相关论文: Solution of the Kac--Bernstein functional equation…

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.…

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…

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 general analytic solution to the functional equation $$ \phi_1(x+y)= { { \biggl|\matrix{\phi_2(x)&\phi_2(y)\cr\phi_3(x)&\phi_3(y)\cr}\biggr|} \over { \biggl|\matrix{\phi_4(x)&\phi_4(y)\cr\phi_5(x)&\phi_5(y)\cr}\biggr|} } $$ is…

Our main result is that we describe the solutions $g,f:S\rightarrow\mathbb{C}$ of the functional equation \[g(x\sigma(y))=g(x)g(y)-f(x)f(y)+\alpha f(x\sigma(y)),\quad x,y\in S,\] where $S$ is a semigroup, $\alpha \in \mathbb{C}$ is a fixed…

The main purpose of this work is to provide the general solutions of a class of linear functional equations. Let $n\geq 2$ be an arbitrarily fixed integer, let further $X$ and $Y$ be linear spaces over the field $\mathbb{K}$ and let…

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…

Let $S$ be a semigroup. We determine the complex-valued solutions $f,g,h$ of the functional equation \begin{equation*}f(xy)=f(x)g(y)+g(x)f(y)+h(x)h(y), x,y\in S,\end{equation*} in terms of multiplicative functions, solutions of the special…

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.…

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,\\…

Making use of nonabelian harmonic analysis and representation theory, we solve the functional equation $$f_1(xy)+f_2(yx)+f_3(xy^{-1})+f_4(y^{-1}x)=f_5(x)f_6(y)$$ on arbitrary compact groups. The structure of its general solution is…

In this paper, we give the general solution of the functional equation $$\big\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\big\}=\big\{\|x+y\|,\|x-y\|\big\}\qquad(x,y\in X)$$ where $f:X\to Y$ and $X,Y$ are inner product spaces. Related equations are also…

We determine the complex-valued solutions of the following functional equation \[f(xy)+\mu (y)f(\sigma (y)x) = 2f(x)g(y),\quad x,y\in S,\] where $S$ is a semigroup and $\sigma$ an automorphism, $\mu :S\rightarrow \mathbb{C}$ is a…

In this paper, we establish the general solution of the functional equation $$f(nx+y)+f(nx-y)=n^2f(x+y)+n^2f(x-y)+2(f(nx)-n^2f(x))-2(n^2-1)f(y)\eqno {0 cm}$$for fixed integers $n$ with $n\neq0,\pm1$ and investigate the generalized…

In \cite{05} B. Ebanks and H. Stetk{\ae}r obtained the solutions of the functional equation $f(xy)-f(\sigma(y)x)=g(x)h(y)$ where $\sigma$ is an involutive automorphism and $f,g,h$ are complex-valued functions, in the setting of a group $G$…

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.…

The main result of the present paper is about the solutions of the functional equation \Eq{*}{ F\Big(\frac{x+y}2\Big)+f_1(x)+f_2(y)=G(g_1(x)+g_2(y)),\qquad x,y\in I, } derived originally, in a natural way, from the invariance problem of…

In this paper, we obtain the general solution and the stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary $t$-norms…

In this paper we find the solutions of the functional equation $$f(xy) = g(x)h(y) + \sum_{j=1}^n g_j(x)h_j(y), \;x,y \in M,$$ where $M$ is a monoid, $n\geq 2$, and $g_j$ (for $j=1,...,n$) are linear combinations of at least $2$ distinct…

We determine the complex-valued solutions of the Kannappan cosine functional law $g(xyz_{0})=g(x)g(y)-f(x)f(y)$, $x,y\in S$, where $S$ is a semigroup and $z_{0}$ is a fixed element in $S.$