Abstract. We will show the general solution of the functionalequationf(x + ay) + f(x ay) + 2(a 2 1)f(x)= a 2 f(x + y) + a 2 f(x y) + 2a 2 (a 2 1)f(y)and investigate the Hyers-Ulam stability of the quartic set-valuedfunctional equation. 1. IntroductionThe theory of set-valued functions in Banach spaces is connected tothe control theory and the mathematical economics. Aumann [4] andDebreu [9] wrote papers that were motivated from the topic. We referthe reader to the papers by [1], [18], [11], [3], [16], [7] and [10].The stability problem of functional equations originated from a ques-tion of Ulam [25] concerning the stability of group homomorphisms. Hy-ers [12] gave a rst armative partial answer to the question of Ulam.Afterwards, the result of Hyers was generalized by Aoki [2] for additivemapping and by Rassias [23] for linear mappings by considering a un-bounded Cauchy di erence. Later, the result of Rassias has provided alot of inuence in the development of what we call Hyers-Ulam stabil-ity or Hyers-Ulam-Rassias stability of functional equations. For furtherinformation about the topic, we also refer the reader to [14], [13], [5]and [6]. Rassias [22] investigated stability properties of the followingquartic functional equation