Let $g$ be an analytic function. My question is simple: How I can solve this functional equation: $$P(s)\exp(g(s))-P(2-s)\exp(g(2-s))=a $$ holds true for all $s∈ℂ$. Here, $a∈ℂ$, $P …

What are the general solutions of the functional equations? $$ f(x,y)+f(y,z)=\frac{1}{f(x,z)} $$ $$ f(x,y)f(y,z)f(x,z)=1 $$

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 …

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 …

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 posit …

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\|^2 …

There are 84 equations, $r-A_DD^5-5A_DD^4D_i-(a_{d_i}+2b_{d_i}D_i)+(1-\alpha)\lambda_i=0,$ $A_L/L-r-(A_L/L^2)L_i-(a_{l_i}+2b_{l_i}L_i)-\lambda_i=0,$ $(1-\alpha)D_i-L_i=0$ wher …

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 expla …

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 fun …

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. …

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- …

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 equation …

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) …

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 n …

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, wh …