Approximation of Mixed-Type Functional Equations in Menger PN-Spaces

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.