# 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).

2,584 questions
Filter by
Sorted by
Tagged with
69 views

### Find $f(x)$ where $f(x)(A-\frac{B}{x+B/A})+Cf(x+\frac{B}{A})=0$.

$A, B, C > 0$, $x$ is complex and $Re(x)>0$. My guess is that $f(x)=0$ but I don't know how to prove it.
23 views

### Pexider's (/ Cauchy's) functional equation over a bounded domain

I am looking at Pexider's equation $f(x+y)=g(x)+h(y)$, where $f,g,h$ are continuous functions but are defined over bounded domains. Specifically, $f,g,h$ each is defined on a real interval (of length ...
31 views

### Cauchy's functional equation from complex to reals

I am looking for the general continuous solutions $f:D \rightarrow \mathbb R$ of the multiplicative Cauchy functional equation $f(x)f(y) = f(xy)$ for the domain $D=\{x \in\mathbb C: |x|<1\}$. (...
44 views

117 views

### Is there a solution to this functional equation?

I was going through my old notebooks and I found a sheet of paper with this problem on it. I thought it would be a shame to let such an unreasonably difficult question go to waste, so I decided I ...
43 views

### How to solve a functional equation involving log?

It's given that $$f(xy)=\frac {f (x)}{y}+\frac {f (y)}{x}$$ Also $x,y>0$ and $f(x)$ is differentiable for $x>0$ such that $f(e)=\frac{1}{e}$. By the look of the functional equation I am sure ...
101 views

### Functions $f:\mathbb R \to \mathbb R$ which satisfy $f(x^2+f(y))=y+(f(x))^2$ [duplicate]

Find all the functions $f:\mathbb R \to \mathbb R$ which satisfy $$f(x^2+f(y))=y+(f(x))^2$$ for all $x, y$ in $\mathbb R$. I have the following proof from my math book and want to see if I can ...
59 views

64 views

### What problems are related with the following type of FDE?

Consider the following type of functional differential equations: \begin{align} \frac{du}{dt} &= F(x,t,u(x,t),u_{x,t}), & (x,t) &\in [a,b] \times [0,T] \end{align} where $u(x,t)$ is ...
655 views

### Minimizing a functional with a free boundary condition

Find the extremals of the functional $$\text{J}(y)= y^2(1) + \int_0^1 y'^2(x)dx , \ \ y(0)=1.$$ Answer: $y(x)=1-\frac{1}{2}x$ My solution: $F (x,y,y')=y'^2(x)$ After solving the Euler ...