Stability for the Mixed Type of Quartic and Quadratic Functional Equations

This method is called a direct method. The theorem of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [4] for linearmappings by considering an unbounded Cauchy difference to be controlled by ε(‖x‖p + ‖y‖p). 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 ε(‖x‖p+‖y‖p) by a general control functionφ(x, y). 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 A of the functional equation and the estimation of the difference with the given mapping f 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:


Introduction
The stability problems of functional equations go back to 1940, when Ulam [1] proposed the following problem concerning group homomorphisms.
Let  1 be a group and let  2 be a metric group with metric (⋅, ⋅) and  a positive number.Does there exist a positive  such that for every  :  1 →  2 with ((), ()()) ≤  there exists a group homomorphism  :  1 →  2 such that ((), ()) <  for all  in  1 ?
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  ≥ 0, 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][6][7][8][9]).In 1994, Gȃvrut ¸a [10] gave a generalization of Rassias' theorem by replacing (‖‖  +‖‖  ) by a general control function (, ).Instead of the direct method, Cȃdariu 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][14][15][16][17][18][19]).In particular, one of the important functional equations studied is the quadratic functional equation: ( We note that the quadratic function () =  2 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][21][22].On the other hand, Rassias [23] investigated stability problems of the following functional equation: It is easy to see that () =  4 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  : R → R is a solution of (3) if and only if () = (, , , ), where the function  : R 4 → R 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  ̸ = 0, ±1.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  ̸ = 0,±1.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  = 2 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.

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).
Proof.Interchanging the role of  and  in ( 9) and using the evenness of  we have for all ,  ∈ .Putting  = 2 in (9) and using (16) we figure out for all ,  ∈ .This shows that ℎ satisfies (2).
From the preceding Lemmas we establish the general solutions of (9) as follows.
Proof.We assume that the mapping  :  →  satisfies (9).Define mappings , ℎ :  →  by for all  ∈ .By Lemmas 1 and 2 we note that the mappings  and ℎ 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.

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 R + denote the set of all nonnegative real numbers and N the set of all positive integers.For convenience, we define the difference operator for a given mapping  :  →  by for all ,  ∈ .
Theorem 4. Let  :  ×  → R + be a mapping satisfying for all ,  ∈ .If a mapping  :  →  with (0) = 0 satisfies      (, )     ≤  (, ) for all ,  ∈ , then there exists a quartic and quadratic mapping  :  →  such that for all  ∈ , where the mapping Φ is given by for all  ∈ .The mapping  is given by for all  ∈ .
Proof.Putting  = 0 and then replacing  by  in (23), one has the approximately even condition of  as follows: for all  ∈ .Substituting  by  in (23) gives      (3) − 6 (2) + 15 ()     ≤  (, ) for all  ∈ .Combining ( 27), (28) for all  ∈ N and for all  ∈ .Actually (30) proves the validity of the inequality (31) for the case  = 1.Assume that inequality (31) holds for some  ∈ N. Using (30) for all  ∈  and  ∈ N.This proves the validity of inequality (31) for the case  + 1.Now let us define a sequence {  ()} by and claim that it is a convergent sequence.For any integers ,  with  >  ≥ 0, we verify by (30) for all  ∈ .Since the right-hand side of the above inequality tends to 0 as  → ∞ by assumption, the sequence {  ()} is a Cauchy sequence in .Thus, we may define a mapping  :  →  by for all  ∈ N and for all  ∈ .We define a sequence {  ()} by and show that it is a Cauchy sequence.For any integers ,  with  >  ≥ 0, we verify by ( 42 for all  ∈ .Since the right-hand side of the above inequality tends to 0 as  → ∞ by assumption, the sequence {  ()} is a convergent sequence.Now we define a mapping  :  →  by for all  ∈ N 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 for all  ∈ .

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.
Proof.It follows from ( 28) and (53) that we have for all  ∈ .Substituting − into  in (56) yields for all  ∈ .Since the right-hand side of the inequality (62) tends to 0 as  tends to infinity, the sequence {(3 for all  ∈ N, ,  ∈ .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 (3  ) = 81  () and (3  ) = 81  () for all  ∈ N,  ∈ .Hence it follows from (54) that for all  ∈ .Letting  → ∞ in the above inequality, we immediately obtain the uniqueness of .  for all  ∈ .

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.