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.