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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Soultion of a particular functional differential equation

I need to solve the following functional differential equation: $(4\mu-\lambda r - \lambda x+\lambda)f'(x) = \lambda( f(x) - f(1-x-r))$, where $x\in (0, 1-r)$, $r\in (\frac{1}{2}, 1)$ Or, a more general version of it. Please help. Thanks
user195925
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Differentiability Problem

Supposing we are given relation that $$f(xy + 1)= f(x).f(y) - f(y) - x +2$$ and also given that $$f(0)=1$$ for a differentiable function then is function one-one onto? I partially differentiated relation first wrt to $x$ then $y$ $$ f'(xy + 1)y=…
Tesla
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How to solve polynomial functional equation $P(x,y)=P(\frac{x-y}{2},\frac{y-x}{2})$?

Given $P(x,y)$ which is a polynomial function, satisfying $P(x,y)=\displaystyle P(\frac{x-y}{2},\frac{y-x}{2})$. Then why should $P(x,y)$ be $\displaystyle\sum^n_{i=0}a_i(x-y)^i$? Is it unique?
Charles Bao
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Solutions to functional equation $f(x_1,x_2)+g(a_1t+x_1,a_2t+x_2)+h(b_1t+x_1,b_2t+x_2) = t^2$

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$, $g:\mathbb{R}^2 \rightarrow \mathbb{R}$ and $h:\mathbb{R}^2 \rightarrow \mathbb{R}$ be twice continuously differentiable functions such…
user103828
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Uniqueness of Pexider's functional equation

Let $f:\mathbb{R}\rightarrow\mathbb{R}$, $g:\mathbb{R}\rightarrow\mathbb{R}$, and $h:\mathbb{R}\rightarrow\mathbb{R}$ and consider Pexider's equation, $$ f(x) + g(y) = h(x + y) \qquad \qquad (1) $$ where $f$, $g$ and $h$ are unknown. I assume (for…
user103828
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Need a equation that defines a certain number

Im programming a function but I just cant structure the equation. I think this is the right place to ask since the problem is completely mathematics. Let me explain three scenarios. There are 4 variables. X, Y, Z and P. They all depend on each…
Jacob
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A logarithm-like functional equation

Suppose we are given that a monotonically decreasing smooth function $f$ on $(0,\infty)$ obeys the functional equation $f(x) = -f(\frac{1}{x})$, and satisfies $f(\frac{1}{3}) = \frac{1}{2}$ and $f(\frac{1}{2}) = \frac{1}{3}$. Furthermore,…
user111187
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Does the functional equation $xyF\left(xy^2,y\right)=F(x,y)$ have a unique solution?

I wish to prove/disprove that there exists a unique solution to the functional equation $$xyF\left(xy^2, y\right) = F(x, y), \quad x \ne 0, \quad |y| < 1, \quad y \ne 0,$$ where $F(x, y)$ is continuous. I tried using the standard technique, i.e.,…
glebovg
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Functional equation question: $ 2 f ( i ) - f ( i + j ) - f ( i - j ) = \lambda j $

The following has come up in the course of my research. I'm looking for a function $ f : \mathbb Z ^ \star \to \mathbb R $ such that $$ 2 f ( i ) - f ( i + j ) - f ( i - j ) = \lambda j $$ for all $ i \ge 0 $ and all $ j $ such that $ 0 \le j \le i…
N. Virgo
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show that $f(x)=c\log x $ for some $c$

Let, $f$: $\mathbb{R^+}$$\rightarrow$$\mathbb{R}$ be a continuous function satisfying $f(xy)=f(x)+f(y)$. Prove that, $f(x)=c\log x$ for some $c>0$.
Topology
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Solving $f(f(x))=g(x)$ equations

Possible Duplicate: Square root of a function (in the sense of composition) I'm interested in solving equations of the form $f(f(x))=g(x)$ for $x\in\mathbb{R}$ where $g(x)$ is a known function. For example: if $g(x)=x$ then we $f(f(x))=x$ so one…
alext87
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check if there are such functions that verify the following functional equation

The statement of the problem: Determine if there are any functions $f : (1, \infty)\rightarrow (1, \infty)$ with the property: $ x^{f(y)^x} $ = $ y^{f(x)^y}\text{for every }x, y > 1$ My approach: I first proved that it is an injective function.…
Last X
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functional equation, continuous functions.

$f:\mathbb{R}\to\mathbb{R}$ such that $\frac {f(x+y)}{f(x-y)}=\frac{f(x)+f(y)}{f(x)-f(y)}$ $\forall$ $x\ne y$. Find all continuous functions $f$. My progress: Denote the assertion as $P(x,y)$. If $P(x,0)$ gives $f(0)=0$ and $P(x,-x)$ gives $f(x) =…
RickSanchez
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Find the number of functions $f:\mathbb Q \rightarrow\mathbb Q$ satisfying the following conditions:

Find the number of functions $f:\mathbb Q \rightarrow\mathbb Q$ satisfying the following conditions: $$f(h+k)+f(hk)=f(h)f(k)+1,\quad \forall \ h,k \in \mathbb Q$$ I really tried using the "general" method for substituting random values per $h$, $k$…
The Revolution
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Proving that powers are the only solutions to the functional relation $g(r)g(1/r)=1$

As stated above, I am trying to prove that the only solutions to the functional relation $$ g(r)g(1/r)=1, \quad \text{for}\; r>0, $$ are of the form $g(r)=r^a$ for $a \in \mathbb{R}$. Other properties that I assume are that $\lim_{r \rightarrow 0}…
thecommutator
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