Stability of a quadratic functional equation in the space of distributions

Abstract

We reformulate a quadratic functional equation of the form f (x + y + z) + f (x− y + z) + f (x + y − z) + f (−x + y + z) = 4f (x) + 4f (y) + 4f (z) and an inequality |f (x + y + z) + f (x − y + z) + f (x + y − z) + f (−x + y + z) − 4f (x) − 4f (y) − 4f (z)| e in the space of distributions. In view of this fact, we use a mollifier and Gauss transform to show that every distributional solution of the inequality is a tempered distribution and finally the stability problem of the equation in the sense of distributions. Mathematics subject classification (2000): 39B82, 46F12.