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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9
votes
3answers
104 views

Mean Value theorem functional equation

I need help solving the following functional equation. Suppose $\,f : \mathbb{R} \to \mathbb{R}$ is differentiable and $$f'\Big(\frac{x + y}{2}\Big) = \frac{f(x) - f(y)}{x -y}$$ holds for all $x ...
0
votes
1answer
20 views

Banach fixed-point theorem for a recursive functional equation

I was asked to prove that the functional equation $$ x(t) = \begin{cases} \frac{1}{2} x(3t) + \frac{1}{2},& 0 \leq t \leq \frac{1}{3}\\\ f(t), & \frac{1}{3} < t\leq \frac{2}{3}\\\ ...
0
votes
2answers
42 views

$f(yf(\frac{x}{y})) = \frac{x^4}{f(y)} $

Solve Functional Equation$$f: \mathbb{R}_+ \to \mathbb{R}_+; \; f(yf(\frac{x}{y})) = \frac{x^4}{f(y)} $$ I'm stuck in the beginning. Any hint will be helpful.
7
votes
3answers
364 views

Functional equation $f(x+y)+f(x-y)=2f(x)f(y)$

Let $f:\mathbb{R}\to\mathbb{R^*}$ be a function such that $f(x+y)+f(x-y)=2f(x)f(y),\forall x,y\in\mathbb{R}$. Prove that $f(x)=1,\forall x\in\mathbb{R}$. I have managed to prove the following: 1) ...
2
votes
2answers
60 views

The 'prime logarithm'

Lately I've been thinking about the functional equation $$f(ab) = f(a) + f(b)$$ but not in the usual sense where continuity or differentiability are assumed. It's clear that $f(1) = 0$, by letting $a ...
2
votes
2answers
106 views

solve equations - find a short piece of a wire

Problem: A piece of wire 20 feet long is cut into two pieces so that the sum of the squares of the lengths of the two pieces is 202 square feet. What is the length, in feet, of the shorter piece of ...
2
votes
4answers
156 views

How to solve $f'(x)=f'(\frac{x}{2})$

How do we solve this given $f'(0)=-1$. It does not look separable. I can integrate both sides but end up with a functional equation with is not helpful.
1
vote
0answers
10 views

Functional extrema and the Euler-Lagrange equation

For a functional of the form: $$S(q)=\int_{t_{1}}^{t_{2}}L(q,\dot{q})dt$$ where $\dot{q}=\frac{\partial q}{\partial t}$ , one finds that extrema are reached (to first order) for the condition : ...
1
vote
1answer
37 views

Solution of a sublinear elliptic problem.

In a lecture notes, the author showed the problem $\tag{$P$}$ $\begin{cases} -\Delta u = |u|^{q-2}u \textrm{ in } \Omega, \\ u(x)= 0 \textrm{ in } \partial\Omega, \end{cases}$ where $\Omega ...
17
votes
4answers
913 views

very elementary proof of Maxwell's theorem

Maxwell's theorem (after James Clerk Maxwell) says that if a function $f(x_1,\ldots,x_n)$ of $n$ real variables is a product $f_1(x_1)\cdots f_n(x_n)$ and is rotation-invariant in the sense that the ...
1
vote
1answer
21 views

Checking my understanding of Cauchy's functional equation.

Cauchy's functional equation is given as $$f(x+y)=f(x)+f(y)$$ Wikipedia states that the solution to this functional equation with $x\in\mathbb Q$ is $f(x)=cx$, where $c$ is an "arbitrary ...
0
votes
1answer
135 views

Prove an additive function has property f(x)=x

So I am given a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x+y)=f(x)+f(y)$ for all real $x$ and $y$, is continuous at $x=0$, and $f(1)=1$. I need to show that $f(x)=x$ for all real ...
23
votes
3answers
2k views

Given $f(f(x))$ can we find $f(x)$?

Given $f(f(x))=x+2$ does it necessarily follow that $f(x)=x+1$? This question comes from a precalculus algebra student.
0
votes
1answer
39 views

Solving an equation involving complex numbers.

I tried solving the problem on my own. I would like to know if I have made any mistakes. If I have indeed made a mistake, I would appreciate it if someone corrects it and explains what it is. Also, I ...
-5
votes
1answer
48 views

$f:R- \{0\} \rightarrow R$ and $f(xy)=f(x)+f(y)$ [closed]

Suppose $f:R- \{0\} \rightarrow R$ and $f(xy)=f(x)+f(y)$ then What are all the features of Function $f$ Suppose you do not know logarithms.
11
votes
1answer
702 views

Equation about generating functions and subfactorial

Suppose $G_n(w)$ is a formal power series (really a probability generating function, see the following explanation) of variable $w$, try to solve out $G_n(w)$ for all $n\ge0$ from the ...
4
votes
1answer
751 views

Limit of a function satisfying an inequality

If $f(x)+f(y)\leq f(x+y)$ and $f:\mathbb{R}\to\mathbb{R}$, then can we find $\lim_{x\to 0} \frac {f(x)}{x}$? I am not sure whether the question is correct.Thank you.(I tried this idea: ...
0
votes
1answer
26 views

Solving functional equation $\frac{f(s, t)}{f(s, m)} = \frac{1}{1+s(m-t)}$

I want to find a functional equation $f(s,x)$ such that $$\frac{f(s, t)}{f(s, m)} = \frac{1}{1+s(m-t)}$$ If it helps the context I need this in is where $t$ is a member of a set of real number and ...
4
votes
3answers
363 views

Strategies to find the set of functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy a given functional equation

My question is as follows: What methods can be used to find the set of functions $f:\mathbb{R}\to\mathbb{R}$ satisfying a certain functional equation. An example of a case where this applies is the ...
1
vote
0answers
16 views

euation of modular functions

In this article at page four: It says that one easily checks the equation $(1)$. But I can't check that. I already tried to use $\eta(-1/z)=(-iz)^{1/2}\eta(z),~z$ in the upper half plane, but this ...
3
votes
0answers
56 views

Find the polynomial $p(x)$

A polynomial $p(x)$ gives a remainder of $1$ when divided by $x^{100}$ and a remainder of $2$ when divided by $(x-2)^3$. Evaluate $p(x)$. By the Remainder Theorem, $p(x)$ can be written as ...
4
votes
1answer
57 views

How to solve $f(x)+f\left(\frac{1}{x}\right)=e^{x+\frac{1}{x}}$

Given that $f:\mathbb{R}_0 \rightarrow \mathbb{R}_0$ find such $f$ that $$f(x)+f\left(\frac{1}{x}\right)=e^{x+\frac{1}{x}}$$ Note that I came up with this question, and personally am not sure ...
0
votes
0answers
15 views

A functional equation with an inequality

I have an increasing function on $[0,1]$, $p \mapsto \Pi(p)$, that has the following properties. $$\Pi(0) = 1 - \Pi(1) = 0$$ $$\Pi(p) + \Pi(1-p) < 1 \quad \forall{p} \in (0,1)$$ $$\Pi(p) > p ...
1
vote
1answer
48 views

Alternate method for finding $f(x)$

Let f be a real valued function $f:(0,\infty)\to(0,\infty)$ such that it satisfies the relation: $$f(xf(y))=x^2\cdot y^a$$ where $a\in\mathfrak{R}$ then find $f(x)$ and the possible values of $a$. ...
2
votes
1answer
50 views

Solve a functional equation involving integrals

Let $a, b \in (0,1)$ with $a+b \le 1$. Find all continuous functions $f:[0,1] \rightarrow \mathbb{R}$ having: $$ \int_{0}^{x} f(t) dt = \int_{0}^{ax} f(t) dt + \int_{0}^{bx} f(t) dt \tag1 $$ ...
2
votes
2answers
70 views

How to show that If $f(x+y)=f(x)f(y)$ then $f(x)\geq 0$ [duplicate]

I would appreciate if somebody could help me with the following problem: Q: Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function and $f(x+y)=f(x)f(y)$. Then $f(x)\geq 0$
12
votes
1answer
272 views

About the derivative of a function defined on rational numbers

I have found this problem: Let $f : \mathbb{Q} → \mathbb{R}$ with property: $$|f(x) − f(y)| \le (x − y)^2 \tag1$$ for all $x, y \in \mathbb{Q}$. Prove $f$ is constant. My idea is to consider ...
2
votes
0answers
33 views

The function $\psi (x+2)=1+\sqrt{2\psi (x)-\psi^2 (x)}$

The function $\psi : \mathbb{R}\rightarrow \mathbb{R}$ satisfies the relation: $$\psi (x+2)=1+\sqrt{2\psi (x)-\psi^2 (x)},$$ for all real $x$. What features it has? Place your example of at ...
0
votes
1answer
29 views

Functional Equations again

Let $f$ be a differentiable function satisfying $f(x+y)=(f(x))^{\cos y}\cdot (f(y))^{\cos x}$ for all $x,y \in R$. $f(0)=1, f'(0)=\ln 2$. If $$\int_{\frac{\pi}{2}}^{\frac{17\pi}{2}} ...
6
votes
4answers
681 views

Find functions $f:\mathbb{R}\to[0,1]$ such that: $\lim_{x \to +\infty} \frac{f(x)^2}{f(2 x)}=1$

Find functions $f:\mathbb{R}\to[0,1]$ that satisfy: $$\lim_{x \to +\infty} \frac{f(x)^2}{f(2 x)}=1,$$ $$f'(x)\leq 0 \, \forall x,$$ and $$\lim_{x \to +\infty} f(x)=0.$$ $$\lim_{x \to -\infty} ...
2
votes
1answer
29 views

Continuous function on compact interval $[a,b]$ with non-negative values

let $f:[a,b]\longrightarrow[0, \infty)$ be a continuous function satisfying the following: $f(\frac{a+x}{2})+f(\frac{2b+a-x}{2})=f(x), \forall x \in [a,b]$. Then the only function that satisfies these ...
5
votes
0answers
42 views

Proving a Functional Equation is Differentiable [duplicate]

Let $f:(0,\infty )\rightarrow \mathbb{R}$ satisfy $f(xy)=f(x)+f(y)$ and let f be differentiable at x=1. Prove f is differentiable over it's entire domain with derivative $f'(x)=\frac{f'(1)}{x}$ Using ...
3
votes
1answer
120 views

If $f\circ f\circ f=id$, then $f=id$ [duplicate]

Let $f$ a continuous function on all $\mathbb R$. How can I prove that if $f\circ f\circ f=id$, then $f=id$ ? I really have no idea.
13
votes
3answers
691 views

Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$

Let $f : [0,1] \to \mathbb{R}$ be a continuous function such that $$\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$$ Determine all such ...
0
votes
2answers
18 views

Number of distinct values

Question: How many possible values of (a, b, c, d), with a, b, c, d real, are there such that abc = d, bcd = a, cda = b and dab = c? I tried multiplying all the four equations to get: $$(abcd)^2 = ...
1
vote
1answer
41 views

Cauchy functional inequality

Given a function on a closed interval $f\colon I\subset \mathbb{R}\to \mathbb{R}$ with $$f(x+y) \leq f(x) + f(y).$$ Moreover, I know that $f$ is monotonic increasing continuous on all points except ...
4
votes
1answer
54 views

A function $f$ satisfies the condition $f[f(x) - e^x] = e + 1$ for all $x \in \Bbb R$.

Let $f$ be a function such that $f[f(x) - e^x] = e + 1$ for all $x \in \Bbb R$. Find $f(\ln 2)$. I've considered two cases: $f(x) = e^x + c$, where $c$ is constant. Then $f(c) = e^c + c = e + 1$, ...
1
vote
1answer
55 views

Functional Equation: $f(x^2-y^2)=xf(x)-yf(y)$

Let $\mathbb{R}$ be the set of Real numbers. Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x^2-y^2)=xf(x)-yf(y)$$ for all pairs of real numbers $x$ and $y$. This is a problem ...
4
votes
2answers
197 views

Does there exist the function $f^2(x)\ge f(x+y)\left(f(x)+y \right) $

Does there exist the function $f:\mathbb R^+\rightarrow \mathbb R^+$, such that $$f^2(x)\ge f(x+y)\left(f(x)+y \right) \forall x,y \in \mathbb R^+$$ My work so far: Assume that a function ...
5
votes
2answers
67 views

Can a conformal map be turned into an isometry?

Let $f: (M, g) \to (M, g)$ be a conformal diffeomorphism of the riemannian manifold $(M, g)$, with $$ g(f(p))(Df(p) \cdot v_1, Df(p) \cdot v_2) = \mu^2(p) g(p)(v_1, v_2), \quad \forall p \in M, \, ...
1
vote
1answer
32 views

Judicious guess for the solution of differential equation $y''-6y'+9y= t^{3/2} e^{3t}$

$(a)$ Let $L[y]=y''-2r_1y'+r_1^2y.$ Show that $$L[e^{r_1t}v(t)]=e^{r_1t}v''(t).$$ $(b)$Find the general solution of the equation $$y''-6y'+9y= t^{3/2} e^{3t}$$ I have problems only in part $(b)$.
2
votes
2answers
70 views

continuous (on 3, 4 and 5) f is constant, if $f(x+2)+f(4x)=f(2x+1)+f(2x+2),\forall x\in\mathbb{R}$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function that is continuous on 3, 4 and 5, such that $f(x+2)+f(4x)=f(2x+1)+f(2x+2),\forall x\in\mathbb{R}$. Show that f is constant. I don't know what to do ...
8
votes
2answers
129 views

Find all functions $f(f(f(…(f(x_1,x_2),x_3),…),x_{2016}))=x_1+x_2+…+x_{2016}$

I am trying to solve the functional equation: Find all functions $f:\mathbb R^2\rightarrow \mathbb R$ such that for all $\left \{x_1,x_2,...,x_{2016} \right \}\subset \mathbb R$: ...
0
votes
0answers
35 views

What can be said about $f$ in $\frac{f(x,y)}{f(y,x)}=\frac{g(x,y)}{g(y,x)}$ if $g$ is known?

Suppose you have the equailty $\frac{f(x,y)}{f(y,x)}=\frac{g(x,y)}{g(y,x)}$, $x,y\in\mathbb{R}$, and $g$ is known. What non-trivial facts about $f$ can be deduced from this? (Assume $f(x,y)\neq ...
0
votes
1answer
56 views

How to solve this functional equation: $f(1-f(x))=1-x^{9}, f(1)=0$

I have managed to guess one solution of this function : $f(x)=1-x^{3}$, but I have no idea how to prove it unique, or get other solutions. If this is not solvable, how can you prove this function ...
2
votes
1answer
25 views

If $g:\mathbb{N}\rightarrow \mathbb{R}$ and $g(m+n)+g(m-n)=2g(m)+2g(n)$ what is $g(x)$

Determine all functions $g:\mathbb{N}\rightarrow \mathbb{R}$ such that $g(1)=1$ and $$g(m+n)+g(m-n)=2g(m)+2g(n), \quad \forall m\ge n \in \mathbb{N}$$ Because of the identity $k\cdot (a+b)^2 ...
2
votes
1answer
57 views

Functional equation.

I'm trying to solve the functional equation $f(x+f(y)) = f(x)-y$ where $f : \mathbb{Z} \to \mathbb{Z}$. What I got so far is: $f$ is injective and $f(0) = 0$. Thanks in advance for your time.
1
vote
1answer
33 views

Find all function that satisfy $(f(x) + xy) \cdot f(x - 3y) + (f(y) + xy) \cdot f(3x - y) = (f(x + y))^2$

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all real numbers $x$ and $y$,$$(f(x) + xy) \cdot f(x - 3y) + (f(y) + xy) \cdot f(3x - y) = (f(x + y))^2.$$
1
vote
1answer
93 views

General solution of a finite-difference equation with real non-commensurable differences.

How to find a general solution of the following functional (recurrence) equation: $$f(x) = c_1 f(x - a_1) + \dots + c_n f(x - a_n), \tag{1}$$ where $c_i, a_i$, $i = 1, ... n$ are arbitrary real ...
6
votes
2answers
81 views

Solution of functional equation $f(x+y)=f(x)+f(y)+y\sqrt{f(x)}$

If $x,y\in \mathbb{R}$ and $f(x+y)=f(x)+f(y)+y\sqrt{f(x)}$ and $f'(0)=0\;,$ Then $f(x)$ is $\bf{My\; Try::}$ Using $$f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} = \lim_{h\rightarrow ...