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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2
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0answers
112 views

A matrix version of Fredholm integral equation of the second kind

Fredholm integral equation are often encountered in physics. I have to deal with the following Fredholm matrix equation \begin{equation} \big[ L (x,y) \big]_{ij} = \big[ A (x,y) \big]_{ij} + \sum_{k} \...
3
votes
3answers
149 views

Find $f(2)$ if $f$ satisfies $2f(x)-3f(\frac1x)=x^2$

The following expression is given, and we are asked to find $f(2)$. \begin{equation} 2f(x)-3f\left(\frac{1}{x}\right) =x^2 \end{equation} Does a unique and well defined answer exist? Why? and what ...
2
votes
4answers
121 views

Determine all functions (functional equation) [closed]

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the functional equation $$f(x + y) + f(z) = f(x) + f(y + z)$$ for all $x, y, z \in \mathbb{R}$.
3
votes
2answers
176 views

Find value of a functional equation

Find $f(x)$ such that $$2 f(n) + \frac{1}{3}f\left(\frac{1}{n}\right) = 12.$$ Can anybody suggest me a way to solve this kind of functional equations?
1
vote
1answer
51 views

Periodic Function With A Given Functional Equation

Let $f$ be a real valued function defined for all real numbers $x$ such that for some positive constant '$a$' the equation $f(x+a)=1/2+\sqrt{f(x)-(f(x))^2}$ holds for all $x$. Then prove that $f$ is ...
1
vote
1answer
36 views

How to solve a set of equations where the unknowns are a function and some parameters?

I'd like to know how to solve something like this: $$\begin{eqnarray} f(f(x_2)-f(x_1)) & = & 27.5\\ f(f(x_3)-f(x_1)) & = & 21.6\\ f(f(x_4)-f(x_1)) & = & 15.1\\ f(f(x_5)-f(x_1)) ...
0
votes
1answer
27 views

Checking that a defined map is satisfied by some given condition

Suppose I have a group homomorphism $f: (\mathbb{R}^{2}, +) \rightarrow (\mathbb{R}, +)$ defined by $$f(x,y)= f(\frac{x+y}{2}, \frac{x+y}{2})+f(\frac{x-y}{2}, \frac{y-x}{2}) \text{,}$$ where $x, y \in\...
0
votes
0answers
103 views

Differential equation involving composition

I've been studying Euler's method to approximate a solution to a differential equation in an algorithm class. I faced a weird differential equation in a mathematics exercise, and I wanted to know if ...
1
vote
1answer
29 views

Functional equation over $f(x) = \int_0^{ax}f(t)dt + g(x)$

Let $a\in(-1,1)$ and $g\in C^{\infty}(\mathbb{R}, \mathbb{R})$. Let $S(a, g)$ the set of all f such that : $$f(x) = \int_0^{ax}f(t)dt + g(x)$$ The first part was to show that : $$S(a, 0) = \{0_{\...
9
votes
1answer
95 views

Function preserving exponentiation [duplicate]

I'm wondering what kind of function preserves exponentiation, i.e., what is an $f$ such that $f(a^b)=f(a)^{f(b)}$?
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votes
1answer
49 views

Technique to compositive functional equation

What is function $f,g:\Bbb R^+\rightarrow\Bbb R$ sought that satisfies $$\forall x\in\Bbb N,\,f_{(r)}(x)=\underbrace{f(f(\dots(f(f(x)))\dots))}_{r \,\mathsf{times}}=2^{(\log x)^c}$$ $$\forall x\in\...
3
votes
2answers
215 views

Find all real functions $f:\mathbb {R} \to \mathbb {R}$ satisfying the equation $f(x^2+y.f(x))=x.f(x+y)$

Find all real functions $f:\mathbb {R} \to \mathbb {R}$ satisfying the equation $f(x^2+y.f(x))=x.f(x+y)$ My attempt - Clearly $f(0)=0$ Putting $x^2=x,y.f(x)=1$, we have $f(x+1)=x.f(x+y)$. Now ...
2
votes
1answer
108 views

Functional equation defined over non-negative real numbers

I'm new to this forum and I don't know how to write mathematical symbols. I have the following functional equation: $f$ defined on $[0, +\infty)$ with values in $[0, +\infty)$ $f$ is bijective and ...
2
votes
4answers
667 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 $...
0
votes
2answers
96 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})] + b[xy(x+y-\frac{1}{...
3
votes
3answers
239 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: $$f(x)f(y) = f(x+y) \quad \forall x,y \in A$$ ...
4
votes
0answers
58 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 ...
17
votes
3answers
613 views

Improvement IMO 1988 $f(f(n))=n+1987$

The following problem was given at IMO 1987. Prove that there is no function f from the set of non-negative integers into itself such that $f(f(n)) = n + 1987$ for every $n$. So I tried to ...
0
votes
1answer
86 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, \...
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vote
1answer
48 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 $h:\mathbb{Z^2}\...
7
votes
2answers
157 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 ...
1
vote
1answer
81 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 ...
4
votes
1answer
132 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
1answer
129 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 $f(0)=f'(0)=h(0)=h'(0)=...
10
votes
4answers
159 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.
1
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1answer
100 views

Role of functional equations in current panorama of pure mathematics

It seems that currently functional equations are greatly explored as a research field. I would like to know what is the importance and role of such a field in the panorama of the current development ...
2
votes
1answer
43 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 $$4f(x)^...
2
votes
3answers
310 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: ...
5
votes
2answers
342 views

Solving functional equation $f(x+y)+f(x-y)=2f(x)\cos y$?

How can I solve this functional equation, where $x,y$ are any real numbers and $f:\mathbb{R}\to \mathbb R$ is a function such that : $$f(x+y)+f(x-y)=2f(x)\cos y$$ I tried substituting $x=0$ to get $...
0
votes
1answer
42 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 number....
1
vote
1answer
73 views

Commutative version of hyper operators.

As I understand it, addition and multiplication are defined on the reals as having identity elements 0 and 1 and being commutative and associative. Multiplication is also distributive over addition. ...
16
votes
4answers
329 views

Find $f''(x)$ if $f\circ f'(x) = 4x^2 + 3$

Can you tell me the solution of this question? If: $f\circ f'(x)=4 x^2 +3$ then what is $f''(x)$? This was a question in math test which I just took yesterday. One function satisfying the ...
3
votes
0answers
62 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, $$ f(xy) = f(x) + f(y), $$ ...
5
votes
2answers
259 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
60 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?
0
votes
1answer
16 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
79 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
125 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 ...
2
votes
1answer
131 views

Geometric meaning of Cauchy functional equation

What is the geometric meaning of Cauchy's functional equation? $$f(x+y) = f(x)+f(y) \quad \forall x,y$$
3
votes
2answers
1k views

To find an extremal of the functional $\int_0^1 [(y')^2 + 12 xy] dx$

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 F}{\...
0
votes
0answers
41 views

A functional equation involving convolution

Let's say I have a smooth function $\varphi:\mathbb R \to \mathbb R$ which is a convolution kernel. I am interested in the following equation: $0=A+B~(\varphi*v)+C~v+D~v~(\varphi*v)$, where $v:\...
1
vote
1answer
25 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 $0<L&...
0
votes
1answer
20 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 ...
2
votes
1answer
66 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 ...
1
vote
1answer
53 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 $T:C[0,...
2
votes
2answers
61 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. ...
3
votes
0answers
54 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 $f(x)=0\,\,\text{for all} \, x\in [-1,1].$
12
votes
2answers
185 views

Functions satisfying $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$

Find all functions $f$ such that $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$ Let us plug in $n=1$ $f(f(1))+f(2)=3$ Since the function is from $\mathbb{N}$ to $\mathbb{N}$, $f(2)...
2
votes
0answers
41 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
49 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)) ...