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.

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I tried substituting some values of $x$. I also know that $f(x)=x$ is a solution to the function but I am just not able to systematically prove it.

Solve the functional equation $f(x+1)+f(x-1)=2f(x)$ for $f(x)$ I tried substituting for $x= x-1; x-2; x+1;x+2$ But I can't seem to get to a systematic method to solve this question. I do know that $ f(x)=x $ works but can't prove it.
Rishav Bhagat
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Analytical results of Schröder's Equation

According to the Wiki page on Schröder's equation, $\Psi(f(x)) = s\Psi(x)$ can be solved for $\Psi(x)$ analytically if there is a fixed point $a$ such that $0 < f'(a) < 1$. That implies that there is an algorithm to put in an arbitrary $f(x)$ and…
HiddenBabel
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Solving functional equations including $f\circ f(x)$ functions

Find all $a\in\mathbb{R} $ for which there exists a function $f : \mathbb{R}\to\mathbb{R} $ , such that (i) $f(f(x))=f(x)+x$, for all $x\in\mathbb{R} $, (ii) $f(f(x)–x)=f(x)+ax$, for all $x\in\mathbb{R} $ Normally in such functional equations, I'd…
Righter
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Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(x^2 + y + f(y)) = 2y + (f(x))^2$

Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $$f(x^2 + y + f(y)) = 2y + (f(x))^2$$ I tried something like this but that inner $f(y)$ makes things worse to my opinion. Any help is appreciated.
Gabrielek
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Finding $f \colon R_+ \to R_+$ such that $sf = f \circ g$, for $g$ strictly increasing

Suppose that $g:R_+\mapsto R_+$ is a strictly increasing and smooth function and let $s$ be a positive real number. I am looking for a function $f:R_+\mapsto R_+$ that satisfies $$sf(x)=f(g(x))$$ for all $x\geq0$. This problem arises if I want to…
Gerhard S.
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exponential additive functional equation

Let $S$ be a semigroup with no identity element and $m:S\to \Bbb C$ be given function($m\not\equiv 0$) satisfying the exponential functional equation $$ m(x+y)=m(x)m(y) $$ for all $x, y\in S$. Find all solutions $f:S\to \Bbb C$ satisfying the…
Chung. J
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The functional equation $f(x) + f\bigl(x+f(y)\bigr) = y + f\Bigl(f(x) + f\bigl(f(y)\bigr)\Bigr)$

I'm having trouble with this functional equation: Find all functions $f:\mathbb R \to \mathbb R$ for which the following is valid for all $x,y\in \mathbb R$: $$f(x) + f\bigl(x+f(y)\bigr) = y + f\Bigl(f(x) + f\bigl(f(y)\bigr)\Bigr)$$ I haven't been…
croraf
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A functial equations problem over nonzero reals: $ f ( x y ) = f ( x ) f ( y ) $ and $ f \big( f ( x ) \big) = f ( x ) $

How can I find all functions $ f : \mathbb R \setminus \{ 0 \} \to \mathbb R \setminus \{ 0 \} $ such that the following two functional equations are fulfilled: $$ f ( x y ) = f ( x ) f ( y ) \quad \text {and} \quad f \big( f ( x ) \big) = f ( x…
Anton
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Question regarding the existence of a general solution to the functional equation $f \circ f = af$

I would like to know how to determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f\circ f =a f$ (where $a$ is any real number) Is this problem completely solved (without or with additional assumptions on $f$ ) A friend sent me the…
Karim Kamil
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What are some real-world applications of functional equations?

My question is simple: What are some real-world applications of functional equations? I recently started working with more and more functional equations, and was pondering this. Most fields in math have some practical application to the real world,…
TheBestMagician
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Another FE, reduced it to $f(x)= - f( \frac 1x)$

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying: $$xf(y)-yf(x)=f(\frac yx)$$ for all $x, y \in \mathbb{R}; x\neq 0$. I found that $f(1)=0$ by plugging $x=y$. Another thing I found is that by putting $y=1$, we get the function…
user829751
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Functional Equation (similar to Cauchy's): $ f ( x + y ) = f ( x ) f ( y ) + k x y ( x + y ) $

Solve the following functional equation: $ k \in \mathbb R $, $ f : \mathbb R \to \mathbb R $ and $$ f ( x + y ) = f ( x ) f ( y ) + k x y ( x + y ) \text , \forall x , y \in \mathbb R \text . $$ I have only got $ f ( 0 ) = 1 $, by letting $ f ( 0…
user829751
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Solving the functional equation $f(x)=f(1/x)$ for $x>0$

I am trying to solve the functional equation $$f(x)=f\left(\frac{1}{x}\right),\quad x>0$$ for a real-valued function $f$, with $f(1)=2$. We may assume $f$ is continuous or infinitely differentiable. There is, obviously, the solution $f(x)=2$ for all…
Ray Bern
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$f^3(x) + f^2(x) \cdot x^2 = 1$

Does there exist a function $f$ such that $\forall x \in \mathbb{R}:f^3(x) + f^2(x) \cdot x^2 = 1$? I haven't studied functional equations so I have no idea how to solve this problem. I think I proved it's impossible if $f$ is polynomial (it would…
user838035
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Some functional equations in two variables: $|f(x)-f(y)|=\frac{1}{|\varphi(x)-\varphi(y)|}$

I have two questions. i) Does there exist a function $\varphi:\mathbb{R}\to\mathbb{R}$ for which the functional equation $$ |f(x)-f(y)|=\frac{1}{|\varphi(x)-\varphi(y)|} $$ has a solution $f:\mathbb{R}\to\mathbb{R}$ for all $x\neq y$? ii) Let…
Asia Fahrieva
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