HYERS-ULAM-RASSIAS STABILITY OF FUNCTIONAL EQUATIONS IN CONNECTION WITH CLASSICAL INEQUALITIES

Abstract

In this paper, the historical backgrounds and important results of the Hyers-Ulam-Rassias stability of various functional equations are mainly surveyed. In the first section, we will introduce the exact de¯nitions of Hyers-Ulam stability, Hyers- Ulam-Rassias stability and superstability which are applicable to a large class of functional equations. Section 2 is devoted to the Hyers-Ulam-Rassias stability of the additive Cauchy equation including the case where the domain is restricted. The stability results of the Jensen's equation are discussed in Section 3. Valuable stability results of quadratic functional equation are presented in Section 4, and signi¯cant stability results of multiplicative Cauchy equation are surveyed in Section 5. Other equations such as cosine equation, sine equation, Hosszu's equation and gamma functional equation are treated in Section 6.