On a General n-Linear Functional Equation

Abstract

Let X, Y be linear spaces over fields $\mathbb {K}$ and $\mathbb {F}$, respectively, and let $f :X^n \rightarrow Y$. For some fixed $a_{ji}\in \mathbb {K}\setminus \{0\}$,   $C_{i_1\ldots i_n}\in \mathbb {F}$, for $j\in \{1,\ldots , n\}$, $i,i_j\in \{1,2\}$, we consider the equation

$$\begin{aligned} \displaystyle f(a_{11}x_{11}+ & {} a_{12}x_{12},\ldots , a_{n1}x_{n1}+a_{n2}x_{n2})\\&=\sum _{1\leqslant i_1,\ldots , i_n\leqslant 2} C_{i_1\ldots i_n}f(x_{1i_1},\ldots ,x_{ni_n}),\qquad \quad {(*)} \end{aligned}$$

for all $x_{ji_j}\in X$, $j\in \{1, \ldots ,n\}$, $i_j\in \{1,2\}$. We determine the general solution of $(*)$ and as special cases we obtain some results for multi-Cauchy, multi-Jensen and multi-Cauchy-Jensen equations.