#### Abstract

We obtain the general solution of the generalized mixed additive and quadratic functional equation , is even; , is odd, for a positive integer . We establish the Hyers-Ulam stability for these functional equations in non-Archimedean normed spaces when is an even positive integer or .

#### 1. Introduction

The basic problem of the stability of functional equations was formulated by Ulam in 1940 in the following form. Suppose that a mapping satisfies the additive functional equation only approximately. Then does there exist an additive function which approximates ? (See also [1].) In 1941, Hyers [2] gave the following answer to this question for Banach spaces. The result of Hyers was generalized in 1950 by Aoki [3] for approximately additive mappings and in 1978 by Rassias [4] for approximately linear mappings, by considering the unbounded Cauchy differences. A further generalization was obtained by GÄƒvruÅ£a [5] in 1994, by replacing the Cauchy differences by a control function satisfying a very simple condition of convergence.

The Hyers-Ulam stability problem for the quadratic functional equation was first proved by Skof for a function , where is a normed space and is a Banach space [6]. One year later, Cholewa [7] demonstrated that Skofâ€™s theorem is still true if relevant domain is replaced by an abelian group. After that, in [8], Czerwik proved the Hyers-Ulam stability of the quadratic functional equation (1) as a special case. In [9], it was shown that a mapping is quadratic if and only if for all . Also, is quadratic if and only ifâ€‰â€‰ for all [10]. CÄƒdariu and Radu investigated the stability of the Cauchy functional equation [11] and for the quadratic functional equation [12]. Stability problems of miscellaneous functional equations have been investigated by several authors during the last decades (see, e.g., [13â€“15]).

In [16], Eskandani et al. determined the general solution of the following mixed additive and quadratic functional equation:

They studied the Hyers-Ulam stability of (2) in non-Archimedean Banach modules over a unital Banach algebra. In [17], Najati and Moghimi established the general solution of the mixed type additive and quadratic functional equation and investigated the stability of this equation in quasi-Banach spaces. The stability of (3) in random normed spaces is proved in [18].

In this paper, we consider the following functional equations: where is an even positive integer and Indeed, (4) and (5) are different from (2) and (3). It is easily verified that the function is a solution of the functional equations (4) and (5). We show that these functional equations are mixed additive and quadratic mappings. We also prove the Hyers-Ulam stability problem for these equations. As a corollary, the hyperstability of (4) and (5) under some conditions in non-Archimedean normed spaces is shown as well.

#### 2. General Solution of (4) and (5)

To achieve our aim in this paper, we need the following lemma which is a fundamental tool.

Lemma 1. *Let and be real vector spaces. *(i)*If an odd mapping satisfies the functional equation (4), then is additive. *(ii)*If an odd mapping satisfies the functional equation (5), then is additive. *(iii)*If an even mapping satisfies the functional equation (4), then is quadratic. *(iv)*If an even mapping satisfies the functional equation (5), then is quadratic. *

*Proof. *(i) Letting in (4), we get for all . This equality implies that
for all . Replacing by in (6), we have
for all . Substituting by in (7), respectively, we obtain
for all . The equalities (7) and (8) show that

(ii) Suppose that satisfies (5). Similar to the part (i), by the oddness of , we have for all . Thus
for all . We substitute by in (10) and then by in (10); we get
for all . Then, by adding (11) to (12), we lead to
for all . Similar to the part (i), we can show that is additive.

(iii) By the assumption, the equality
holds for a fixed even positive integer . Putting in (14), we get . Once more, by letting in (14), we have
for all . Interchanging into in (14), respectively, we deduce that
for all . Plugging (15) into (16), we have for all and thus for all . Using the last equality and (14), we have
for all . Setting in (17), we obtain for all . Applying this equality and putting by in (17), we get
for all . This shows that is a quadratic mapping.

(iv) Suppose that satisfies (5). Replacing by and in (5), respectively, we have
for all . The equalities (19) and (20) imply that
for all . Now, the above equality is a special case of the part (iii) when .

In the following theorem, we solve (4) in which is an even positive integer and where is an odd positive integer.

Theorem 2. *Let and be real vector spaces. Then a mapping satisfies the functional equation (2) if and only if it satisfies
**
for all , where is a fixed positive integer with . *

*Proof. *Suppose that satisfies the functional equation (2). Putting in (2), we get . Replacing by and in (2), respectively, we have
Similar to the above, we get
Using the above method, we can deduce that
for all for which , and
Solving the above recurrence equations, we get
for all positive integers .

Conversely, assume that satisfies the functional equations (4) and (22) for each . Firstly, we assume that is even. For and for each , we have
for all . On the other hand,
for all . It follows from (29) and (30) that
for all . Since is an odd number, we have
for all . Also,
for all . Plugging (32) into (33) and using (31), we get
For the odd case , we have
for all . Also
for all . The comparison of (35) and (36) shows that
for all . For and for each , we have
for all . On the other hand,
for all . Now, by comparing (38) with (39) and applying (37), we obtain
for all . This completes the proof.

Theorem 3. *Let and be real vector spaces. A mapping satisfies either (4) or (5) if and only if there exist a symmetric biadditive mapping and an additive mapping such that for all . *

*Proof. *Assume that there exist a symmetric biadditive mapping and an additive mapping such that for all . A simple computation shows that the mappings and given by satisfy the functional equations (4) and (5). Therefore the mapping satisfies (4) and (5).

Conversely, we decompose into the even part and odd part by setting
for all . Obviously, for all . One can easily check that the mappings and satisfy (4) and (5). It follows from Lemma 1 that the mappings and are quadratic and additive, respectively. Since is quadratic, by [19], there exists a symmetric biadditive mapping such that for all . Thus for all , where for all .

#### 3. Hyers-Ulam Stability of (4) and (5)

We recall some basic facts concerning non-Archimedean spaces and some preliminary results.

By a non-Archimedean field, we mean a field equipped with a function (valuation) from into such that if and only if ,â€‰â€‰, and for all . Clearly and for all .

Let be a vector space over a scalar field with a non-Archimedean nontrivial valuation . A function is a non-Archimedean norm (valuation) if it satisfies the following conditions: (i) if and only if ; (ii), ; (iii)the strong triangle inequality (ultrametric), namely, Then is called a non-Archimedean normed space. Due to the fact that a sequence is Cauchy if and only if converges to zero in a non-Archimedean normed space . By a complete non-Archimedean normed space, we mean one in which every Cauchy sequence is convergent.

In [20], Hensel discovered the -adic numbers as a number theoretical analogue of power series in complex analysis. The most interesting example of non-Archimedean normed spaces is -adic numbers. A key property of -adic numbers is that they do not satisfy the Archimedean axiom; for all , 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 . It is easy to check that is a non-Archimedean norm on . The completion of with respect to which is denoted by is said to be the -adic number field. One should remember that if , then for all integers . The stability of some functional equations in non-Archimedean spaces was investigated, for instance, in [21â€“24] (see also [13, 25]).

Let be an even positive integer. We use the abbreviation for the given mapping as follows:

From now on, we assume that is a real vector space and is a complete non-Archimedean space unless otherwise stated explicitly. In the upcoming theorem, we prove the stability of functional equations (4) and (5).

Theorem 4. *Let be a function such that
**
for all . Suppose that is an odd mapping satisfying the inequality
**
for all , where is an even positive integer or . Then there exists a unique additive mapping such that
**
for all , where . *

*Proof. *We prove the result when is even; another case is similar. Putting in (46), we have
for all . Replacing by in (48) and then dividing both sides by , we get
for all and all nonnegative integers . Thus the sequence is Cauchy by (45) and (49). Completeness of the non-Archimedean space allows us to assume that there exists a mapping , so that
For each and nonnegative integers , we have
Taking that tends to approach infinity in (51) and applying (50), we can see that inequality (47) holds when is even. It follows from (45), (46), and (50) that for all ,
Hence, the mapping satisfies (4). Part (i) of Lemma 1 shows that the mapping is additive. Now, let be another additive mapping satisfying (47). Then we have
for all . This shows the uniqueness of .

We have the following result which is analogous to Theorem 4 for the functional equations (4) and (5). We include the proof for (4). The proof of (5) is similar.

Theorem 5. *Let be a function such that
**
for all . Suppose that is an odd mapping satisfying the inequality
**
for all , where is an even positive integer or . Then there exists a unique additive mapping such that
**
for all , where . *

*Proof. *We only obtain the result for the even integers. Similar to the proof of Theorem 4, we have
for all . If we replace by in inequality (57) and then multiply both sides of the result to , we get
for all and all nonnegative integers . Thus, we conclude from (54) and (58) that the sequence is Cauchy. Since the non-Archimedean normed space is complete, this sequence converges in to the mapping . Indeed,
Using induction and (57), one can show that
for all and nonnegative integers . Since the right-hand side of inequality (60) tends to be as to approach infinity, by applying (59), we deduce inequality (56). Now, similar to the proof of Theorem 4, we can complete the rest of the proof.

Corollary 6. *Let , and be positive real numbers such that and . Suppose that is a non-Archimedean normed space and is an odd mapping fulfilling
**
for all , where is an even positive integer or . Then there exists a unique additive mapping such that
**
for all . *

*Proof. *The result follows from Theorems 4 and 5 by letting .

In the next result, we prove the hyperstability of the functional equations (4) and (5) under some conditions. Recall that a functional equation is called *hyperstable* if every approximate solution is an exact one (see, e.g., [13, 26â€“29]).

Corollary 7. *Let , and be positive real numbers such that and . Suppose that is a non-Archimedean normed space and is an odd mapping fulfilling
**
for all , where is an even positive integer or . Then is an additive mapping. *

*Proof. *Taking in Theorems 4 and 5, we can obtain the desired result.

Theorem 8. *Let be a function such that
**
for all . Suppose that is an even mapping satisfying the inequality
**
for all , where is an even positive integer. Then there exists a unique quadratic mapping such that
**
for all , where . *

*Proof. *It follows from (64) that . Thus (65) implies that . Putting in (65) and interchanging into , we have
for all . Substituting by in (65), respectively, we get
for all . It follows from (67) and (68) that
for all . Thus we have
for all . Replacing by in (70) and then dividing both sides by , we get
for all and all nonnegative integers . Thus the sequence is Cauchy by (64) and (71). Since is complete, the sequence converges in for all . So one can define the mapping by
For each and nonnegative integers , we have
Taking to approach infinity in (73) and using (64) and (72), we find (66). Employing (64), (65), and (72), we obtain
Hence, the mapping satisfies (4). It follows from part (iii) of Lemma 1 that the mapping is quadratic. If is another quadratic mapping satisfying (66), then
for all . Therefore . This completes the proof of the uniqueness of .

Theorem 9. *Let be a function such that
**
for all . Suppose that is an even mapping satisfying and the inequality
**
for all , where is an even positive integer. Then there exists a unique quadratic mapping such that
**
for all , where . *

*Proof. *Similar to the proof of Theorem 8, we have
for all . Then we get
for all . Replacing by in (80) and multiplying both sides to , we get
for all and all nonnegative integers . Thus the sequence is Cauchy by (76). The completeness of implies that the mentioned sequence is convergent. So we consider the mapping by
For each and nonnegative integers , we have
Letting approach infinity in (83) and using (76) and (82), we can see that (78) holds. The rest of the proof is similar to the proof of Theorem 8.

Corollary 10. *Let , and be nonnegative real numbers and . Suppose that is a non-Archimedean normed space and is an even mapping fulfilling
**
for all , where is an even positive integer. Then there exists a unique quadratic mapping such that
**
for all . *

Theorem 11. *Let be a function such that
**
for all . Suppose that is an even mapping satisfying the inequality
**
for all . Then there exists a unique quadratic mapping such that
**
for all , where . *

*Proof. *Similar to the proof of Theorem 8, we can show that . Replacing by and in (87), respectively, we get
for all . Inequalities (89) and (90) imply that
for all . Interchanging into in (91), respectively, we obtain
for all . On the other hand, by putting in (87), we can deduce that
for all . It follows from (92) and (93) that
for all . Thus
for all . Substituting by in (95) and then dividing both sides by , we obtain
for all and all nonnegative integers . Thus the sequence is Cauchy by (86) and (96). The completeness of implies that the sequence is convergent. Define the mapping via
By a simple computation, one can show that
for all and for all . Taking to approach infinity in (98) and applying (86) and (97), we find (88). By (86), (87), and (97), we have
Hence, the mapping satisfies (5). It follows from part (iv) of Lemma 1 that the mapping is quadratic. Similar to the proof of Theorem 8, one can show that is unique.

Theorem 12. *Let be a function such that
**
for all . Suppose that is an even mapping satisfying the inequality
**
for all . Then there exists a unique quadratic mapping such that
**
where . *

*Proof. *Similar to the proof of Theorem 8, one can obtain
for all . Thus
for all . Substituting by in (104) and then multiplying both sides to , we have
for all and all nonnegative integers . The last inequality and (100) imply that the sequence is Cauchy. So, this sequence converges to the mapping . In other words,
We can also show that