Cubic-Quartic Functional Equation

and Applied Analysis 3 In 2008, Gordji et al. 17 provided the solution as well as the stability of a mixed type cubic-quartic functional equation. We only mention here the papers 19, 32, 33 concerning the stability of the mixed type functional equations. In this paper, we deal with the following general cubic-quartic functional equation: f ( x ky ) f ( x − ky) k2(f(x y) f(x − y)) 2 ( 1 − k2 ) f x k4 − k2 4 × (f(2y) − 8f(y)) ̃ f 2x − 16 ̃ f x , where ̃ f x : f x f −x . 1.7 Then it follows easily that the function f x ax4 bx3 satisfies 1.7 . We investigate the general solution and the generalized Hyers-Ulam-Rassias stability of the functional equation 1.7 . 2. General Solution In this section, we establish the general solution of functional equation 1.7 . Theorem 2.1. Let X, Y be vector spaces and let f : X → Y be a function. Then f satisfies 1.7 if and only if there exists a unique symmetric multiadditive function Q : X × X × X × X → Y and a unique function C : X ×X ×X → Y such that f x Q x, x, x, x C x, x, x for all x ∈ X, where the function C is symmetric for each fixed one variable and is additive for fixed two variables. Proof. Let f satisfies 1.7 . We decompose f into the even part and odd part by setting fe x 1 2 ( f x f −x ), fo x 12 ( f x − f −x ) 2.1 for all x ∈ X. By 1.7 , we have fe ( x ky ) fe ( x − ky) 1 2 [ f ( x ky ) f (−x − ky) f(x − ky) f(−x ky)] 1 2 [ f ( x ky ) f ( x − ky)] 1 2 [ f ( −x (−ky)) f( −x − (−ky))] 1 2 [ k2 ( f ( x y ) f ( x − y)) 2 ( 1 − k2 ) f x k4 − k2 4 ( f ( 2y ) − 8f(y)) ̃ f 2x − 16 ̃ f x ]