The functional-equations tag has no wiki summary.
2
votes
1answer
141 views
Survey on functional equations and inequalities
Where can I find a comprehensive survey monograph on functional equations and inequalities from sketch to current research trends with some focus on applications (both inside and outside mathematics)?
...
0
votes
0answers
46 views
System of integral equations
Let $K_1,K_2,K_3,K_4$ be integral operators. I'm interested in the following system of integral equations.
$$\begin{cases}
g_1 = K_1f_1 + K_2f_2 \\
g_2 = K_3f_1 + K_4f_2
\end{cases}$$
I'm ...
0
votes
0answers
90 views
Distributive law between Kleisli triples
A distributive law of a monad $S$ over a monad $T$ is a natural transformation $l : T S \to S T$ such that:
$l \circ T \eta^S = \eta^S T$
$l \circ \eta^T S = S \eta^T$
$\mu^S T \circ S l \circ l S = ...
0
votes
1answer
63 views
Functions with special separability
Suppose we have differentiable functions $F$, $f_1, \dots, f_n$, and $g_1, \dots, g_n$ satisfy the following relation
$$ F(x+y) = \sum_{i=1}^n f_i(x) g_i(y).$$
What are the possible forms of $F$?
0
votes
1answer
53 views
Uniqueness of solutions of functional equations [closed]
A solution to $f(2x)=\alpha f(x)$ with a boundary condition $f(\beta) = \beta$ is
$$ f(x) = \left( \frac{\beta}{\alpha^{\log_2 \beta}} \right) \alpha^{\log_2 x}. $$
Do we know whether or not the ...
3
votes
1answer
134 views
Existence of solution for this set of polynomial equations
We are given a number $n$ and a vector $p=(p_1,p_2,\ldots,p_r)$, where
$$p_1\geq p_2 \geq \ldots \geq p_r > 0 ; \ \ \ \ \sum_{i\in [r]} p_i \leq 1$$
I'm interested in proving that a solution for ...
5
votes
2answers
311 views
Does this equation has a closed-form solution for $t$? ($(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i)$)
We are given $n\in \mathbb N^+$ and $p\in[\frac{1}{2},\frac{n+1}{n+2}]$.
Our goal is to find $t\in[0,1]$ such that
$$(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i$$
Is there a closed-form ...
3
votes
2answers
309 views
Consistent price index
This question came out of a discussion with a colleague from economics about price indices. Here is MattF's formulation of the question which differs somehow from the original problem.
Let ...
2
votes
1answer
81 views
Counterexample for the Generalized Associativity Equation
The Generalized Associativity Equation is given by
$$ F(G(x,y),z)=K(x,H(y,z)),$$
where the functions $F,G,H$ and $K$ are all from $\mathbb{R}^2$ to $\mathbb{R}$. In his book "Lectures on Functional ...
4
votes
0answers
55 views
Archimedean $\varepsilon$-factors
Let $K$ be either $\bf R$ or $\bf C$. Let $p$ and $q$ be integers with $p \leq -1$, $q \geq 0$, and $p+q=-1$. Consider the Hodge structure $M = M(p,q)$ over $K$ with coefficients in $\bf R$, defined ...
1
vote
0answers
29 views
Family of (Cumulative Distribution) Functions
I'm looking for a 2 (or more)-parameter family of functions $F$ with the following properties:
For each $f \in F$, $f(0)=0$, $f(1)=1$, and $f$ is (weakly) increasing.
$F$ is closed under products.
...
2
votes
0answers
154 views
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 ...
2
votes
1answer
330 views
A functional inequality
$g:[0,1]\to[0,1]$ continuously differentiable and increasing such that for
all integers $t>0$ and for all $r\in(0,1)$, $g(r^{t+1})>g(r)\cdot g(r^t)$. Does this imply
that for all ...
0
votes
2answers
135 views
Solving a functional equation
I would like to consider the following simple problem. I want to find two functions $f,g : \mathbb R \to \mathbb R$ such that, being given a collection $(h_v)_{v\in V}$ of real functions indexed by ...
4
votes
1answer
324 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]:
...
5
votes
1answer
160 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
111 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
1answer
266 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 ...
10
votes
3answers
530 views
Relating the roots of polynomials to the solution sets of certain functional equations
Consider a functional equation of the following form:
$$\sum_{k=0}^n a_k\,\underbrace{f(f(\cdots f}_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a_i\in ...
2
votes
2answers
150 views
Unusual Differential Equation for CDF
Consider the following differential equation
$$F(cx) = F(x) + x F'(x)$$
for $c>1$.
Does this differential equation belong to a some well known class?
Is there a way to find all the solutions ...
0
votes
1answer
167 views
How to solve for the nonlinear functional equation? [closed]
I got a nonlinear functional equation like:
$f(x) = g(x) + h(f(Ax))$,
where $A$ is a constant, $x$ is a scalar, $g()$ and $h()$ are given. The objective is to solve for the expression of $f(x)$.
...
2
votes
1answer
103 views
A Recursive Maximization Problem
Let $A\ge B>0$ be real constants. I say that a function $f:[0,1]\rightarrow[0,1]$ satisfies the $(A,B)$-condition if for all $p\in [0,1]$, the expression
$$q(A-Bp-Bf(q))$$
is maximized (not ...
0
votes
1answer
182 views
An Integral Functional Equation
Let $f$ be a non-negative function supported and integrable on the positive real axis, such that
$$\int_0^\infty f(x+y)p(y) dy = c[p] f(x), $$
where $c[p]$ a number (functional) dependent on function ...
4
votes
1answer
372 views
A Differential Equation with Nested Functions
This was posted to Math Stackexchange, but got no useful answers, and the more I think about it, the harder it seems.
I would like to know whether there exists a differentiable function from the ...
2
votes
0answers
101 views
A Convolution Integral Equation
Is there any close-form solution for a function $f(t)$ satisfied the below equation:
$f(t)=g(t)+\frac{1}{t^2}(h(t)*f(t))$. Operator $*$ is convolution integral, and $g(t)$ and $h(t)$ are known ...
3
votes
0answers
257 views
The functional equation of Hofstadter's Q sequence
Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and
$Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything
on this sequence has been proved -- not even that $Q(n)$ is ...
2
votes
2answers
354 views
Functional equations
What are the general solutions of the functional equations?
$$
f(x,y)+f(y,z)=\frac{1}{f(x,z)}
$$
$$
f(x,y)f(y,z)f(x,z)=1
$$
0
votes
1answer
175 views
Some functional equations in two variables
I have two questions.
i) Does there exist a function $\varphi:\mathbb{R}\to\mathbb{R}$ for which the functional equation
$$
|f(x)-f(y)|=\frac{1}{|\varphi(x)-\varphi(y)|}
$$
has a solution ...
0
votes
1answer
159 views
Is still it weakly continuous ？
If $\{u_n\}$ is bounded in $H$（real Hilbert space）with inner product such that $(\cdot,\cdot)$, then ${\|u_n\|^2u_n}$ is bounded also. Passing a subsequence, one has that $\{\|u_n\|^2u_n\}$ converges ...
0
votes
0answers
72 views
Solving Multivariate and multi power Equations
There are 84 equations,
$r-A_DD^5-5A_DD^4D_i-(a_{d_i}+2b_{d_i}D_i)+(1-\alpha)\lambda_i=0,$
$A_L/L-r-(A_L/L^2)L_i-(a_{l_i}+2b_{l_i}L_i)-\lambda_i=0,$
$(1-\alpha)D_i-L_i=0$
where $i=1,\cdots,28$, ...
9
votes
2answers
450 views
Series defined by a fixed-point functional equation
Description
I my work, I met fixed-point functional equations of a very specific form. Since I am not expert in the domain of functional equations, I have not pushed the study of these very far. Here ...
-2
votes
1answer
296 views
why we are finding the stability for functional equations? [closed]
We know why we are finding stability of differential equation. but i need the answer for the question "why we are finding the stability for functional equations?" if possible explain with some sutable ...
2
votes
0answers
145 views
Critical case linear autonomous functional differential equation
I am looking for asymptotic ($t\to\infty$) behavior of the general
solution $g(t)$ to a following linear functional differential equation
$$
\text{(1)} \quad\quad\quad g'(t)-g'(t-T)=-g(t)
$$
with ...
24
votes
3answers
859 views
Rational functions with a common iterate
Let $f$ and $g$ be two rational functions. To avoid trivialities, we suppose that their degrees are
at least $2$. We say that they have a common iterate if $f^m=g^n$ for some positive integers $m,n$,
...
5
votes
1answer
355 views
Hahn-Banach theorem with real extended valued function
Hello to everyone,
My problem is the following: I have this version of the Hahn-Banach theorem:
Let V be a vector space and let $p:V\rightarrow \mathbb{R}$ be any
convex function. Let $W$ be a vector ...
0
votes
3answers
183 views
References for functional equations in more general settings than the reals
Hi there -
I'm Manny, a soon to be MSC thesist. I'm looking for a subject to write my thesis about - and recently I was caught by functional differential equations. Is there any neat reference for ...
2
votes
2answers
825 views
Techniques to solve equations involving a definite integral [closed]
Are there any well known techniques to solve a problem of the following form: $$\int_a^b f(x,\alpha) dx = g(\alpha),$$ where $a,b\in\mathbb{R}$ are fixed, $f$ and $g$ are known functions, ...
9
votes
2answers
844 views
Which trigonometric identities involve trigonometric functions?
Another question that's getting no answers on stackexchange:
Once upon a time, when Wikipedia was only three-and-a-half years old and most people didn't know what it was, the article titled ...
3
votes
3answers
1k views
Finding f such that f(f(x))=g(x) given g
Suppose $g(x)$ is a smooth increasing function defined for $x \ge 0$ such that $g(x) \ge x$ for all $x$. Does there exist a function $f$ with similar properties such that $f(f(x))=g(x)$ for all $x \ge ...
2
votes
3answers
347 views
solvability of an elementary functional equation
Is there some other way to characterize the functions $f:\mathbb Z\times \mathbb Z\to \mathbb Z$ which are expressible as
$$f(x,y)=g(x)+g(y)-g(x+y)$$
for some $g:\mathbb Z\to\mathbb Z$?
Easy facts: ...
10
votes
1answer
2k views
Are there any non-linear solutions of Cauchy's equation ($f(x+y)=f(x)+f(y)$) without assuming the Axiom of Choice?
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be s.t. $f(x+y) = f(x) + f(y), \ \forall x, y$
It is quite obvious that this implies $f(cx)=cx$ for all $c \in \mathbb{Z}$ and even further: $\forall c \in ...
3
votes
1answer
684 views
Restriction of a linear functional equation to surface of a sphere
Let $f_i : R \rightarrow R$ and $g_j: R \rightarrow R$ be unknown functions, for $i = 1, \cdots, N$ and $j = 1, \cdots, K$. Let $A$ be a $K \times N$ matrix whose columns are unit-length vectors ...
5
votes
5answers
759 views
Are there functions satisfying the following integral condition?
Can we find two functions $f$ and $g$ that are reasonably defined nontrivial(not everywhere zero, $f\neq g$, not linear polynomials) functions such that the following condition is satisfied?
$$ f( ...
41
votes
14answers
5k 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 ...
42
votes
10answers
6k 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 ...
5
votes
1answer
502 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
534 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
554 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
289 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
300 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 ...
