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

Let X and Y be vector spaces. We show that a function f:X→Y with f(0)=0 satisfies Δf(x1,…,xn)=0 for all x1,…,xn∈X, if and only if there exist functions C:X×X×X→Y, B:X×X→Y and A:X→Y such that f(x)=C(x,x,x)+B(x,x)+A(x) for all x∈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 Δf(x1,…,xn)=∑k=2n(∑i1=2k∑i2=i1+1k+1⋯∑in-k+1=in-k+1n)f(∑i=1,i≠i1,…,in-k+1nxi-∑r=1n-k+1xir)+f(∑i=1nxi)-2n-2∑i=2n(f(x1+xi)+f(x1-xi))+2n-1(n-2)f(x1) (n∈N, n≥3) for all x1,…,xn∈X. Furthermore, we solve the stability problem for a given function f satisfying Δf(x1,…,xn)=0, in the Menger probabilistic normed spaces