## Abstract

Let $X$ and $Y$ be vector spaces. We show that a function $f:X\to Y$ with $f\left(0\right)=0$ satisfies $\Delta f({x}_{1},\dots ,{x}_{n})=0$ for all ${x}_{1},\dots ,{x}_{n}\in X$, if and only if there exist functions $C:X\times X\times X\to Y$, $B:X\times X\to Y$ and $A:X\to Y$ such that $f\left(x\right)=C(x,x,x)+B(x,x)+A\left(x\right)$ for all $x\in X$, where the function $C$ is symmetric for each fixed one variable and is additive for fixed two variables, $B$ is symmetric bi-additive, $A$ is additive and $\Delta f({x}_{1},\dots ,{x}_{n})=$ ${\sum}_{k=2}^{n}({\sum}_{{i}_{1}=2}^{k}{\sum}_{{i}_{2}={i}_{1}+1}^{k+1}\cdots {\sum}_{{i}_{n-k+1}={i}_{n-k}+1}^{n})f({\sum}_{i=1,i\ne {i}_{1},\dots ,{i}_{n-k+1}}^{n}{x}_{i}-{\sum}_{r=1}^{n-k+1}{x}_{{i}_{r}})+$ $f\left({\sum}_{i=1}^{n}{x}_{i}\right)-{2}^{n-2}{\sum}_{i=2}^{n}\left(f\right({x}_{1}+{x}_{i})+f({x}_{1}-{x}_{i}\left)\right)$ $+{2}^{n-1}(n-2)f\left({x}_{1}\right)$ ($n\in \mathbb{N}$, $n\ge 3$) for all ${x}_{1},\dots ,{x}_{n}\in X$. Furthermore, we solve the stability problem for a given function $f$ satisfying $\Delta f({x}_{1},\dots ,{x}_{n})=0$, in the Menger probabilistic normed spaces.

## Citation

M. Eshaghi Gordji. H. Khodaei. Y. W. Lee. G. H. Kim. "Approximation of Mixed-Type Functional Equations in Menger PN-Spaces." Abstr. Appl. Anal. 2012 1 - 17, 2012. https://doi.org/10.1155/2012/392179

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