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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 ...