Measure zero stability problem of a new quadratic functional equation

Let RR be the set of real numbers, Y   a Banach space and f:R→Yf:R→Y. We prove the Ulam–Hyers stability theorem for the new quadratic functional equation∑j=2k(f(x1+xj)+f(x1−xj))=2(k−1)f(x1)+2∑j=2kf(xj) for all (x1,…,xk)∈Γ(x1,…,xk)∈Γ, where Γ⊂RkΓ⊂Rk is of Lebesgue measure 0. Using the result we obtain an asymptotic behavior of the equation.