相关论文: Regular solutions of a functional equation derived…
In this paper we describe the solutions of the functional equation \begin{equation*} F\Big(\frac{x+y}2\Big)+f_1(x)+f_2(y)=G \big(g_1(x)+g_2(y)) \end{equation*} defined on an open subinterval of $ \mathbb{R} $. Improving previous results we…
In this paper, we are going to describe the solutions of the functional equation $$ \varphi\Big(\frac{x+y}{2}\Big)(f(x)+f(y))=\varphi(x)f(x)+\varphi(y)f(y) $$ concerning the unknown functions $\varphi$ and $f$ defined on an open interval.…
The purpose of this paper is to investigate the following invariance equation involving two $2$-variable generalized Bajraktarevi\'c means, i.e., we aim to solve the functional equation $$…
In this paper, we are dealing with the solution of the functional equation $$ \varphi\Big(\frac{x+y}2\Big)(f(x)-f(y))=F(x)-F(y), $$ concerning the unknown functions $\varphi,f$ and $F$ defined on a same open subinterval of the reals.…
The purpose of this paper is to investigate the invariance of the arithmetic mean with respect to two weighted Bajraktarevi\'c means, i.e., to solve the functional equation $$…
In this paper we determine the solutions $(\varphi,f_1,f_2)$ of the Pexider functional equation \[\varphi\Big(\frac{x+y}2\Big)\big(f_1(x)-f_2(y)\big)=0,\qquad (x,y)\in I_1\times I_2,\] where $I_1$ and $I_2$ are nonempty open subintervals.…
This paper examines various aspects related to the Cauchy functional equation $f(x+y)=f(x)+f(y)$, a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to…
The purpose of this paper is to investigate the equality problem of generalized Bajraktarevi\'c means, i.e., to solve the functional equation \begin{equation}\label{E0}\tag{*}…
We study the problem of the existence of increasing and continuous solutions $\varphi\colon[0,1]\to[0,1]$ such that $\varphi(0)=0$ and $\varphi(1)=1$ of the functional equation \begin{equation*}…
We continue our study started in "On a problem of Janusz Matkowski and Jacek Weso{\l}owski" (see arXiv:1703.08459) of the functional equation \begin{equation*} \varphi(x)=\sum_{n=0}^{N}\varphi(f_n(x))-\sum_{n=0}^{N}\varphi(f_n(0))…
In this paper we investigate the functional equation \[ \varphi \left( \frac{x+y}{2} \right) \left( \psi_1(x) - \psi_2(y) \right) = 0 \hspace{20mm} \left( \mbox{ for all } x \in I_1 \mbox{ and } y \in I_2 \right) \] where $ I_1 \,, I_2 $…
Given a semigroup $S$ generated by its squares, we determine the complex-valued solutions of the following system of cosine-sine functional equations \begin{align*} f(xy)=f(x)g_{1}(y)+g_{1}(x)f(y)+\lambda_{1}^{2}\,h(x)h(y),\; x,y\in S,\\…
The purpose of the present paper is to solve (under some assumption on the domain) the equation $$ g(x+y)-g(x)-g(y)=xf(y)+yf(x). $$ After determining the general solutions, we will investigate the so--called alien solutions. %More…
The equality problem of the two-variable Bajraktarevi\'c means can be expressed as the functional equation $$ \left(\frac{f}{g}\right)^{-1}\bigg(\frac{f(x)+f(y)}{g(x)+g(y)}\bigg)…
Given a semigroup $S$ equipped with an involutive automorphism $\sigma$, we determine the complex-valued solutions $f,g,h$ of the functional equation \begin{equation*}f(x\sigma(y))=f(x)g(y)+g(x)f(y)+h(x)h(y),\,\,x,y\in S,\end{equation*} in…
In this work, we prove the existence of integrable solutions for the following generalized mixed-type nonlinear functional integral equation $$x(t)=g\left(t,(Tx)(t)\right)+f\left(t,\int_0^t…
In this short note, we introduce probabilistic Cauchy functional equations, specifically, functional equations of the following form: $$ f(X_1 + X_2) \stackrel{d}{=} f(X_1) + f(X_2), $$ where $X_1$ and $X_2$ represent two independent…
This paper investigates functional equations arising from perturbations of Cauchy differences. We study equations of the form \[ f(x+y)-f(x)-f(y)=B(x,y) \quad \text{or} \quad f(xy)-f(x)f(y) = B(x,y) \] where $B$ is a biadditive mapping, and…
In this paper we study the solutions and stability of the generalized Wilson's functional equation $\int_{G}f(xty)d\mu(t)+\int_{G}f(xt\sigma(y))d\mu(t)=2f(x)g(y),\; x,y\in G$, where $G$ is a locally compact group, $\sigma$ is a continuous…
This paper offers a solution of the functional equation $$ \big(tf(x)+(1-t)f(y)\big)\varphi(tx+(1-t)y)=tf(x)\varphi(x)+(1-t)f(y)\varphi(y) \qquad(x,y\in I), $$ where $t\in\,]0,1[\,$ is a fixed number, $\varphi:I\to\mathbb{R}$ is strictly…