General 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 ]


Introduction
The stability problem of functional equations originated from a question of Ulam 1 in 1940, concerning the stability of group homomorphisms.Let G 1 , • be a group and let G 2 , * be a metric group with the metric d •, • .Given > 0, does there exist a δ > 0, such that if a mapping h : G 1 → G 2 satisfies the inequality d h x • y , h x * h y < δ for all x, y ∈ G 1 , then there exists a homomorphism H : G 1 → G 2 with d h x , H x < for all x ∈ G 1 ?In other words, under what condition does there exists a homomorphism near an approximate homomorphism?The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation.In 1941, Hyers 2 gave the first affirmative answer to the question of Ulam for Banach spaces.Let f : E → E be a mapping between Banach spaces such that for all x, y ∈ E and for some δ > 0. Then there exists a unique additive mapping T : E → E such that Abstract and Applied Analysis for all x ∈ E.Moreover, if f tx is continuous in t ∈ R for each fixed x ∈ E, then T is linear.In 1978, Rassias 3 proved the following theorem.
Theorem 1.1.Let f : E → E be a mapping from a normed vector space E into a Banach space E subject to the inequality for all x, y ∈ E, where and p are constants with > 0 and p < 1.Then there exists a unique additive mapping T : E → E such that for all x ∈ E. If p < 0 then inequality 1.3 holds for all x, y / 0 and 1.4 for x / 0. Also, if the function t → f tx from R into E is continuous in real t for each fixed x ∈ E, then T is linear.
In 1990, Rassias during the 27th International Symposium on Functional Equations asked the question whether such a Theorem can also be proved for all real values of p that are greater or equal to one.In 1991, Gajda 4 , following the same approach as that of Rassias, provided an affirmative solution to this question for all real values of p that are strictly greater than one.The new concept of stability of the linear mapping that was inspired by Rassias' stability theorem is called Hyers-Ulam-Rassias stability of functional equations.
Jun and Kim 5 introduced the following cubic functional equation: and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation 1.5 .The function f x x 3 satisfies the functional equation 1.5 , which is thus called a cubic functional equation.Every solution of the cubic functional equation is said to be a cubic function.Jun and Kim proved that a function f between real vector spaces X and Y is a solution of 1.5 if and only if there exists a unique function C : X × X × X → Y such that f x C x, x, x for all x ∈ X and C is symmetric for each fixed one variable and is additive for fixed two variables.The stability of the quartic functional equations was studied by Park and Bae 6 , when In fact, they proved that a function f between real vector spaces X and Y is a solution of 1.6 if and only if there exists a unique symmetric multi-additive function x for all x ∈ X see also 7, 8 .It is straightforward to verify that the function f x x 4 satisfies the functional equation 1.6 , which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic function.
The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem.see 9-45 .
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: where f x : f x f −x .

1.7
Then it follows easily that the function f x ax 4 bx 3 satisfies 1.7 .We investigate the general solution and the generalized Hyers-Ulam-Rassias stability of the functional equation 1.7 .

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 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 for all x ∈ X.By 1.7 , we have Abstract and Applied Analysis for all x, y ∈ X.This means that f e satisfies 1.7 , or 1.5 e for all x, y ∈ X. Applying the fact that the function f e is even for all x, y ∈ X, 1.5 e can be written in the form for all x, y ∈ X.Now be setting x y 0 in 2.3 , we get f e 0 0. Similarly, by setting y 0 in 2.3 , we obtain for all x ∈ X. Hence 2.3 can be written as for all x, y ∈ X.By substituting x by x y in 2.5 , we have for all x, y ∈ X. Substituting −y for y in 2.6 , we get by evenness of f for all x, y ∈ X. Adding 2.6 to 2.7 , we obtain for all x, y ∈ X.By substituting x by x − ky in 2.5 , we have 2.9 for all x, y ∈ X. Substituting −x for x in 2.9 , we get by evenness of f e f e x f e x 2ky for all x, y ∈ X. Adding 2.9 to 2.10 , we obtain x for all x ∈ X.On the other hand, we can show that f o satisfies 1.7 , or for all x, y ∈ X.By oddness of f o for all x, y ∈ X, 1.5 o can be written as for all x, y ∈ X.Now by setting x y 0 in 3.2 , we get f o 0 0, and by setting x 0 in 2.16 , we obtain 8f o y 2.17 for all y ∈ X. Hence 2.16 can be written as for all x, y ∈ X. Replacing x by x − y in 2.18 , we obtain for all x, y ∈ X. Substituting −x for x in 2.19 , we get by oddness of for all x, y ∈ X.If we subtract 2.19 from 2.20 , we obtain for all x, y ∈ X.By substituting x by x ky in 2.18 , we have for all x, y ∈ X. Substituting −y for y in 2.22 , we get for all x, y ∈ X. Adding 2.22 to 2.23 , we obtain for all x, y ∈ X.Now, by using 2.18 , 2.21 , and 2.24 , we lead to for all x, y ∈ X.If we replace y by 2y in 2.18 , we get for all x, y ∈ X.If we compare 2.25 with 2.26 , then we conclude that for all x, y ∈ X. Replacing x by 2x in 2.27 , we get for all x, y ∈ X.Finally, it follows from 2.17 and 2.28 that for all x, y ∈ X.By multiplying both sides of 2.29 by 1/4, we get for all x, y ∈ X.This means that f o is a cubic function and that there exits a unique function C : X × X × X → Y such that f o x C x, x, x for all x ∈ X and C is symmetric for each fixed one variable and is additive for fixed two variables.Thus for all x ∈ X, we have

2.31
The proof of the converse is trivially.
The following corollary is an alternative result of above Theorem 2.1.
Corollary 2.2.Let X, Y be vector spaces, and let f : X → Y be a function satisfying 1.7 .Then the following assertions hold.
Abstract and Applied Analysis 9

Stability
We will investigate the generalized Hyers-Ulam-Rassias stability problem for the functional equation 1.7 .In the following, let X be a real vector space and let Y be a Banach space.Given f : X → Y , we define the difference operator D f : X × X → Y by for all x, y ∈ X.
Theorem 3.1.Let j ∈ {−1, 1} be fixed and let ϕ : X × X → 0, ∞ be a function such that for all x, y ∈ X. Suppose that an even function f : X → Y with f 0 0 satisfies the inequality for all x, y ∈ X.Then the limit exists for all x ∈ X and Q : X → Y is a unique quartic function satisfying for all x ∈ X, where Proof.Let j 1.It follows from 3.3 and using evenness of f that Abstract and Applied Analysis for all x, y ∈ X. Replacing x and y by 0 and x in 3.7 , respectively, we see that for all x ∈ X.If we divide both sides of 3.8 by 2, we get for all x ∈ X. Putting y 0 in 3.7 , we obtain 2f 2x − 32f x ≤ ϕ x, 0 3.10 for all x ∈ X.If we multiply both sides of 3.10 by k 4 − k 2 /16, then we have for all x ∈ X.It follows from 3.9 and 3.11 that for all x ∈ X.Thus by 3.12 , we get 14 for all x ∈ X.If we replace x in 3.14 by x/k n 1 and multiply both sides of 3.14 by k 4n , we see that for all x ∈ X and all nonnegative integers n.So for all nonnegative integers n and m with n ≥ m and all x ∈ X.By 3.2 , we infer that for all x ∈ X.It follows from 3.16 and 3.17 that the sequence {k 4n f x/k n } is a Cauchy sequence for all x ∈ X.Since Y is complete, the sequence {k 4n f x/k n } converges for all x ∈ X.So one can define a mapping Q : X → Y by 3.4 for all x ∈ X. Letting m 0 and passing the limit n → ∞ in 3.16 , we obtain 3.5 .It follows from 3.4 , 3.15 , and 3.17 that for all x ∈ X.On the other hand, it follows from 3.2 , 3.3 , and 3.4 that for all x ∈ X.So it follows from 3.5 and 3.22 that for all x ∈ X. Hence Q Q .For j −1, the proof of the theorem is similar.
Theorem 3.2.Let j ∈ {−1, 1} be fixed, and let ϕ : X × X → 0, ∞ be a function such that for all x, y ∈ X. Suppose that an odd function f : X → Y with f 0 0 satisfies the inequality 3.3 .Then the limit exists for all x ∈ X and C : X → Y is a unique cubic function satisfying for all x ∈ X.
Proof.Let j 1.It follows from 3.3 and using oddness of f that for all x, y ∈ X. Replacing x and y by 0 and x in 3.27 , respectively, we see that for all x ∈ X.If we multiply both sides of 3.28 by 4/ k 4 − k 2 , we get for all x ∈ X.If we replace x in 3.29 by x/2 n 1 and multiply both sides of 3.29 by 2 3n , we see that for all x ∈ X and all nonnegative integers n.So for all nonnegative integers n and m with n ≥ m and all x ∈ X.By 3.24 , we infer that for all x, y ∈ X.It follows from 3.31 and 3.32 that the sequence {2 3n f x/2 n } is a Cauchy sequence for all x ∈ X.Since Y is complete, the sequence {2 3n f x/2 n } converges for all x ∈ X.So one can define a mapping C : X → Y by 3.25 for all x ∈ X. Letting m 0 and passing the limit n → ∞ in 3.31 , we obtain 3.26 .It follows from 3.25 , 3.30 , and 3.32 that for all x ∈ X.On the other hand, it follows from 3.3 , 3.24 , and 3.25 that for all x ∈ X. Hence C C .For j −1, the proof of the theorem is similar.
for all x, y ∈ X, then there exists a unique quartic function Q : X → Y and a unique cubic function for all x ∈ X, where

3.41
Proof.Let f e x 1/2 f x f −x for all x ∈ X.Then f e 0 0 and f e is even function satisfying D f e x, y ≤ 1/2 φ x, y φ −x, −y for all x, y ∈ X.By Theorem 3.1, there exists a unique quartic function for all x ∈ X, where Then f e 0 0 and f o is an odd function satisfying D f o x, y ≤ 1/2 φ x, y φ −x, −y for all x, y ∈ X.Hence, in view of Theorem 3.2, there exists a unique cubic function C : X → Y such that for all x ∈ X, where for all x, y ∈ X.On the other hand, we have f x f e x f o x for all x ∈ X.Then by combining 3.42 and 3.44 , it follows that We are going to investigate the Hyers-Ulam-Rassias stability problem for functional equation 1.7 .for all x ∈ X.

Corollary 3 . 4 .Corollary 3 . 5 .Corollary 3 . 6 .
Let p ∈ −∞, 3 ∪ 4, ∞ , θ > 0. Suppose f : X → Y satisfies f 0 0 and inequality D f x, y ≤ θ x p y p 3.47for all x, y ∈ X.Then there exist a unique quartic function Q : X → Y and a unique cubic functionC : X → Y satisfying f x − Q x − C x for all x ∈ X.Proof.In Theorem 3.3, put φ x, y θ x p y p for all x, y ∈ X.Similarly, one can solve Ulam stability problem for functional equation 1.7 when the norm of the Cauchy difference is controlled by the mixed type product-sum function x, y −→ θ x Let u, v, p be real numbers such that u v, p ∈ −∞, 3 ∪ 4, ∞ and θ > 0. Suppose f : X → Y satisfies f 0 0 and inequalityD f x, y ≤ θ xfor all x, y ∈ X.Then there exist a unique quartic function Q : X → Y and a unique cubic functionC : X → Y satisfying f x − Q x − C x for all x ∈ X.Applying Corollary 3.4, one can obtain the stability of the functional equation 1.7 in the following form.Let be a positive real number.Suppose f : X → Y satisfies f 0 0 and D f x, y ≤ for all x, y ∈ X.Then there exists a unique quartic function Q : X → Y and a unique cubic function C : X → Y satisfyingf x − Q x − C x ≤ Now, by using 2.5 , 2.8 , and 2.11 , we lead to ky f e x ky 4 k 4 − k 2 f e y − 2f e x 2.11 for all x, y ∈ X.for all x, y ∈ X.It follows from 2.12 and 2.13 that