Stability for the Mixed Type of Quartic and Quadratic Functional Equations

Abstract

We establish the general solutions of the following mixed type of quartic and quadratic functional equation: . Moreover we prove the Hyers-Ulam-Rassias stability of this equation under the approximately quartic and the approximately quadratic conditions.

1. Introduction

The stability problems of functional equations go back to 1940, when Ulam [1] proposed the following problem concerning group homomorphisms.

Let be a group and let be a metric group with metric and a positive number. Does there exist a positive such that for every with there exists a group homomorphism such that for all in ?

In 1941, Hyers [2] had affirmatively answered the question of Ulam for Banach spaces. He proved that if is a mapping between Banach spaces satisfying for some fixed , then there exists the unique additive mapping such that . Actually, the additive mapping is explicitly constructed from the given function by the formular This method is called a direct method. The theorem of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference to be controlled by . In addition, Rassias generalized the Hyers’ stability result by introducing two weaker conditions controlled by the product of different powers of norms and mixed product-sum of powers of norms, respectively (see [5–9]). In 1994, Găvruţa [10] gave a generalization of Rassias’ theorem by replacing by a general control function . Instead of the direct method, Cădariu and Radu [11] introduced another approach for proving the stability of functional equations (see also [12]) via the fixed point theory. They observed that the existence of a solution of the functional equation and the estimation of the difference with the given mapping can be obtained from the fixed point alternative. This method is called a fixed point method.

As of now, both the direct method and the fixed point method have been intensively used in the study of stability problems of various types of functional equations (see [13–19]). In particular, one of the important functional equations studied is the quadratic functional equation: We note that the quadratic function is a solution of (2). So one usually calls the above functional equation quadratic and every solution of (2) is said to be a quadratic mapping. Stability results of quadratic functional equations can be found in [20–22]. On the other hand, Rassias [23] investigated stability problems of the following functional equation: It is easy to see that is a solution of (3) by virtue of the algebraic identity For this reason, (3) is called a quartic functional equation and every solution of (3) is said to be a quartic mapping. Chung and Sahoo [24] determined the general solutions of (3) without assuming any regularity conditions on the unknown function. In fact, they proved that the function is a solution of (3) if and only if , where the function is symmetric and additive in each variable. Since the solution of (3) is even, we can rewrite (3) as Lee et al. [25] obtained the general solutions of (5) and proved the Hyers-Ulam-Rassias stability of this equation (see also [26]). Lee and Chung [27] proved the stability of the following quartic functional equation, which is a generalization of (5), for fixed integer with . Also Kim [28] solved the general solutions and proved the Hyers-Ulam-Rassias stability for the mixed type of quartic and quadratic functional equation: Gordji et al. [29] introduced another mixed type of quartic and quadratic functional equation: for fixed integers with . They established the general solutions and proved the Hyers-Ulam-Rassias stability of this equation in quasi-Banach spaces.

In this paper, we deal with the following mixed type of quartic and quadratic functional equations, for the case in (8), In Section 2, we solve the general solutions of (9) using another way as in [29]. As a matter of fact, satisfies (9) if and only if there exists a quartic mapping and a quadratic mapping which satisfy (5) and (2), respectively; the mapping can be written as . Using the idea of Gãvruta [10] we prove the Hyers-Ulam-Rassias stability of (9) in Section 3. Applying the different approaches as in [29] we prove the Hyers-Ulam-Rassias stability of (9) under the approximately quartic condition and the approximately quadratic condition in Sections 4 and 5, respectively.

2. General Solutions of (9)

Throughout this section, we denote both and by real vector spaces.

It is well-known [30] that a mapping satisfies the quadratic functional equation (2) if and only if there exists a unique symmetric biadditive mapping such that for all . The biadditive mapping is given by Similarly, a mapping satisfies the quartic functional equation (5) if and only if there exists a symmetric biquadratic mapping such that for all (see [25]). The biquadratic mapping is given by

Now we are going to establish the general solutions of (9).

Lemma 1. If a mapping satisfies (9), then the mapping defined by is a quartic mapping satisfying (5).

Proof. Putting in (9) gives . Letting in (9) we have for all . Substituting by in (9) yields for all . Replacing by in (9) and using the evenness of we obtain for all . Combining (12) and (13) to eliminate the term gives for all . By the definition of and using (9), (14) we have for all . This shows that satisfies (5).

Lemma 2. If a mapping satisfies (9), then the mapping defined by is a quadratic mapping satisfying (2).

Proof. Interchanging the role of and in (9) and using the evenness of we have for all . Putting in (9) and using (16) we figure out for all . This shows that satisfies (2).

From the preceding Lemmas we establish the general solutions of (9) as follows.

Theorem 3. A mapping satisfies (9) for all if and only if there exists a symmetric biquartic mapping and a symmetric biadditive mapping such that for all .

Proof. We assume that the mapping satisfies (9). Define mappings by for all . By Lemmas 1 and 2 we note that the mappings and satisfy (5) and (2), respectively, and for all . According to the results as in [25, 30] there exists a symmetric biquadratic mapping and a symmetric biadditive mapping such that for all . Conversely, one can easily verify that the mappings and satisfy (9) by a simple computation.

3. Stability of (9)

Now we are going to prove the Hyers-Ulam-Rassias stability for the mixed type quartic and quadratic functional equation. In what follows, we denote by a real vector space and by a Banach space. Let denote the set of all nonnegative real numbers and the set of all positive integers. For convenience, we define the difference operator for a given mapping by for all .

Theorem 4. Let be a mapping satisfying for all . If a mapping with satisfies for all , then there exists a quartic and quadratic mapping such that for all , where the mapping is given by for all . The mapping is given by for all .

Proof. Putting and then replacing by in (23), one has the approximately even condition of as follows: for all . Substituting by in (23) gives for all . Replacing by in (23) yields for all . Combining (27), (28), and (29) to eliminate the terms and we have the following relation for all .
Making use of induction arguments in (30) we obtain for all and for all . Actually (30) proves the validity of the inequality (31) for the case . Assume that inequality (31) holds for some . Using (30) and (31) we have the following relation for all and . This proves the validity of inequality (31) for the case .
Now let us define a sequence by and claim that it is a convergent sequence. For any integers with , we verify by (30) that for all . Since the right-hand side of the above inequality tends to as by assumption, the sequence is a Cauchy sequence in . Thus, we may define a mapping by for all . By virtue of the inequality (23) we figure out for all and for all . Letting in the above inequality we see that which shows that satisfies (9). Finally letting in (31) we have the result (24). This completes the proof.

Theorem 5. Let be a mapping satisfying for all . If a mapping satisfies for all , then there exists a quartic and quadratic mapping such that for all . The mapping is given by for all .

Proof. Replacing by in (30) we have for all . Using the induction argument in (42) we obtain for all and for all . We define a sequence by and show that it is a Cauchy sequence. For any integers with , we verify by (42) that for all . Since the right-hand side of the above inequality tends to as by assumption, the sequence is a convergent sequence. Now we define a mapping by for all . From (39) we figure out for all and for all . Letting in the above inequality we see that satisfies (9). Letting in (43) we finally obtain the result (40). This completes the proof.

From the previous Theorem 4, we obtain the following corollary concerning the stability of (9) immediately.

Corollary 6. Suppose that for some , a mapping satisfies for all . Then there exists a quartic and quadratic mapping such that for all .

4. Stability of (9) under the Approximately Quartic Condition

In the next part, we state and prove the Hyers-Ulam-Rassias stability of (9) under the approximately quartic condition.

Theorem 7. Let be a mapping satisfying for all and let be a mapping satisfying for all . If a mapping with satisfies for all and for all , then there exists a unique quartic mapping such that for all , where . The mapping is given by for all .

Proof. It follows from (28) and (53) that we have for all . Substituting into in (56) yields for all . Combining (56) and (57) to eliminate the term we obtain for all . Making use of induction arguments in (58) we have for all and for all . Actually, (58) proves the validity of the inequality (59) for . Using (56), (57), and the following relation one can easily verify (59) for . It follows from (27) and (59) that for all and for all . We show that the sequence is a convergent sequence. For any integers with , we figure out for all . Since the right-hand side of the inequality (62) tends to as tends to infinity, the sequence is a Cauchy sequence in . Now we define for all . Letting in (61) we arrive at (54).
Let us prove that the mapping satisfies (9). Replacing and by and , respectively in (52) and dividing by yields for all . Taking the limit as in the above inequality, we see that satisfies (9) for all .
Finally we prove the uniqueness of the mapping . Assume that there exists another quartic mapping which satisfies (9) and the inequality (54). Obviously, we have and for all . Hence it follows from (54) that for all . Letting in the above inequality, we immediately obtain the uniqueness of .

Theorem 8. Let be a mapping satisfying for all and let be a mapping satisfying for all . If a mapping satisfies for all and for all , then there exists a unique quartic mapping such that for all . The mapping is given by for all .

Proof. Replacing by in (54) gives for all . Using induction arguments in (72) we have for all and for all . It follows from (27) and (73) that for all and for all . The rest of the proof is similar to the proof of Theorem 7.

By Theorem 7, we have the following corollary immediately.

Corollary 9. Suppose that for some , a mapping satisfies for all and for all . Then there exists a unique quartic mapping such that for all .

5. Stability of (9) under the Approximately Quadratic Condition

Now we state and prove the Hyers-Ulam-Rassias stability of (9) under the approximately quadratic condition.

Theorem 10. Let be a mapping satisfying for all and let be a mapping satisfying for all . If a mapping with satisfies for all and for all , then there exists a unique quadratic mapping such that for all . The mapping is defined by for all .

Proof. It follows from (28) and (81) that we have for all . Substituting for in (84) yields for all . Combining (84) and (85) to eliminate the term we obtain for all . Making use of induction arguments in (86) we have for all and for all . It follows from (86) and (87) that for all . From (88) one can easily show that the sequence is a Cauchy sequence in . Define a mapping for all . It follows from (80) and (88) that we verify the mapping is the unique mapping satisfying (9) and (82). Letting in (88) we have the result (82).

Theorem 11. Let be a mapping satisfying for all and let be a mapping satisfying for all . If a mapping satisfies for all and for all , then there exists a unique quadratic mapping such that for all . The mapping is given by for all .

Proof. Putting in (84) gives for all . Making use of induction arguments in (96) we have for all and for all . It follows from (27) and (73) that for all and for all . The rest of the proof goes through in the similar way as that of the proof of Theorem 10.

It follows from Theorem 10 that we obtain the following corollary immediately.

Corollary 12. Suppose that for some , a mapping satisfies for all and for all , for all , and for some . Then there exists a unique quartic mapping such that for all .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.