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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6
votes
3answers
133 views

Finding a unique continuous function

Let $f$ be a given continuous function on $[0,1]$. How do you prove that there is a unique continuous function $g$ on $[0,1]$ satisfying $$g(x) = \frac{1}{2}g\left(\frac{x+1}{2}\right) + f(x)$$ for ...
1
vote
0answers
26 views

Do I need to verify solutions to functional equations?

I am studying the (basics of) solving functional equations. My teacher stipulates that we check any solutions obtained by substitution. Similar guidelines are given in this IMO training material. For ...
0
votes
0answers
56 views

The functional equation $ f(x-f(y))=f(f(y))+xf(y)+f(x)-1$

I came across the functional equation: $f(x-f(y))=f(f(y))+xf(y)+f(x)-1$ So far I tried plugging $x=f(y)$ and got $f(x)=\frac{f(0)-x^2+1}{2}$ which holds for every $x = f(y)$. I suppose that $f(0)=1$ ...
0
votes
1answer
47 views

d'Alembert functioal equation: $f(x+y)+f(x-y)=2f(x)f(y)$

The d'Alembert functioal equation is: $$f(x+y)+f(x-y)=2f(x)f(y)\tag0$$ This equation plays a central role in determining the sum of two vectors in Euclidean and non-Euclidean geometries. Is there a ...
0
votes
1answer
73 views

The functional equation $f(f(x)+xf(y))=xf(y+1)$

I'm trying to solve the functional equation $f(f(x)+xf(y))=xf(y+1)$. Up to now I found that $f(f(0))=0$ when $x=0$ and that $f(y+1)=f(f(y)+f(1))$ by setting $x=1$. Also $f(x)=x$ is an apparent ...
-1
votes
4answers
56 views

How can one show that $f(n)>n$

Given a function $f\colon \Bbb N^\ast\to \Bbb N^\ast$ such that we have $f(f(n))=f(n+1)-f(n)~\forall~n\in\Bbb N^\ast$. Show that: If $f(n+1)-f(n)>1$ then $f(n)>n$. Since $f(n+1)>f(n)+1$ I ...
7
votes
5answers
155 views

Find all functions $f$ such that $f(x-f(y)) = f(f(x)) - f(y) - 1$

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $f(x-f(y)) = f(f(x)) - f(y) - 1$. So far, I've managed to prove that if $f$ is linear, then either $f(x) = x + 1$ or $f(x) = -1$ must be ...
4
votes
2answers
150 views

Functional equation $f(xy)=f(x)+f(y)$ and continuity

Prove that if $f:(0,\infty)→\mathbb{R}$ satisfying $f(xy)=f(x)+f(y)$, and if $f$ is continuous at $x=1$, then $f$ is continuous for $x>0$. I let $x=1$ and I find that $f(x)=f(x)+f(1)$ which ...
2
votes
1answer
37 views

discontinuous solutions of functional equation

This is a followup to this question. It's well known that Cauchy's functional equation, $$f(x+y) = f(x) + f(y),$$ has discontinuous solutions. In fact, any discontinuous solution is discontinuous ...
0
votes
2answers
56 views

Can you solve the following functional equation?

I found the following functional equation: Find all functions $f : \Bbb R \rightarrow \Bbb R $ such that: $xf(x) - yf(y) = (x - y)f(x + y) $ for all $x, y \in \mathbb R $ Could you please help me? I ...
2
votes
1answer
47 views

Functional Equation similar to Cauchy's

Find all functions $f:\mathbb{Q}\rightarrow \mathbb{Q}$ such that for any $x,y\in{}\mathbb{Q}$ we have $$\{f(x)\}+\{f(y)\}=\{f(x+y)\}.$$ Note that $\{t\}$ denotes the fractional part of $t$ for ...
2
votes
3answers
101 views

Functional equation $f\left(\frac{1}{x}\right)+(x+1)f(x)=1$

Find all functions $f$ such that $f\left(\frac{1}{x}\right)+(x+1)f(x)=1,\space x\neq0$.
0
votes
2answers
40 views

Does induction find all solutions?

Induction shows that an equality holds for all values of $n$. It doesn't show that this is the only equality or formula for the expression that may hold true, correct? For example, say a question asks ...
1
vote
0answers
58 views

Is it possible to compute $\Gamma(\rho_k)$ or $\zeta(3+2\omega_k)$, when $\rho_k=\frac{1}{2}+i\omega_k$ is a nontrivial zero of Riemann zeta function?

I believe that I can to use the equation (1) (from reference [1]), a well known equation involving the Gamma function, Riemann zeta function and the theta function $\psi(x)=\sum_{n=1}^\infty e^{-n^2\...
3
votes
0answers
61 views

Functional Equation $f(m + f(n)) = f(m) - n$

Well, similar questions have already been asked. But they are not identical and the solution methods offered there are not the same one as here. Anyway, I want to solve functional equation $f(m + f(n))...
2
votes
1answer
31 views

Functional equation $f(x+y)-f(x)-f(y)=\alpha(f(xy)-f(x)f(y))$ is solvable without regularity conditions

I was reviewing this question and got motivated to solve this general problem: Find all functions $f:\mathbb R\to\mathbb R$ such that for all real numbers $x$ and $y$, $$f(x+y)-f(x)-f(y)=\alpha(...
2
votes
2answers
78 views

Differentiability of $f(x+y) = f(x)f(y)$ [duplicate]

Let $f$: $\mathbb R$ $\to$ $\mathbb R$ be a function such that $f(x+y)$ = $f(x)f(y)$ for all $x,y$ $\in$ $\mathbb R$. Suppose that $f'(0)$ exists. Prove that $f$ is a differentiable function. This is ...
2
votes
1answer
53 views

Solution of the functional equation $g(x)g(z) = g(x+z)+g(x-z)$

What is the solution for the following functional equation? $g(x)g(z) = g(x+z)+g(x-z)$ The solution given is: $g(z) = 2\cos(z)$. In the derivation of the result (using Taylor expansion), there is ...
4
votes
4answers
61 views

Functions satisfying the functional equation $[1-f(x)f(y)]f(x+y)=f(x)+f(y)$

How to prove that there is no real function defined on $\mathbb{R}$, continuous at $0$ and not always vanishing satisfying the functional equation $$[1-f(x)f(y)]f(x+y)=f(x)+f(y) \tag{E}$$
2
votes
2answers
57 views

Prove that functional equation doesn't have range $\Bbb R.$

Prove that any solution $f: \mathbb{R} \to \mathbb{R}$ of the functional equation $$ f(x + 1)f(x) + f(x + 1) + 1 = 0 $$ cannot have range $\mathbb{R}$. I transformed it into $$ f(x) = \frac {-1}...
1
vote
2answers
74 views

Finding $f(n)=f(f(n-1))+f(f(n+1))$

Determine whether a function exists from the positive integers to the positive integers which satisfies the equation: $$f(n)=f(f(n-1))+f(f(n+1))$$. My guess is that this function does not exist, as ...
0
votes
1answer
44 views

continuous function and functional equation

Let $ g:\mathbb{R} \to \mathbb{R}$ be a continuous function: $g(x) = g(x+1)$. Show that $g$ satisfies the equation: $g(x)=\frac{1}{4} \left(g(\frac{x}{2})+g(\frac{x+1}{2})\right)$ for all $x$. Is it ...
1
vote
0answers
32 views

Brute force a formula based on numbers and the result?

I have thought of a very funny thing. In World of Warcraft, all weapons have some Damage Per Second variable specified. I want to know how they calculate that result, based on the result and some ...
1
vote
2answers
35 views

Does the equation $f(x)+g(y)=x^2+xy+y^2$ have solutions in real functions $f$ and $g$?

Does the equation $$f(x)+g(y)=x^2+xy+y^2 \mbox{ } \forall x,y \in \mathbb{R}$$ have solutions in real functions $f$ and $g$?
1
vote
2answers
57 views

if $2f\left(\frac{x}{x^2+x+1}\right)=\frac{x^2}{x^4+x^2+1}$ then what is $f(x)$?

assume that: $$2f\left(\frac{x}{x^2+x+1}\right)=\frac{x^2}{x^4+x^2+1}$$ Then what is $f(x)$?
3
votes
4answers
81 views

How to solve the functional equation $f(x+a)=f(x)+a$

Are there any other solutions of the functional equation $f(x+a)=f(x)+a$ ($a=\mathrm{const}$, $a\in\mathbb{R}\setminus\left\{0\right\}$) apart from $f(x)=x+C$ ($C=\mathrm{const}$)? Edit: $a$ is a ...
2
votes
2answers
216 views

Find all the functions satisfying this criterion

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$\left|f(x)-f(y)\right|=2\left|x-y\right|$$
1
vote
2answers
69 views

Find all functions such that $f: R \longrightarrow R, \forall x \in R, f(x)f(x^2-1)=\sin(x) $.

Find all functions such that $$f: \mathbb{R} \longrightarrow \mathbb{R} \\ f(x)f(x^2-1)=\sin(x), \quad\forall x \in \mathbb{R}$$ That is a difficult problem for me. Help me please.
4
votes
2answers
106 views

Functional equation $x\space f(x^2) = f(x)$

How can I logically lead to the answer from the following conditions? $$ \left\{ \begin{align} & x \, f(x^2) = f(x) \text{ for all } x > 0, \\ & f(x) \text{ is continuous}, \\ &f(1) = ...
4
votes
3answers
143 views

Find $f$ if $ f(x)+f\left(\frac{1}{1-x}\right)=x $

Find $f$ if $$ f(x)+f\left(\frac{1}{1-x}\right)=x $$ I think, that I have to find x that $f(x) = f\left(\frac{1}{1-x}\right)$ I've tried to put x which make $x = \frac{1}{1 - x}$, but this equation ...
7
votes
1answer
105 views

Prove $f(x)=0$ when $f(2x^2-1)= f(x)\cdot 2x$

Prove that $f(x)=0$ for $$x\in[-1, 1]$$ $f-continuous$ and for all $x$: $$f(2x^2-1)= f(x)\cdot 2x$$ It is simple for integer numbers. Another fact that I've noticed that $$f(2(-x)^2-1)= f(-x)\cdot (-...
1
vote
0answers
36 views

Functional equation for $\sum_{n=1}^{\infty} \sinh(cn)^{-s}$?

Does anyone know of any kind of functional equation (or closed form) for $\sum\limits_{n=1}^{\infty}\sinh(cn)^{-s}$, where $c$ is an arbitrary constant? I've been messing around with it off and on for ...
0
votes
0answers
30 views

Time delay equation

If $x(t)=\displaystyle\frac{1+(x(t-T))^m}{(x(t-T)))^m}$ for all $t$ where $T$ is constant and $x(t)=x_s$ is the solution to the above equation, why can I write that: $x_s=\displaystyle\frac{1+{...
3
votes
1answer
57 views

Given $f(x+y)=f(x)+f(y)$ and $f(0)=0$, what can we say about f(x)

Given $f(x+y)=f(x)+f(y)$ and $f(0)=0$, what can we deduce about $f(x)$? I intend to say that $f(x)=x$, but find difficult to prove it. Is my guess correct, or wrong?
1
vote
3answers
86 views

Find all function satisfying $f(f(n))+f(n)=2n+3k$

Find all functions $f:\mathbb{N_{0}} \rightarrow \mathbb{N_{0}}$ satisfying the equation $f(f(n))+f(n)=2n+3k,$ for all & $n\in \mathbb{N_{0}}$, where $k$ is a fixed natural number. A friend of ...
0
votes
1answer
42 views

Integral functional equation.

$f(x)= \Big(\int _1^x g_1(t)g_2(t)dt\Big)\Big(\int _1^x g_3(t)g_4(t)dt\Big)-\Big(\int _1^x g_1(t)g_3(t)dt\Big)\Big(\int _1^x g_2(t)g_4(t)dt\Big) $ $\forall$ $x$ $\in R$ Where $g_1(x),g_2(x),g_3(x)$ ...
1
vote
0answers
90 views

Equidistant sets and extremal points

Let $\lVert\cdot \rVert$ be any norm on $\mathbb{R}^n$ and let $x\in \mathbb{R}^n$ be a point. Let $\textrm{ext}(B_{\lVert x\rVert})$ denote the set of all extremal points of $B_{\lVert x \rVert}:=\{...
0
votes
1answer
112 views

Is $f(x)=e^x$ the only solution to $f(f'(x))=f'(f(x))$? [duplicate]

Is $f(x)=e^x$ the only solution to $f(f'(x))=f'(f(x))$? In particular I'm interested in the qualitative properties of the such solutions.
1
vote
1answer
40 views

How to prove $X(t)$ is differentiable?

Suppose $X(t)\in M_n(\Bbb R), t\in\Bbb R$ and is continuous, invertible at every point on the real line, if the equation $$X(t)X(s)=X(t+s)$$ holds for all $t,s\in\Bbb R$, prove that there exists a ...
2
votes
0answers
25 views

Show that retarded argument functional operator is surjective

I want to show that for every $\lambda \neq 0$ and $g\in L^{2}(\mathbb{R})$, there is a $f \in L^{2}(\mathbb{R})$ such that $$e^{-|x+1|}f(x) - \lambda f(x+1) = g(x)$$ I tried looking at the operator ...
2
votes
1answer
50 views

Find all real polynomials $P(x)$ which satisfy the equation$ P(x)P(-x)=P(x^2-1)$

Find all real polynomials $P(x)$ having only real zeros and which satisfy the equation $$P(x)P(-x)=P(x^2-1)$$ Please explain me the process and refer some books to learn polynomials. Thanks ...
0
votes
0answers
20 views

How to convert $\ max \int_{-\infty}^{\infty} f(x) dx =\ min \int_{-\infty}^{\infty} g(x) dx $

Let $f$ or $g$ have a given condition. ( example $\int_{- \infty}^{\infty} f(x) exp(-x^2) dx = 1$ Or $g(g(x)) = x^3$ Or a differential equation for one of $f,g$. ) I want to find a general way - if ...
1
vote
0answers
69 views

Minimizing $\int_{0}^{1} (1+x^2)f(x)^2 dx$ for $f(f(x)) = x^2$

What is $$\min_{f\in D} \int_{0}^{1} (1+x^2)f(x)^2\mathrm dx,$$ where $D$ is the collection of all continuous real functions from $[0,1]$ such that in the interval $[0,1]$ we have $f(f(x)) = x^2 $. ...
-1
votes
1answer
31 views

Functional equaliton on continuous function $f:[0,1]\to \mathbb{R}$

Let $f:[0,1]\to \mathbb{R}$ be continuous, and suppose that $f(0)=f(1).$ Show that there is a value $x\in [0, 1998/1999]$ satisfying $f(x)=f(x+1/1999)$.
6
votes
1answer
94 views

How can I find $f(3)$ if $f(2x)=f^2(x)-2f(x)-\frac{1}{2}$?

If $f(2x)=f^2(x)-2f(x)-\frac{1}{2}$ and $f(1) = 2$ then find $f(3)$. Can you give me any hint that I can start with?
1
vote
2answers
70 views

Express $y$ from $\ln(x)+3\ln(y) = y$

i am finding a inverse function to $$y=\frac{e^\frac{x^3}{3}}{x^3}$$ First step: $$x=\frac{e^\frac{y^3}{3}}{y^3}$$ Next: ... $$(xy^3)^3 = e^{{y}^3}$$ and now in this step i have no idea how can i ...
0
votes
0answers
22 views

Existence of an inverse function in this functional equation

Let $V,W$ each be connected and separable sets. Suppose we have continuous functions $F:\mathbb{R}\times V\rightarrow \mathbb{R}$, $G:\mathbb{R}\times V\rightarrow \mathbb{R}$, $f:V\times W\rightarrow ...
2
votes
1answer
25 views

Functional inequality $f(y)+xf(x)≤yf(x)+f(f(x))$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function sucht that: $$ f(y)+xf(x)≤yf(x)+f(f(x)) $$ for all $x,y\in\mathbb{R}$. Show that $$ f(x)+yf(x+y)≤0 $$ for all $x,y\in\mathbb{R}$. I tried some ...
2
votes
1answer
32 views

Cauchy's functional equation for involutions

It is well known that Cauchy's functional equation for $f:\mathbb{R}\to\mathbb{R}$ $$ f(x+y)=f(x)+f(y)\space\forall x,y\in\mathbb{R} $$ admits highly pathological solutions if no further conditions ...
2
votes
1answer
49 views

Functional equation $f(x^3) - f(x^3 - 2) = (f(x))^2 + 12$

$$f(x^3) - f(x^3 - 2) = (f(x))^2 + 12$$ Given the functional equation above, I am trying to find the value of $f(3)$. I do not remember the exact statement of the problem precisely, so I am not sure ...