Abstract

We obtain the general solution and the generalized Ulam-Hyers stability of the cubic and quartic functional equation .

1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms. Let be a group and let be a metric group with the metric Given does there exist a such that if a mapping satisfies the inequality for all then there exists a homomorphism with for all ? In the other words, under what condition does there exist 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 be a mapping between Banach spaces such thatfor all and for some Then there exists a unique additive mapping such thatfor all Moreover, if is continuous in for each fixed then is linear. Finally, in 1978, Th. M. Rassias [3] proved the following theorem.

Theorem 1.1. Let be a mapping from a normed vector space into a Banach space subject to the inequalityfor all where and p are constants with and Then there exists a unique additive mapping such thatfor all If then inequality (1.3) holds for all and (1.4) for Also, if the function from into is continuous in real for each fixed then is linear.

In 1991, Gajda [4] answered the question for the case which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias stability of functional equations (see [2, 4–13]). On the other hand, J. M. Rassias [14–16] generalized the Hyers stability result by presenting a weaker condition controlled by a product of different powers of norms. According to J. M. Rassias theorem.

Theorem 1.2. If it is assumed that there exist constants and such that and is a map from a norm space into a Banach space such that the inequalityfor all then there exists a unique additive mapping such thatfor all If in addition for every is continuous in real for each fixed then is linear (see [14, 15, 17–22]).

The oldest cubic functional equation, and was introduced by J. M. Rassias [23, 24] is as follows:Jun and Kim [25] introduced the following cubic functional equation:and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.7). The function satisfies the functional equation (1.7), 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 between real vector spaces and is a solution of (1.7) if and only if there exists a unique function such that for all and is symmetric for each fixed one variable and is additive for fixed two variables. The oldest quartic functional equation, and was introduced by J. M. Rassias [16, 26], and then was employed by Park and Bae [27], such that In fact, they proved that a function between real vector spaces and is a solution of (1.8) if and only if there exists a unique symmetric multiadditive function such that for all (see also [27–33]). It is easy to show that the function satisfies the functional equation (1.8), which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic function.

We deal with the following functional equation deriving from quartic and cubic functions:It is easy to see that the function is a solution of the functional equation (1.9). In the present paper, we investigate the general solution and the generalized Hyers-Ulam-Rassias stability of the functional equation (1.9).

2. General solution

In this section, we establish the general solution of functional equation (1.9).

Theorem 2.1. Let be vector spaces, and let be a function. Then satisfies (1.9) if and only if there exists a unique symmetric multiadditive function and a unique function such that is symmetric for each fixed one variable and is additive for fixed two variables, and that for all

Proof. Suppose there exists a symmetric multiadditive function and a function such that is symmetric for each fixed one variable and is additive for fixed two variables, and that for all Then it is easy to see that satisfies (1.9). For the convlet satisfy (1.9). We decompose into the even part and odd part by settingfor all By (1.9), we havefor all This means that satisfies (1.9), orNow, putting in (1.9e), we get Setting in (1.9e), by evenness of we obtain for all Hence, (1.9e) can be written asfor all With the substitution in (2.4), we haveReplacing by in (2.4), we obtainSubstituting for in (2.6) givesBy utilizing (2.5), (2.6), and (2.7), we obtainInterchanging and in (2.5), we getIf we add (2.5) to (2.9), we haveAnd by utilizing (2.6), (2.7), and (2.10), we arrive atLet us interchange and in (2.11). Then we see thatComparing (2.12) with (2.5), we getIf we compare (2.13) and (2.8), we conclude thatThis means that is quartic function. Thus, there exists a unique symmetric multiadditive function such that for all On the other hand, we can show that satisfies (1.9), orNow setting in (1.9o) gives Putting in (1.9o), then by oddness of we haveHence, (1.9o) can be written asfor all Replacing by and by in (2.16) we haveand interchanging and in (2.16) yieldsWhich on substitution of for in (2.16) givesReplace by in (2.16). Then we haveFrom the substitution in (2.20) it follows thatIf we add (2.20) to (2.21), we haveLet us interchange and in (2.22). Then we see thatWith the substitution in (2.16), we haveand replacing by givesIf we add (2.24) to (2.25), we haveBy comparing (2.19) with (2.26), we arrive atand replacing by in (2.16) givesBy comparing (2.28) with (2.22), we arrive atLet us interchange and in (2.28). Then we see thatThus combining (2.30) with (2.23) yieldsBy comparing (2.31) with (2.18), we arrive atWhich, by putting in (2.17), leads toReplacing by in (2.33) givesIf we subtract (2.33) from (2.34), we obtainSetting instead of and instead of in (2.27), we getCombining (2.35) and (2.36) yieldsand subtracting (2.21) from (2.20), we obtainBy comparing (2.37) with (2.38), we arrive atInterchanging with in (2.32) gives the equationWe obtain from (2.39) and (2.40)By using (2.31) and (2.41), we lead toAnd interchanging with in (2.42) givesIf we compare (2.43) and (2.29), we conclude thatThis means that is cubic function and that there exits a unique function such that for all and is symmetric for each fixed one variable and is additive for fixed two variables. Thus for all we haveThis completes the proof of theorem.

The following corollary is an alternative result of Theorem 2.1.

Corollary 2.2. Let be vector spaces, and let be a function satisfying (1.9). Then the following assertions hold.
(a)If is even function, then is quartic. (b)If is odd function, then is cubic.

3. Stability

We now investigate the generalized Hyers-Ulam-Rassias stability problem for functional equation (1.9). From now on, let be a real vector space and let be a Banach space. Now before taking up the main subject, given we define the difference operator byfor all We consider the following functional inequality: for an upper bound

Theorem 3.1. Let be fixed. Suppose that an even mapping satisfies andfor all If the upper bound is a mapping such that the series converges, and that for all then the limit exists for all and is a unique quartic function satisfying (1.9), andfor all

Proof. Let Putting in (3.3), we get Replacing by in (3.5), yields Interchanging with in (3.6), and multiplying by 16 it follows that Combining (3.6) and (3.7), we lead to From the inequality (3.6) we use iterative methods and induction on to prove our next relation: We multiply (3.9) by and replace by to obtain thatThis shows that is a Cauchy sequence in by taking the limit Since is a Banach space, it follows that the sequence converges. We define by for all It is clear that for all and it follows from (3.3) thatfor all Hence, by Corollary 2.2, is quartic. It remains to show that is unique. Suppose that there exists another quartic function which satisfies (1.9) and (3.4). Since and for all we conclude thatfor all By letting in this inequality, it follows that for all which gives the conclusion. For we obtain from which one can prove the result by a similar technique.

Theorem 3.2. Let be fixed. Suppose that an odd mapping satisfies for all If the upper bound is a mapping such that converges, and that for all then the limit exists for all and is a unique cubic function satisfying (1.9), andfor all

Proof. Let Set in (3.14). We obtain Replacing by in (3.16) to get An induction argument now implies Multiply (3.18) by and replace by we obtain thatThe right hand side of the inequality (3.19) tends to as because ofby assumption, and thus the sequence is Cauchy in as desired. Therefore we may define a mapping as The rest of proof is similar to the proof of Theorem 3.1.

Theorem 3.3. Let be fixed. Suppose a mapping satisfies and for all If the upper bound is a mapping such thatfor all Then there exists a unique quartic function and a unique cubic function satisfying for all

Proof. Let for all Then and is even function satisfying for all From Theorem 3.1, it follows that there exists a unique quartic function satisfies for all Let now for all Then is odd function satisfying for all Hence, in view of Theorem 3.2, it follows that there exists a unique cubic function such that for all On the other hand, we have for all Then by combining (3.23) and (3.25), it follows thatfor all and the proof of theorem is complete.

We are going to investigate the Hyers-Ulam-Rassias stability problem for functional equation (1.9).

Corollary 3.4. Let Suppose satisfies and inequalityfor all Then there exists a unique quartic function and a unique cubic function satisfying for all

Proof. Let in Theorem 3.3. Then by taking for all the relations (3.21) hold for Then there exists a unique quartic function and a unique cubic function satisfying for all Let now in Theorem 3.3 and put for all Then the relations (3.21) hold for Then there exists a unique quartic function and a unique cubic function satisfying for all

Similarly, we can prove the following Ulam stability problem for functional equation (1.9) controlled by the mixed type product-sum functionintroduced by J. M. Rassias (e.g., [34]).

Corollary 3.5. Let be real numbers such that and let Suppose satisfies and inequalityfor all Then there exists a unique quartic function and a unique cubic function satisfyingfor all

By Corollary 3.4, we solve the following Hyers-Ulam stability problem for functional equation (1.9).

Corollary 3.6. Let be a positive real number. Suppose satisfies and for all Then there exists a unique quartic function and a unique cubic function satisfying for all

Acknowledgments

The authors would like to express their sincere thanks to the referee for his invaluable comments. The first author would like to thank the Semnan University for its financial support. Also, the third author would like to thank the office of gifted students at Semnan University for its financial support.