The functional-equations tag has no wiki summary.
3
votes
1answer
173 views
solution of functional equation $f^{\circ k}(x) = x$
The equation $f^{\circ k}(x) = \mathrm{Id}$ for $x\in E$ is called the Babbage equation and the general solution is given in the following way [M. Kuczma, Functional equations in a single variable]:
...
3
votes
1answer
113 views
General additive function of probability
Let $H$ be a function of finite sequences of probabilities (non-negative numbers summing up to 1) into real numbers, such that:
$H$ is continuous,
$H$ is symmetric w.r.t. the order of its arguments,
...
2
votes
0answers
92 views
A $GL_1$ Voronoi formula
I want a functional equation for the function defined by the Dirichlet series,
$$ D(s,a/q)= \sum_{n=1}^\infty \frac{e^{2\pi i n a/q}}{n^s}. $$
which sends $s$ to $1-s$ and preferably sends $a$ to ...
0
votes
0answers
42 views
Functional equation of Ramanujan type
For a given positive integer $k$, can one find a function $\phi(k;n)$ such that the following functional equation $$\phi(k;n)+\phi\left (k;\frac{1}{n}\right )=\zeta(2k)$$ is satisfied for every ...
0
votes
1answer
257 views
Is there any mathematical study about |a-b|=f(|g(a)-g(b)|)? or does there exist f() and g() satisfy this equation? [closed]
The problem is just as the title. It is clear that the linear function $f(x)=kx$ and $g(x)=(1/k)x$ can meet it. Is there any other function pairs f(x) and g(x) can meet this equation? or the equation ...
9
votes
3answers
487 views
Relating the roots of polynomials to the solution sets of certain functional equations
Consider a functional equation of the following form:
$$\sum_{k=0}^n a_k\,\underbrace{f(f(\cdots f}_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a_i\in ...
2
votes
2answers
124 views
Unusual Differential Equation for CDF
Consider the following differential equation
$$F(cx) = F(x) + x F'(x)$$
for $c>1$.
Does this differential equation belong to a some well known class?
Is there a way to find all the solutions ...
0
votes
1answer
135 views
How to solve for the nonlinear functional equation? [closed]
I got a nonlinear functional equation like:
$f(x) = g(x) + h(f(Ax))$,
where $A$ is a constant, $x$ is a scalar, $g()$ and $h()$ are given. The objective is to solve for the expression of $f(x)$.
...
2
votes
1answer
94 views
A Recursive Maximization Problem
Let $A\ge B>0$ be real constants. I say that a function $f:[0,1]\rightarrow[0,1]$ satisfies the $(A,B)$-condition if for all $p\in [0,1]$, the expression
$$q(A-Bp-Bf(q))$$
is maximized (not ...
0
votes
1answer
138 views
An Integral Functional Equation
Let $f$ be a non-negative function supported and integrable on the positive real axis, such that
$$\int_0^\infty f(x+y)p(y) dy = c[p] f(x), $$
where $c[p]$ a number (functional) dependent on function ...
4
votes
1answer
303 views
A Differential Equation with Nested Functions
This was posted to Math Stackexchange, but got no useful answers, and the more I think about it, the harder it seems.
I would like to know whether there exists a differentiable function from the ...
0
votes
0answers
118 views
Solving a Volterra integral equation with both limits variable
I would like to obtain an analytical solution to the following Volterra integral equation: $\int_{\alpha(x)}^{\beta(x)}y(t)(t^2-g(x))\,\mathrm dt=0$.
The functions $\alpha(x)$, $\beta(x)$ and $g(x)$ ...
2
votes
0answers
85 views
A Convolution Integral Equation
Is there any close-form solution for a function $f(t)$ satisfied the below equation:
$f(t)=g(t)+\frac{1}{t^2}(h(t)*f(t))$. Operator $*$ is convolution integral, and $g(t)$ and $h(t)$ are known ...
3
votes
0answers
188 views
The functional equation of Hofstadter's Q sequence
Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and
$Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything
on this sequence has been proved -- not even that $Q(n)$ is ...
2
votes
2answers
329 views
Functional equations
What are the general solutions of the functional equations?
$$
f(x,y)+f(y,z)=\frac{1}{f(x,z)}
$$
$$
f(x,y)f(y,z)f(x,z)=1
$$
0
votes
1answer
151 views
Some functional equations in two variables
I have two questions.
i) Does there exist a function $\varphi:\mathbb{R}\to\mathbb{R}$ for which the functional equation
$$
|f(x)-f(y)|=\frac{1}{|\varphi(x)-\varphi(y)|}
$$
has a solution ...
0
votes
1answer
155 views
Is still it weakly continuous ？
If $\{u_n\}$ is bounded in $H$（real Hilbert space）with inner product such that $(\cdot,\cdot)$, then ${\|u_n\|^2u_n}$ is bounded also. Passing a subsequence, one has that $\{\|u_n\|^2u_n\}$ converges ...
0
votes
0answers
63 views
Solving Multivariate and multi power Equations
There are 84 equations,
$r-A_DD^5-5A_DD^4D_i-(a_{d_i}+2b_{d_i}D_i)+(1-\alpha)\lambda_i=0,$
$A_L/L-r-(A_L/L^2)L_i-(a_{l_i}+2b_{l_i}L_i)-\lambda_i=0,$
$(1-\alpha)D_i-L_i=0$
where $i=1,\cdots,28$, ...
9
votes
2answers
423 views
Series defined by a fixed-point functional equation
Description
I my work, I met fixed-point functional equations of a very specific form. Since I am not expert in the domain of functional equations, I have not pushed the study of these very far. Here ...
-2
votes
1answer
222 views
why we are finding the stability for functional equations? [closed]
We know why we are finding stability of differential equation. but i need the answer for the question "why we are finding the stability for functional equations?" if possible explain with some sutable ...
2
votes
0answers
137 views
Critical case linear autonomous functional differential equation
I am looking for asymptotic ($t\to\infty$) behavior of the general
solution $g(t)$ to a following linear functional differential equation
$$
\text{(1)} \quad\quad\quad g'(t)-g'(t-T)=-g(t)
$$
with ...
23
votes
3answers
756 views
Rational functions with a common iterate
Let $f$ and $g$ be two rational functions. To avoid trivialities, we suppose that their degrees are
at least $2$. We say that they have a common iterate if $f^m=g^n$ for some positive integers $m,n$,
...
4
votes
1answer
325 views
Hahn-Banach theorem with real extended valued function
Hello to everyone,
My problem is the following: I have this version of the Hahn-Banach theorem:
Let V be a vector space and let $p:V\rightarrow \mathbb{R}$ be any
convex function. Let $W$ be a vector ...
0
votes
3answers
178 views
References for functional equations in more general settings than the reals
Hi there -
I'm Manny, a soon to be MSC thesist. I'm looking for a subject to write my thesis about - and recently I was caught by functional differential equations. Is there any neat reference for ...
1
vote
1answer
679 views
Techniques to solve equations involving a definite integral
Are there any well known techniques to solve a problem of the following form: $$\int_a^b f(x,\alpha) dx = g(\alpha),$$ where $a,b\in\mathbb{R}$ are fixed, $f$ and $g$ are known functions, ...
9
votes
2answers
772 views
Which trigonometric identities involve trigonometric functions?
Another question that's getting no answers on stackexchange:
Once upon a time, when Wikipedia was only three-and-a-half years old and most people didn't know what it was, the article titled ...
3
votes
3answers
1k views
Finding f such that f(f(x))=g(x) given g
Suppose $g(x)$ is a smooth increasing function defined for $x \ge 0$ such that $g(x) \ge x$ for all $x$. Does there exist a function $f$ with similar properties such that $f(f(x))=g(x)$ for all $x \ge ...
2
votes
3answers
333 views
solvability of an elementary functional equation
Is there some other way to characterize the functions $f:\mathbb Z\times \mathbb Z\to \mathbb Z$ which are expressible as
$$f(x,y)=g(x)+g(y)-g(x+y)$$
for some $g:\mathbb Z\to\mathbb Z$?
Easy facts: ...
8
votes
1answer
2k views
Are there any non-linear solutions of Cauchy's equation ($f(x+y)=f(x)+f(y)$) without assuming the Axiom of Choice?
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be s.t. $f(x+y) = f(x) + f(y), \ \forall x, y$
It is quite obvious that this implies $f(cx)=cx$ for all $c \in \mathbb{Z}$ and even further: $\forall c \in ...
3
votes
1answer
643 views
Restriction of a linear functional equation to surface of a sphere
Let $f_i : R \rightarrow R$ and $g_j: R \rightarrow R$ be unknown functions, for $i = 1, \cdots, N$ and $j = 1, \cdots, K$. Let $A$ be a $K \times N$ matrix whose columns are unit-length vectors ...
5
votes
5answers
704 views
Are there functions satisfying the following integral condition?
Can we find two functions $f$ and $g$ that are reasonably defined nontrivial(not everywhere zero, $f\neq g$, not linear polynomials) functions such that the following condition is satisfied?
$$ f( ...
41
votes
14answers
4k views
Does any research mathematics involve solving functional equations?
This is a somewhat frivolous question, so I won't mind if it gets closed. One of the categories of Olympiad-style problems (e.g. at the IMO) is solving various functional equations, such as those ...
35
votes
10answers
5k views
The functional equation $f(f(x))=x+f(x)^2$
I'd like to gather information and references on the following functional equation for power series $$f(f(x))=x+f(x)^2,$$$$f(x)=\sum_{k=1}^\infty c_k x^k$$
(so $c_0=0$ is imposed).
First things that ...
4
votes
1answer
452 views
Is this method of “fractional sums” using a Fourier series viable?
Hi.
I have this idea about developing what I call a "continuum sum", that is, a method to "add up a non-integer number of terms", i.e. to see if there is a "natural" way to assign a meaning to the ...
0
votes
2answers
525 views
Approach to solving a differential-functional equation
What could be an approach to solving such equations?
$$f'(x)=C \prod_{k=0}^x f(k)$$
$$\frac{g'(x)}{g(x)}=C+ \sum_{k=0}^{x-1} g(k)$$
Here the product and the sum are understood as indefinite sum and ...
2
votes
1answer
472 views
What are conditions to make f(x) defined by f(x)=f(x-1)*x + 1/e unique(for instance convex)?
[Background:]
Looking at the powerseries for the gamma-function
$ \Gamma(1+x) = 1 + a_1 x + a_2 x^2 - a_3 * x^3 + ... $
then we can arrive at a decomposition
$ \Gamma(1+x) = r(x) + g(x) $
...
3
votes
0answers
285 views
elementary Abel function of a polynomial
Is there an elementary real function $F$ such that
$F(1+F^{-1}(x))$ is a polynomial of degree at least 2 without real fixpoints.
1
vote
1answer
297 views
Does any iterative equation of n-th order have exactly n independent solutions?
Does any iterative equation of n-th order which does not inclute derivatives of order higher than 1 have exactly n independent solutions?
Let's designate n-th iterate of a function $y(x)$ as ...
0
votes
1answer
881 views
how to solve general integral equations with both variable lower and upper bounds
I am interested in solving the following equations:
$f(x) + \int_{\alpha(x)}^{\beta(x)}K(x,t)u(t)dt = 0$
and
$f(x) + \int_{\alpha(x)}^{\beta(x)}K(x,t)u(t)dt = u(x)$
when $u(x)$ is the unknown function ...
3
votes
1answer
2k views
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 ...
13
votes
9answers
5k views
What is the indefinite sum of tan(x)?
What is the indefinite sum of the tangent function, that is, the function $T$ for which
$\Delta_x T = T(x + 1) - T(x) = \tan(x)$
Of course, there are infinitely many answers, who all differ by a ...
1
vote
1answer
638 views
Infinite graphs as functional operators
Original Question
Consider an infinite tree of constant degree $k$. For such a tree we can consider the total number of nodes at depth $n$, $g(f)$, and the total number of paths from the root, ...
12
votes
2answers
918 views
Getting a differential equation for a function from a functional equation of its Mellin transform
If $f$ is a locally integrable function then its Mellin transform
$\mathcal{M}[f]$ is defined by
$$ \mathcal{M}[f] (s) = \int_0^{\infty} x^{s - 1} f (x) dx . $$
This integral usually converges in a ...
6
votes
1answer
449 views
“inversion” of a convolution
I have the following relation:
$$
\sum_{d|n} (1+1/x)^{d-1} F_{n/d}(x^d)=L_n(x)
$$
where the right hand side is (for every $n$) a polynomial in $x$, which I have an expression for, but it's not ...
6
votes
3answers
554 views
$n$th root of $(a,b) \mapsto (gm, am)$
Suppose $0 < a < b$, and let GM and AM be respectively the geometric and arithmetic means of $a$ and $b$. Does the mapping $(a,b) \mapsto (\mathrm{GM}, \mathrm{AM})$ have a well-behaved ...
2
votes
0answers
217 views
Algebraic Dirichlet series and beyond
I wonder what the "right" notion of "algebraic Dirichlet series" might be. Here I'm thinking of formal Dirichlet series $D(s)=\sum_{n\geq 1} a_n/n^s$, say with $a_n$ being rational numbers.
I'm ...
3
votes
1answer
266 views
approximately linear functions — more
Suppose $f,g$ are continuous functions from $\mathbb R$ to $\mathbb R$, with the property that
$$f(x)+f(y)=g(x+y)$$
for all $x,y$. Taking $x=y=z/2$ implies that $g(x)=2f(x/2)$ so that the above ...
2
votes
1answer
382 views
What kind of uniqueness can I conclude for solutions to a simple functional equation?
I'm going to ask a very vague question, and then give specifics for the version I particularly care about. I'm interested in answers at all levels of vagueness.
At the most vague version, I am in ...
11
votes
1answer
582 views
Modular equations for quasimodular forms
This problem is motivated by this question and by teaching
modular polynomials for the classical modular invariant $j(\tau)$.
The latter implies
that if we consider the fields of modular functions ...
5
votes
1answer
703 views
beyond differentially algebraic power series
In a recent question, we learned about the existence of functions that do not satisfy any algebraic differential equation.
One nice property of such equations is that there is a good way to ...
