Questions tagged [functional-equations]
213 questions
5
votes
5 answers
Are there functions satisfying the following integral condition?
Can we find two functions $f$ and $g$ that are reasonably defined nontrivial(not everywhere zero, $f\neq g$, not linear polynomials) functions such that the following condition is satisfied?
$$ f( \left(\int_{0}^{t} g(x) \ \text{d}x\right)) = g(…
Chulumba
- 789
4
votes
0 answers
A functional equation: Functional families that are "weakly" closed under product?
Suppose that for any real number $a$, we have a function $f_a:\mathbb R \to \mathbb R$ or such that $f_a(x)$ is monotonically strictly increasing in $x$ and hence invertible on its image. We also assume continuity in $a$.
We want to find conditions…
user56834
- 91
4
votes
1 answer
Cauchy-like functional equation $f(h(y)x+y)=g(y)f(x)+f(y)$
I am looking for the solution to the following two variable functional equation:
(*) $f(h(y)\cdot x+y)= g(y)f(x)+f(y)$
where:
$h$ is some given continuous function,
$f, g,$ unknown functions on some interval $[0,\alpha]$ for some $\alpha>0$,
$f$…
mike
- 63
3
votes
1 answer
Solution to the following functional equation
I would sincerely appreciate if anyone can tell me how to solve g(x) defined by the following functional equation:
$h(t) = \int_0^t f(2t-x)g(x)dx$
for
$0\leq t\leq \infty$?
where: f(x) is a known function (actually a probability density functions…
Haining Yu
- 43
3
votes
0 answers
Criteria for $f(f(x))=g(x)$
I 'm searching about the solvability of the functional equation $f(f(x))=g(x)$. I have three questions about it:
Let's be $g$ an arbitrary function and the functional equation $f(f(x))=g(x)$. Are there any specific criteria to ensure us that there…
Kώστας Κούδας
- 189
3
votes
1 answer
Is it true that the only solutions are $f(x)=0$ and $f(x)=x$?
Suppose that $f: \mathbb R \rightarrow \mathbb R$ such that
$$f(x^3+y^3)=f(x+y)((f(x-y))^2+f(xy)),$$ for all $x,y$ real numbers. Is it true that the only solutions are $f(x)=0$ and $f(x)=x$? I did not come to any good result, but I think the…
Noram Sadir
- 33
2
votes
2 answers
solvability of an elementary functional equation
Is there some other way to characterize the functions $f:\mathbb Z\times \mathbb Z\to \mathbb Z$ which are expressible as
$$f(x,y)=g(x)+g(y)-g(x+y)$$
for some $g:\mathbb Z\to\mathbb Z$?
Easy facts: (1) $f$ must satisfy $f(x,y)=f(y,x)$ and…
Mircea
- 2,031
2
votes
2 answers
Non-exponential functions $f(x)$ satisfying $f(x+c)=\gamma(c)f(x)$
Question:
what can be said about the existence of functions
\begin{align}
f:x\mapsto f(x)&\implies x+c\mapsto \gamma(c)f(x)\\ f(x)\ne f(y)&\iff\frac{f(x)}{\gamma(x)}\ne\frac{f(y)}{\gamma(y)}
\end{align}
These functions would generalize the…
Manfred Weis
- 12,594
- 4
- 34
- 71
2
votes
0 answers
book recommendation about iterative functional equations
I would like to learn about iterative functional equations.
I read this book, but it doesn't include such functional equations. I tried this, but it was too general for my purpose. Finally I read this, witch suggests that book.
I tried to find a…
Kώστας Κούδας
- 189
2
votes
0 answers
Solve this functional equation with respect to $f$
Let $v\not= 1$ be a real number. Let $f(s)$ be real analytic on an open interval containing $v$ and $1$, with a zero of order $m\ge 1$ at $s=1$.
My question is: Can we solve this functional equation with respect to…
China-Hong Kong
- 403
2
votes
2 answers
Functional equations
What are the general solutions of the functional equations?
$$
f(x,y)+f(y,z)=\frac{1}{f(x,z)}
$$
$$
f(x,y)f(y,z)f(x,z)=1
$$
Umar
- 21
- 1
1
vote
1 answer
Is there a systematic procedure to Solve Abel's, Böttcher's, or Schröder's Equation
I've been interested greatly in the study of functional equations for some time now, I've learnt many different techniques for their solution. Currently I have been studying superfunctions and fractional iterations of functions. In all these…
1
vote
1 answer
General solution of a linear functional equation
As we know, general solution of the linear functional equation $f(x+1)-f(x)=g(x)$ ($g$ is a known function) is $f=f_0+\phi$, where $f_0$ is an its special solution and $\phi$ any 1-periodic function.
Now, does any one know general solution of the…
M.H.Hooshmand
- 995
1
vote
1 answer
The functional equation $f(xy) - 2 f(\frac{x+y}{2}) + f(x+y- x\cdot y) = 0$
Incidentally, I came across the following functional equation
$$f(xy) - 2 f(\frac{x+y}{2}) + f(x+y- x\cdot y) = 0$$
that is to hold for all $x,y\in \mathbb R$. Is there a neat way to find all solutions $f:\mathbb R\to \mathbb R$?
J. Netzing
- 13
1
vote
2 answers
A functional equation with a quadratic solution
I have the following problem. I have a function $v(x, \theta)$ that can be expressed in two ways, for all $x, \theta \in \Re$:
$v(x, \theta) = u(x - \theta)$, where $u$ is strictly concave and symmetric about $0$, and
$v(x, \theta) = g_1 (x) f_1…
user118723
- 13