We establish some stability results over -adic fields for the generalized quadratic functional equation where and .

1. Introduction and Preliminaries

In 1899, Hensel [1] discovered the -adic numbers as a number of theoretical analogue of power series in complex analysis. Fix a prime number . For any nonzero rational number , there exists a unique integer such that , where and are integers not divisible by . Then, -adic absolute value defines a non-Archimedean norm on . The completion of with respect to the metric is denoted by , and it is called the -adic number field. In fact, is the set of all formal series , where are integers (see, e.g., [2, 3]). Note that if , then for each integer .

During the last three decades, -adic numbers have gained the interest of physicists for their research, in particular, in problems coming from quantum physics, -adic strings, and superstrings [4, 5]. A key property of -adic numbers is that they do not satisfy the Archimedean axiom: For , there exists such that .

Let denote a field and function (valuation absolute) from into . A non-Archimedean valuation is a function that satisfies the strong triangle inequality; namely, for all . The associated field is referred to as a non-Archimedean field. Clearly, and for all . A trivial example of a non-Archimedean valuation is the function taking everything except 0 into 1 and . We always assume in addition that is nontrivial, that is, there is a such that .

Let be a linear space over a field with a non-Archimedean nontrivial valuation . A function is said to be a non-Archimedean norm if it is a norm over with the strong triangle inequality (ultrametric); namely, for all . Then, is called a non-Archimedean space. In any such a space, a sequence is Cauchy if and only if converges to zero. By a complete non-Archimedean space, we mean one in which every Cauchy sequence is convergent.

The study of stability problems for functional equations is related to a question of Ulam [6] concerning the stability of group homomorphisms, which was affirmatively answered for Banach spaces by Hyers [7]. Subsequently, the result of Hyers was generalized by Aoki [8] for additive mappings and by Rassias [9] for linear mappings by considering an unbounded Cauchy difference. The paper by Rassias has provided a lot of influences in the development of what we now call the generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. Rassias [10] considered the Cauchy difference controlled by a product of different powers of norm. The above results have been generalized by Forti [11] and Găvruţa [12] who permitted the Cauchy difference to become arbitrary unbounded (see also [1322]). Arriola and Beyer [23] investigated stability of approximate additive functions . They showed that if is a continuous function for which there exists a fixed such that for all , then there exists a unique additive function such that for all . For more details about the results concerning such problems, the reader is referred to [2445].

Recently, Khodaei and Rassias [46] introduced the generalized additive functional equation and proved the generalized Hyers-Ulam stability of the above functional equation. The functional equation is related to symmetric biadditive function and is called a quadratic functional equation [47, 48]. Every solution of the quadratic equation (1.2) is said to be a quadratic function.

Now, we introduce the generalized quadratic functional equation in -variables as follows: where . Moreover, we investigate the generalized Hyers-Ulam stability of functional equation (1.3) over the -adic field .

As a special case, if in (1.3), then we have the functional equation (1.2). Also, if in (1.3), we obtain that is,

2. Stability of Quadratic Functional Equation (1.3) over -Adic Fields

We will use the following lemma.

Lemma 2.1. Let and be real vector spaces. A function satisfies the functional equation (1.3) if and only if the function is quadratic.

Proof. Let satisfy the functional equation (1.3). Setting () in (1.3), we have that is, or but , and also so .
Putting () in (1.3) and then using , we get that is, for all , this shows that satisfies the functional equation (1.2). So the function is quadratic.
Conversely, suppose that is quadratic, thus satisfies the functional equation (1.2). Hence, we have and is even.
We are going to prove our assumption by induction on . It holds on . Assume that it holds on the case where ; that is, we have for all . It follows from (1.2) that for all . Replacing by in (2.7), we obtain for all . Adding (2.7) to (2.8), we have for all . Replacing by in (2.9), we get for all . Adding (2.9) to (2.10), one gets for all . By using the above method, for until , we infer that for all . Now, by the case , we lead to for all , so (1.3) holds for . This completes the proof of the lemma.

Corollary 2.2. A function satisfies the functional equation (1.3) if and only if there exists a symmetric biadditive function such that for all .

Now, we investigate the stability of the functional equation (1.3) from a Banach space into -adic field . For convenience, we define the difference operator for a given function :

Theorem 2.3. Let be a Banach space and let be real numbers. Suppose that a function with satisfies the inequality for all . Then there exists a unique quadratic function such that for all nonzero .

Proof. Letting and () in (2.15), we obtain for all . Hence, for all nonnegative integers and with and for all . It follows from (2.18) that the sequence is a Cauchy sequence for all . Since is complete, the sequence converges. Therefore, one can define the function by for all . It follows from (2.15) and (2.19) that for all . So . By Lemma 2.1, the function is quadratic.
Taking the limit in (2.18) with , we find that the function is quadratic function satisfying the inequality (2.16) near the approximate function of (1.3).
To prove the aforementioned uniqueness, we assume now that there is another additive function which satisfies (1.3) and the inequality (2.16). So which tends to zero as for all nonzero . This proves the uniqueness of , completing the proof of uniqueness.

The following example shows that the above result is not valid over -adic fields.

Example 2.4. Let be a prime number and define by . Since , for all . Hence, the conditions of Theorem 2.3 for and hold. However for each , we have for all . Hence is not convergent for all nonzero .

In the next result, which can be compared with Theorem 2.3, we will show that the stability of the functional equation (1.3) in non-Archimedean spaces over -adic fields.

Theorem 2.5. Let be fixed. Let be a non-Archimedean space and be a complete non-Archimedean space over , where is a prime number. Suppose that a function satisfies the inequality for all , where and are nonnegative real numbers. Then, the limit exists for all and is a unique quadratic function satisfying for all .

Proof. By (2.24), for all , where . Putting () in (2.27) to obtain , setting () in (2.27), we obtain for all . So for all . Letting in (2.29), we have for all . By induction on , we will show that for each , for all . It holds on ; see (2.30). Let (2.31) hold for . Replacing and by and in (2.29), respectively, we get for all . It follows from (2.32) and our induction hypothesis that for all . This proves (2.31) for each . In particular, for all . So for all . Hence, for all . Since the right side of the above inequality tends to zero as , is a Cauchy sequence in complete non-Archimedean space , thus it converges to some function for all . Using (2.35) and induction, one can show that for any , we have for all . Letting in this inequality, we see that for all . Moreover, for all . So . By Lemma 2.1, the function is quadratic.
Now, let be another quadratic function satisfying (1.3) and (2.38). So which tends to zero as for all . This proves the uniqueness of .
The rest of the proof is similar to the above proof, hence it is omitted.


The third author of this work was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant number: 2011-0005197).