Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [functional-equations]

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (functions) or (elementary-set-theory).

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation(s) and possibly other conditions; e.g., the goal can be to find all continuous solutions. Solving the equation means finding all functions satisfying the given equation(s) and any additional conditions.This is different from the more common use of the word "equation", where the solutions are numbers. It is also different from the more common use of the word "functional", referring to a mapping from a space into the reals or complexes. For basic questions about functions use more suitable tags like or .

A common technique used in solving functional equations is finding some properties of satisfying functions by substituting variables for certain values in the equation. Proving properties of satisfying functions is also helpful - finding that a function is injective, surjective, involutive, and so on, is often a key step in finding all possible solutions. Other techniques such as exploiting symmetry, considering fixed points, and even using certain properties of domains (e.g. well-ordering) sometimes help.

Some well-known functional equations are:

More information can be found at Wikipedia.

3976 questions
0
votes
1 answer

Boundedness ,Additive and Cauchy

Suppose a function f is additive. Also $f(x)$ is positive whenever $x$ is positive and $f(x)$ is negative whenever $x$ is negative. My question is "Is it necessary that f is linear?" If yes then please give a proof and if no then I am looking for a…
Sufaid Saleel
  • 3,789
0
votes
1 answer

A functional equation problem calculus

If $f(x)+f(1-1/x)=\arctan(x)$, find $f(x)+f(1-x)$.
sandy
  • 1
0
votes
1 answer

Solving general solution of recurrence relation by iteration

$$a_{n-1} = ca_{n-2} $$ Hence $$a_n = c \cdot c \cdot a_{n-2} $$ $$ = c \cdot c \cdot c \cdot a_{n-3} $$ ...... $$ = c^na_0 $$ Why is there a iteration on the constant $c$ ?
ilovetolearn
  • 1,341
0
votes
1 answer

Find all functions $f$ such that $f(xf(y)+y)+f(xy+x)=f(x+y)+2xy$.

I know that $f(x)=x$ and f(x)=-2x are solutions to this functional equation and I doubt there are any other solutions but I'm not entirely sure. Also I don't really know how to prove that if $f$ is a solution to this equation, then $f$ must be a…
user765855
0
votes
1 answer

Cyclic function in functional equation

$f(x) + f \left( \frac{x - 1}{x} \right) = \frac{5x^2 - x - 5}{x}$ I proved $f(\frac{x-1}{x})$ was cyclic in cycles of 3, becoming $f(\frac{-1}{x-1})$, then becoming $f(x)$ but can't see how to apply this and get solutions.
user118161
  • 234
0
votes
1 answer

finding an explicit solution to a functional equation

I am concerned in finding an explicit solution to the following functional equation, where $P$ is the unknown function: $$P(q) = \lambda_1(q)+ \lambda_2(q)P(q+2 \chi_s),$$ where: $$\lambda_1(q)=\frac{\alpha_2(q) + \alpha_2(q+2\chi_s)}{\cosh q -…
Juan Gustavo Wouchuk Schmidt
  • 11
0
votes
2 answers

function equation $f(x)+f(y)=f(\frac {x+y} {1+xy})$

Is there exist $f:(-1,1)\to R$ such that $f(x)+f(y)=f(\frac {x+y} {1+xy})$? how to find $f$?
추민서
  • 560
0
votes
1 answer

What values can take expression

What values can take the expression $T=\frac{f(t)-f(0)}{f(t^2)+f(t)-2f(0)+2}\mspace{20mu},\mspace{15mu} $ where $f(2x+y)-f(x+y)=2x , \mspace{20mu}x,y\in \mathbb{R}$ No idea what to do
Hrackadont
  • 545
  • 2
  • 7
0
votes
1 answer

Solving functional equation $f(x+y) = f(x)f(y)$ under some constraints

Suppose that $x, y \in \{1,\dots, n\}$. Is it possible to find a nonzero $f \in \mathbb{C}$ such that $$f(x+y) = f(x)f(y) \text{ for } x \neq y$$ $$f(x)^2 - \frac{1}{2x-1} = f(2x)$$ This problem is related to solving $f(x+y) = f(x)f(y) $ with…
KRL
  • 1,170
0
votes
1 answer

Functional equations towers of $f$

If $f(f(1))=f(1)$ does it imply that $f(1)=1$? The problem I'm solving is $(x+y)f(y(f(x))=x^2(f(x)+f(y))$ where $f$ maps from $R^{>0}$ to $R^{>0}$.
user243558
  • 11
0
votes
2 answers

Solving $f(3x)=f\Big(\frac{x+y}{(x+y)^2+1}\Big)+f\Big(\frac{x-y}{(x-y)^2+1}\Big)$ and $f\big(x^2-y^2\big)=(x+y)f(x-y)+(x-y)f(x+y)$

I need help solving this functional equations problem. Find all $ f : \mathbb R \to \mathbb R $ such that for all $ x , y \in \mathbb R $, the two following equations hold: $$f(3x)=f\left(\frac{x+y}{(x+y)^2+1}\right) +…
Hadi Hamdan
  • 31
0
votes
2 answers

How to find number of distinct solutions of equation $f(x) + f(x/2) =x$.

How to find number of distinct solutions of equation $f(x) + f(x/2) =x$. by replacing x by x/2? and so on, at end i m left with f(0).
maveric
  • 2,168
0
votes
2 answers

What functions satisfy $f \circ g = g \circ f\ne \text{id}$?

What functions satisfy $f \circ g = g \circ f\ne \text{id}$ and $f \ne g$? Some thoughts: This is fulfilled for $f(x) := \alpha x$ and $g(x) := \beta x$ for $\alpha, \beta \in \mathbb{R}$ such that $\alpha \beta \ne 1$ and $x \in \mathbb{R}$…
ViktorStein
  • 4,838
0
votes
0 answers

Functional eqation problem

Is there a function $f: \mathbb N\to \mathbb N$ that satisfies $$f(f(n-1))=f(n+1)-f(n)$$ for $n \geq 2$? So far I just know that $f(n)>f(n-1)$ for $n \geq 2$.
Papi Papagalov
  • 9
0
votes
0 answers

Show that the functional equation $f(f(x))=-x^{3}+g(x)$ has no continuous solution.Here $g$ is a continuous periodic function with positive period.

I want to show $f(f(x))=-x^{3}+g(x)$ has no continuous solution $f:\mathbb{R}\to\mathbb{R}$.Here $g$ is a continuous periodic function with its period $T>0.$ Any help will be thanked.
Tree23
  • 1,176
1 2 3
…
60 61