentiable functions satisfying $f'(f(x))=f(f'(x))$

I am wondering whether or not there is a reasonable characterization of differentiable functions $f: \mathbb{R}\to \mathbb{R}$ such that $f'(f(x))=f(f'(x))$ for each $x\in\mathbb{R}$. (Or, if you like ...
5
votes
0answers
138 views

Solution(s) to $f(x+f(y))+f(y+f(x))=2f(f(x))f(f(y))$

I would like to know the solutions of the functional equation: $$f(x+f(y))+f(y+f(x))=2f(f(x))f(f(y)), \forall x,y\in\mathbb{R}$$ where $f:\mathbb{R}\rightarrow\mathbb{R}$. I have already determined ...
4
votes
0answers
30 views

Wronskian different from zero and solutions of ODE.

Let $a_0, \ldots , a_{n-1}$ continuous functions in an interval $I$.Consider the equation $$x^{(n)} = a_{n-1}(t)x^{(n-1)}+\cdots+a_0(t)x. \tag 1$$ Let $\phi_1, \phi_2, \ldots,\phi_n$ $n$ are ...
4
votes
0answers
80 views

$f(a+b,\lambda) = f (a,\lambda) \cdot f(b,\lambda)$

Consider the equation: $f(a+b,\lambda) = f (a,\lambda) \cdot f(b,\lambda)$, for $a \geq 0$ and $b \geq 0$. Is my understanding that this simple functional equation is important in analysis. Can ...
4
votes
0answers
175 views

Vector valued contraction

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 ...
3
votes
0answers
26 views

Convergence of sum of antiderivative and derivative

This question is inspired by this question: Solutions for $ \frac{dy}{dx}=y $?. It makes me wonder if there are any function where the sum of all antiderivative and derivative converges. The ...
3
votes
0answers
72 views

Is this a field of study?

Is there a name for an equation that takes the following form? $$F(f(x),f^{-1}(x),x)=0$$ A nice example being $$f(x)-f^{-1}(x)=0$$ because the solutions of this equation are their own inverses. ...
3
votes
0answers
51 views

Could we compute $P(t^2)$?

Let $P$ be an operator such that $P(kx)=kP(x)$, $k \in \mathbb{C}$, $x$ is a variable, $P(xy)=P(xP(y))+P(P(x)y)-P(x)P(y)$, $x, y$ are variables. All variables commute. Let $P(t)=t$. Then ...
3
votes
0answers
69 views

Order of Recursion?

Define an extended algebraic function $f(a)$ as a function on $a$ that utilizes any combination of recursive extensions and inverses of sequentiation. Example: $a + 1$ , sequentiation. $a + a$, ...
3
votes
0answers
49 views

secants, exponentials, quotient structures

Trigonometric functions are . . . . . . somewhat like exponential functions. If $f$ is an exponential function, then $\displaystyle \frac{\prod_i ...
3
votes
0answers
242 views

A function $f(x)$ such that $f(f(x))=\ln(x)$

How to find a continuous function $f(x)$ for $x > 12$, such that $f(f(x))=\ln(x)$? Preferably analytic too.
3
votes
0answers
143 views

Analytic function that provide $f^2(z)=z$

I am trying to solve this problem: Does there exist a function $f(z)$, that is analytic at $E=\{x+iy :x>y\}$ and provides $f^2(z)=z$ for every $z \in \mathbb C$. I have seen a solution that ...
3
votes
0answers
210 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 ...
3
votes
0answers
77 views

Function Shape Reference

I'm wondering if their exists a visual/behavioral reference for the fundamental families of functions. I'm not a mathematician so excuse my language if I'm being overly vague. I would like to have a ...
2
votes
0answers
20 views

Correct math notation to show min and max of a list of numbers and sign?

I am working on a paper that uses a calculation from DIN 15018. The calculation is only described in text and not as a math equation. I would like to display as an actual equation if possible. The ...
2
votes
0answers
22 views

Is determining a non-constant solution to a functional inequality with polynomial arguements decidable?

So suppose we have a functional inequality with polynomial arguments in $2$ variables, $\sum_i c_i f(p_i(x,y)) \geq 0$, where $c_i$ are say integer constants and $p_i$ are polynomials, say with ...
2
votes
0answers
64 views

Creating a monotonic function

I have $n$ functions $f_i(x) \{i = 1 ,...,n\} $that does not preserve the monotonic mapping order. i.e. if $x_1 < x_2$, then in general, $f_i(x_1)$ is not less than $f_i(x_2)$ (for all $i = 1 ... ...
2
votes
0answers
51 views

Solving functional equation

Problem:find all continuous functions $f:[0,+\infty)\to [0,+\infty)$ such that $$f(y^2f(x)+x^2f(y))=xy(f(x)+f(y)),\;\forall x,y\in [0,+\infty)$$
2
votes
0answers
61 views

On a specific non-linear partial differential equation

Given an $n$-dimensional variable $\mathbf{x}\in\mathbb{R}^n$ and the functions $h_i: \mathbb{R}^n \rightarrow \mathbb{R}$, $i=1,\dots,l$, we would like to find a solution of the following equation: ...
2
votes
0answers
69 views

How prove this sequences have limts and find this value?

let $f:R\longrightarrow R$,and $f(x)=x-\dfrac{x^2}{2}$,and $$L_{1}(x)=f(x),L_{2}(x)=f(f(x)),L_{3}(x)=f(L_{2}(x)),\cdots,L_{n}(x)=f(L_{n-1}(x))$$ and let $$a_{n}=L_{n}\left(\dfrac{17}{n}\right)$$ ...
2
votes
0answers
83 views

Unicity of solution to an integral equation

Suppose we have the following integral equation $$f(x)=\int_0^\infty f(y)(x+y)e^{-x^2/2-xy}\text{d}y$$ where $f: [0,\infty)\rightarrow[0,\infty)$. The function $f(x)=Ce^{-x^2/2}$ solves the equation. ...
2
votes
0answers
26 views

A question about scaling

One wants the function $\Delta ^2$ to be such that, $\Delta^2(k,\tau) = \Delta^2(\frac{k}{\lambda ^{\frac{4}{n+3}}}, \lambda \tau )$. Now from this how does this follow that, the following holds, ...
2
votes
0answers
65 views

How to solve the functional equation : $T(n)=(\log n)T(\log n)+n$

I want to solve the following functional equation using any ways: $$T(n)=(\log n)T(\log n)+n$$
2
votes
0answers
24 views

Multiple integrals satisfying functional equations

If $f:\mathbb{R}_+\to\mathbb{R}_+$ is a positive function which is locally $L^1$ with $\int_0^t{f(s)ds}=at^n$ for all $t\geq0$ then, by differentiating, it follows that $f(t)=a$ if $n=1$ and $f=0=a$ ...
2
votes
0answers
100 views

Convexity conditions for $f$ and $\dfrac {1} {f}$

Let $f:\mathbb{R} _{+}\rightarrow \mathbb{R} _{+}$ be a real function. Find all conditions on $f$ under which $f$ is convex on $\left( a;b\right) \subset \mathbb{R} _{+}$$\Leftrightarrow$ $\dfrac ...
2
votes
0answers
73 views

Indeterminacy in second degree equations of Spencer-Brown's “Laws of Form”

I am trying to follow G. Spencer-Brown's argumentation regarding 'Equations of the Second Degree' in the Laws of Form. He shows how infinitely growing self-referential forms can be created by using a ...
2
votes
0answers
55 views

Matrix functional equation

Could someone give me some hint about a possible method to find the function $f$ which solve this equation: $$f(H^2)=\alpha f(H)$$ where $\alpha$ a constant with $\alpha \in C$ and: ...
1
vote
0answers
14 views

Need a equation that defines a certain number

Im programming a function but I just cant structure the equation. I think this is the right place to ask since the problem is completely mathematics. Let me explain three scenarios. There are 4 ...
1
vote
0answers
39 views

General Solution to Linear Difference Equation: Is this correct

Notation: Let $$Q_1[f(x)] = \lim _{h \rightarrow 1} \frac{f(x + h) - f(x) }{h}$$ And let $$Q_1^{-1}\left[f(x)\right] = G(x)|Q_1\left[g(x)\right] = f(x)$$ Consider the equation $$a_0(x) + ...
1
vote
0answers
20 views

Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which ...
1
vote
0answers
61 views

Are even Dirichlet series constants?

I consider a Dirichlet series with an absolute abscissa of convergence $\sigma$ that can be meromorphically extend to $\mathbb{C}$: $$\phi(s)=\sum_{n=1}^{+\infty}{\frac{a_n}{n^s}}$$ and the analytic ...
1
vote
0answers
15 views

Implementing Equation on current data

I am trying to implement Personality, Gender, and Age in the Language of Social Media equation. I have 5 patterns and one list of 100 text = 900 words. The result of find a Match in the 900 to the ...
1
vote
0answers
22 views

What is the function $f$ verifying : $f(\frac{x}{2}+\frac{x}{2}\cos(\frac{v\pi}{x}))=\frac{x}{2}\sin(\frac{v\pi}{x})$

What are the solutions to the functional equality (for a constant $v$): $$ \forall\, x > 0, \ \ \ \ ...
1
vote
0answers
18 views

I want to solve these inequalities with respect to $f$

Let $f\colon\mathbb C\to\mathbb C$ be an entire non constant function. We consider its values on the real line. The function $f$ has infinitely many real zeros and there is infinitely many real ...
1
vote
0answers
43 views

Proof that $\forall x \in \mathbb{R}, \forall n \in \mathbb{N}: (f(x))^n = f(nx)$ only has solutions of the form $f(x) = c^x$ for some constant $c$

I am looking for a proof (or a counterexample I suppose) that the recurrence equation (is that the right term here?) $(f(x))^n = f(nx)$ only has a solution of the form $ f(x)=c^x$ for some ...
1
vote
0answers
33 views

Can a linear solution make a non-linear (functional) differential equation linear?

I was inspired by this question Does a non-trivial solution exist for $f'(x)=f(f(x))$? And tried coming up with similar problems, one interesting case I found was $f'(x) +f(x)=f(f(x))$ which has ...
1
vote
0answers
23 views

Functional equation on a ring

Let $R$ be a ring with identity. Find all functions $f: R \to \Bbb R$ satisfying the functional equation \begin{equation} f(xp+yq, yp+xq)=f(x, y)f(p, q) \end{equation} for all $x, y, p, q\in R$, and ...
1
vote
0answers
28 views

The uniqueness of a solution of a system of differential equations

Suppose I have a system of delay differential equations $\frac{d G_1(\phi_1(s))}{ds}= F_1(\phi_1(s),\phi_2(1-s),1-s)$ $\frac{d G_2(\phi_2(1-s))}{ds}= F_2(\phi_1(s),\phi_2(1-s),s)$ where $G_1, G_2, ...
1
vote
0answers
28 views

I tried this problem in the following way; is it right?

Let, f(x) be a twice differentiable function defined on (-1, 1) and f(0) = 1. Let, f(x) ≥ 0, f'(x) ≤ 0 and f''(x) ≤ f(x) for all x ≥ 0. Show that, f'(0) ≥ -√2. I am telling you what I did. First, ...
1
vote
0answers
57 views

Another functional equation

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
1
vote
0answers
42 views

Can the Euler-Lagrange equations be derived from a variation over a time of order $dt$ rather than $t$?

In the calculus of variations, the solution of the Euler-Lagrange equations gives those functions for which a given functional is stationary. Now all derivations I've come across up to now, carry out ...
1
vote
0answers
104 views

Galois Theory for Differential Equations?

Consider the set of algebraically primitive functions: that is the set of functions who can be created through some recursive expression involving arithmetic operators. Example $$y = x +1$$ $$y = ...
1
vote
0answers
33 views

Solution of equation with special function using maths

I have the following equation with β∈[0,1] and δ∈[0,1] are 2 parameters and $\mu$, $\sigma$ are the mean and standard deviation of a random variable. Is it possible to use software to get the explicit ...
1
vote
0answers
49 views

Differential Equation for CDF

Consider the following differential equation $$F(cx) = F(x) + x F'(x)$$ for $c>1$. Does this differential equation belong to a some well known class? Is there a way to find all the solutions ...
1
vote
0answers
40 views

regarding a set of integro-PDE

Let say I have a minimal example that contains most of the mathematics I am looking for. The example is as follows: \begin{align} &\frac{\partial u^0}{\partial t}(x,t)+\frac{\partial^2 ...
1
vote
0answers
96 views

solving a functional equation using given values

Moderator Note: This is a current contest question on Brilliant.org. The current contest ends on 13 October 2013, after which time this question will be unlocked. A Function $f$ from the ...