Quadratic functional equations in a set of Lebesgue measure zero

摘要

Let R be the set of real numbers, Y a Banach space and f:R →Y. We prove the Hyers-Ulam stability theorem for the quadratic functional inequality||f(x+y) + f(x-y)-2f(x)-2f(y)|| ≤ ε for all (x, y) ? Ω, where ?? R~2 is of Lebesgue measure 0. Using the same method we dealt with the stability of two more functional equations in a set of Lebesgue measure 0.