Stability of a Quartic Functional Equation

We obtain the general solution of the generalized quartic functional equation f(x + my) + f(x − my) = 2(7m − 9)(m − 1)f(x) + 2m 2(m 2 − 1)f(y)−(m − 1)2 f(2x) + m 2{f(x + y) + f(x − y)} for a fixed positive integer m. We prove the Hyers-Ulam stability for this quartic functional equation by the directed method and the fixed point method on real Banach spaces. We also investigate the Hyers-Ulam stability for the mentioned quartic functional equation in non-Archimedean spaces.


Introduction
We say a functional equation F is stable if any function satisfying the equation F approximately is near to exact solution of F. Moreover, a functional equation F is hyperstable if any function satisfying the equation F approximately is a true solution of F.
The study of stability problems for functional equations is related to a question of Ulam [1] concerning the stability of group homomorphisms, affirmatively answered for Banach spaces by Hyers [2]. Subsequently, the result of Hyers was generalized by a number of authors. For example, Bodaghi et al. investigated the Hyers-Ulam stability of Jordan *derivation pairs for the Cauchy additive functional equation and the Cauchy additive functional inequality in [3]. For some results on the stability of various functional equations, see also [4][5][6][7][8][9].
Bodaghi et al. [15] applied the fixed point alternative theorem (Theorem 8 of the current paper) to establish Hyers-Ulam stability of (3). They also showed that the functional equation (3) can be hyperstable under some conditions. This The Scientific World Journal method which is different from the "direct method," initiated by Hyers in 1941, had been applied by Cȃdariu and Radu for the first time in [16]. In other words, they employed this fixed point method to the investigation of the Cauchy functional equation [17] and of the quadratic functional equation [16] (for more applications of this method, see [18][19][20]).
In this paper, we consider the following functional equation which is somewhat different from (2), (3), and (4): for a fixed positive integer . In case = 2, then (5) is the celebrated Jordan-von Neumann equation. Then we find out the general solution of (5). We also prove the Hyers-Ulam stability problem and the hyperstability for (5) by the directed method and the fixed point method.

General Solution of (5)
To achieve our aim in this section, we need the following lemma.

Lemma 1.
Let X and Y be real vector spaces. If a function : X → Y satisfies the functional equation (5) for all integers ≥ 3, then satisfies ( ) = 4 ( ) for all integers ≥ 2.
There is a gap in its proof. In fact, in the proof, the author only showed that the functional equation (1) implies (9) but the converse is not proved. Theorem 3 resolved this problem. Indeed, we solve the equation of (5).
In Similar way to the above, we get Using the above method, we can deduce that for which = − −2 + 28 2 − 120 + 156, 2 = 10, 3 = 48, Solving the above recurrence equations is routine, and so we get for all , ∈ X and each positive integer ≥ 2.

Hyers-Ulam Stability of (5)
Let be an integer with ≥ 2. We use the abbreviation for the given mapping : X → Y as follows: Throughout this section, we assume that X is a normed real linear space with norm ‖ ⋅ ‖ X and Y is a real Banach space with norm ‖ ⋅ ‖ Y . We are going to prove the stability of the quartic functional equation (5).

Theorem 4.
Let be a real number and let : X → Y be a mapping for which there exists a function : for all , ∈ X, where is an integer with ≥ 2. Then there exists a unique quartic mapping T : X → Y such that for all ∈ X. Thus The Scientific World Journal for all ∈ X. Replacing by 2 in (25) and continuing this method, we get On the other hand, we can use induction to obtain for all ∈ X, and > ≥ 0. Thus the sequence { (2 )/2 4 } is Cauchy by (21) and (27). Completeness of Y allows us to assume that there exists a map T so that Taking the limit as → ∞in (26) and applying (28), we can see that inequality (23) holds. Now, we replace , by 2 , 2 , respectively, in (22); then Letting the limit as → ∞, we obtain D T( , ) = 0 for all positive integers ≥ 2 and all , ∈ X. Hence, by Theorem 3, it indicates that T : X → Y is a quartic mapping. Now, let T : X → Y be another quartic mapping satisfying (23). Then we have for all ∈ X. Taking → ∞ in the preceding inequality, we immediately find the uniqueness of T. This completes the proof.
Corollary 5. Let , , , , and be nonnegative real numbers such that > 0 and , < 4. Suppose that : X → Y is a mapping fulfilling for all , ∈ X, where is an integer with ≥ 2. Then there exists a unique quartic mapping T : X → Y such that for all ∈ X and all ∈ X \ {0} if < 0.
Proof. Setting ( , ) = ‖ ‖ X + ‖ ‖ X in Theorem 12, we have It follows from (31) that By these statements we can get the result.
We have the following result which is analogous to Theorem 12 for the quartic functional equation (5). We include its proof. Theorem 6. Suppose that : X → Y is a mapping for which there exists a function : X × X → [0, ∞) such that The Scientific World Journal 5 for all , ∈ X, where is an integer with ≥ 2. Then there exists a unique quartic mapping T : X → Y such that for all ∈ X.
Proof. It follows from (35) that (0, 0) = 0. Thus from (36) we have (0) = 0. Putting = 0 in (36), we get for all ∈ X. If we replace by /2 in the above inequality and divide both sides by ( − 1) 2 , we have Using triangular inequality and proceeding this way, we obtain for all ∈ X. If we show that the sequence {2 4 ( /2 )} is Cauchy, then it will be convergent by the completeness of Y.
For this, if we replace by /2 in (40) and then multiply both sides by 2 4 , then we get for all ∈ X, and > > 0. Thus the mentioned sequence is convergent to the mapping T; that is, Now, in a similar way to the proof of Theorem 12, we can complete the rest of the proof.

Corollary 7.
Let , , , and be nonnegative real numbers such that , > 4. Suppose that : X → Y is a mapping fulfilling for all , ∈ X, where is an integer with ≥ 2. Then there exists a unique quartic mapping T : X → Y such that for all ∈ X.
We are going to investigate the hyperstability of the given quartic functional equation (5) by using the fixed point method. First, we bring the next theorem which was proved in [22]. This result plays a fundamental role to achieve our goal.
for all , ∈ X, where is an integer with ≥ 2. If there exists a constant ∈ (0, 1), such that for all , ∈ X, then there exists a unique quartic mapping T : X → Y such that for all ∈ X.
Proof. We wish to make the conditions of Theorem 8. We consider the set and define the mapping D on Δ × Δ as follows: for all ∈ X. Using (46), we get for all ∈ X. The above inequality shows that D(J , Jℎ) ≤ D( , ℎ) for all , ℎ ∈ Δ. Hence, J is a strictly contractive mapping on Δ with a Lipschitz constant . We now show that D(J , ) < ∞. Putting = 0 in (45), we obtain for all ∈ X. We conclude from the last inequality that (54) Theorem 8 shows that D(J , J +1 ) < ∞ for all ≥ 0, and thus in this theorem we have 0 = 0. Consequently, the parts (iii) and (iv) of Theorem 8 hold on the whole Δ. Hence there exists a unique mapping T : X → Y such that T is a fixed point of J and that J → T as → ∞. Thus for all ∈ X, and so The above inequalities show that (47) is true for all ∈ X. Now, it follows from (46) that Substituting and by 2 and 2 , respectively, in (45), we get Taking the limit as → ∞, we obtain D T( , ) = 0 for all integers ≥ 2 and all , ∈ X. It follows from Theorem 3 that T : X → Y is a quartic mapping which is unique.

Corollary 10.
Let , , and be nonnegative real numbers with , < 4 and let : X → Y be a mapping such that for all , ∈ X. Then there exists a unique quartic mapping T : X → Y satisfying for all ∈ X.
In the next result, we prove the hyperstability of quartic functional equations under some conditions. Corollary 11. Let , , and be nonnegative real numbers with 0 < + ̸ = 4 and let : X → Y be a mapping such that for all , ∈ X. Then is a quartic mapping on X.

Stability of (5) in Non-Archimedean Spaces
We recall some basic facts concerning non-Archimedean spaces and some preliminary results. By a non-Archimedean field we mean a field K equipped with a function (valuation) | ⋅ | from K into [0, ∞) such that | | = 0 if and only if = 0, | | = | || |, and | + | ≤ max{| |, | |} for all , ∈ K. Clearly |1| = | − 1| = 1 and | | ≤ 1 for all ∈ N. Let X be a vector space over a scalar field K with a non-Archimedean nontrivial valuation | ⋅ |. A function ‖ ⋅ ‖ : X → R is a non-Archimedean norm (valuation) if it satisfies the following conditions: (ii) ‖ ‖ = | |‖ ‖, ( ∈ X, ∈ K); (iii) the strong triangle inequality (ultrametric); namely, Then (X, ‖ ⋅ ‖) is called a non-Archimedean space. Due to the fact that a sequence { } is Cauchy if and only if { +1 − } converges to zero in a non-Archimedean normed space X. By a complete non-Archimedean normed space we mean one in which every Cauchy sequence is convergent.
In [23], Hensel discovered the -adic numbers as a number theoretical analogue of power series in complex analysis. The most interesting example of non-Archimedean spaces is -adic numbers. A key property of -adic numbers is that they do not satisfy the Archimedean axiom: for all , > 0, there exists an integer such that < . Let be a prime number. For any nonzero rational number = ( / ) in which and are coprime to the prime number . Consider the -adic absolute value | | = on Q. It is easy to check that | ⋅ | is a non-Archimedean norm on Q. The completion of Q with respect to | ⋅ | which is denoted by Q is said to be the -adic number field. One should remember that if > 3, then |2 | = 1 for all integers . In [24], the stability of some functional equations in non-Archimedean normed spaces is investigated (see also [25]).
Here and subsequently, we assume that X is a normed space and Y is a complete non-Archimedean space unless otherwise stated explicitly. In the upcoming theorem, we prove the stability of the functional equation (5).
Proof. Putting = 0 in (65), we get for all ∈ X. Thus we have for all ∈ X. Replacing by 2 in (68) and then dividing both sides by |16| +1 , we have for all ∈ X and all nonnegative integers . Thus the sequence { (2 )/16 } is Cauchy by (64) and (69). Due to the completeness of Y as a non-Archimedean space, there exists a mapping so that For each ∈ X and non-negative integers , we have Taking → ∞ in (71) and applying (70), we can see that the inequality (66) holds when ≥ 2. It follows from (64), (65), and (70) that, for all , ∈ X, Hence, the mapping satisfies (5). Now, let : X → Y be another quartic mapping satisfying (66). Then we have for all ∈ X. Now, Theorem 12 implies the desired result.
We have the following result which is analogous to Theorem 12 for the functional equation (5).
Proof. In a similar way to the proof of Theorem 12, we have (2 ) − 16 ( ) ≤ 1 | − 1| 2 ( , 0) for all ∈ X. If we replace by /2 +1 in the above inequality and multiply both sides of (81) to |16| , we get for all ∈ X and all non-negative integers . Thus, we conclude from (78) and (82) that the sequence {2 ( /2 )} is Cauchy. Since the non-Archimedean space Y is complete, this sequence converges in Y to the mapping . Indeed, for all ∈ X and non-negative integers . Since the right hand side of inequality (84) goes to 0 as → ∞, by applying (83), we deduce inequality (80). Now, in a similar way to the proof of Theorem 12, we can complete the rest of the proof. for all ∈ X.
Proof. The proof is a direct consequence of Theorem 14 and similar to the proof of Corollary 13.