Stabilities of a 3-dimensional sine functional equation

Abstract

The stability of functional equations is currently active mathematical research problems. A number of research papers after the pioneer paper of Th. M. Rassias in 1978 were published. In this thesis, we study the solution of a 3-dimensional sine functional equation. cf (x)f(y)f(z) = f(x+y+z) – f(x+y-z) – f(x-y+z) +f(x-y-z) and investigate its stabilities. We consider it into two cases c = 0 and c [is not equal to] 0. For the case c [is not equal to] 0, it will be called the non-degenerate form. We can prove that its general solution is contained in the class of the general solution of the sine functional equation and we give its superstability. For the case c = 0, we will call the degenerate form. We get that its general solution is the same as that of the Jensen's functional equation and we prove the Hyers-Ulam-Rassias stability of the degenerate case then treat the Hyers-Ulam-Rassias stability and Hyers-Ulam stability as special cases.