AQCQ-Functional Equation in Non-Archimedean Normed Spaces

Gordji, M. Eshaghi; Khodabakhsh, R.; Jung, S.-M.; Khodaei, H.

doi:https://doi.org/10.1155/2010/741942

Abstract and Applied Analysis

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Volume 2010 | Article ID 741942 | https://doi.org/10.1155/2010/741942

AQCQ-Functional Equation in Non-Archimedean Normed Spaces

M. Eshaghi Gordji,^1,2,3R. Khodabakhsh,^1,2,3S.-M. Jung,⁴and H. Khodaei^1,2,3

Academic Editor: John M. Rassias

Received30 Jun 2010

Revised11 Sept 2010

Accepted11 Oct 2010

Published13 Dec 2010

Abstract

We prove the generalized Hyers-Ulam stability of generalized mixed type of quartic, cubic, quadratic and additive functional equation in non-Archimedean spaces.

1. Introduction and Preliminaries

In 1897, Hensel [1] has introduced a normed space which does not have the Archimedean property.

During the last three decades, theory of non-Archimedean spaces has gained the interest of physicists for their research in particular in problems coming from quantum physics, -adic strings, and superstrings [2]. Although many results in the classical normed space theory have a non-Archimedean counterpart, their proofs are essentially different and require an entirely new kind of intuition [3–10].

Let be a field. A non-Archimedean absolute value on is a function such that for any we have that (i) and equality holds if and only if , (ii), (iii).

Condition (iii) is called the strict triangle inequality. By (ii), we have . Thus, by induction, it follows from (iii) that for each integer . We always assume in addition that is non trivial, that is, there is an such that .

Let be a linear 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: (NA1) if and only if , (NA2) for all and , (NA3)the strong triangle inequality (ultrametric), namely,

Then is called a non-Archimedean space.

It follows from NA3 that therefore a sequence is Cauchy in if and only if converges to zero in a non-Archimedean space. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent.

The concept of stability of a functional equation arises when one replaces a functional equation by an inequality which acts as a perturbation of the equation. The first stability problem concerning group homomorphisms was raised by Ulam [11] in 1940 and affirmatively solved by Hyers [12]. Perhaps Aoki was the first author who has generalized the theorem of Hyers (see [13]).

Theorem 1.1 (Aoki [13]). If a mapping between two Banach spaces satisfies for all , where with , , then there exists a unique additive function such that

Moreover, Bourgin [14], Rassias [15], and Găvruta [16] have considered the stability problem with unbounded Cauchy differences (see also [17]). On the other hand, Rassias [18–23] considered the Cauchy difference controlled by a product of different powers of norm. However, there was a singular case; for this singularity a counterexample was given by Găvruta [24]. This stability phenomenon is called the Ulam-Găvruta-Rassias stability (see also [25]).

Theorem 1.2 (Rassias [18]). Let be a real normed linear space and a real complete normed linear space. Assume that is an approximately additive mapping for which there exist constants and such that and satisfies the inequality for all . Then there exists a unique additive mapping satisfying for all . If, in addition, is a mapping such that the transformation is continuous in for each fixed , then is an -linear mapping.

Very recently, Rassias [26] in inequality (1.5) replaced the bound by a mixed one involving the product and sum of powers of norms, that is, .

For more details about the results concerning such problems and mixed product-sum stability (Rassias Stability) the reader is referred to [27–42].

The functional equation is related to a symmetric biadditive function [43, 44]. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.7) is said to be a quadratic function. It is well known that a function between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function such that for all . The biadditive function is given by The Hyers-Ulam stability problem for the quadratic functional equation was solved by Skof [45]. In [46], Czerwik proved the Hyers-Ulam-Rassias stability of (1.7). Later, Jung [47] has generalized the results obtained by Skof and Czerwik.

Jun and Kim [48] introduced the following cubic functional equation: and they established the general solution and the generalized Hyers-Ulam stability for the functional equation (1.9). They proved that a function between two real vector spaces and is a solution of (1.9) if and only if there exists a unique function such that for all ; moreover, is symmetric for each fixed variable and is additive for fixed two variables. The function is given by for all (see also [47, 49–55]).

Lee et al. [56] considered the following functional equation: In fact, they proved that a function between two real vector spaces and is a solution of (1.11) if and only if there exists a unique symmetric biquadratic function such that for all . The biquadratic function is given by Obviously, the function satisfies the functional equation (1.11), which is called the quartic functional equation.

Eshaghi Gordji and Khodaei [49] have established the general solution and investigated the Hyers-Ulam-Rassias stability for a mixed type of cubic, quadratic, and additive functional equation (briefly, AQC-functional equation) with , in quasi-Banach spaces, where is nonzero integer with . Obviously, the function is a solution of the functional equation (1.13). Interesting new results concerning mixed functional equations have recently been obtained by Najati et al. [57–59] and Jun and Kim [60, 61] as well as for the fuzzy stability of a mixed type of additive and quadratic functional equation by Park [62]. The stability of generalized mixed type functional equations of the form for fixed integers , where , in quasi-Banach spaces was investigated by Eshaghi Gordji et al. [63]. The mixed type functional equation (1.14) is additive, quadratic, cubic, and quartic (briefly, AQCQ-functional equation).

This paper is organized as follows. In Section 2, we prove the generalized Hyers-Ulam stability of the functional equation (1.14) in non-Archimedean normed spaces, for an odd case. The generalized Hyers-Ulam stability of the functional equation (1.14) in non-Archimedean normed spaces, for an even case, is discussed in Section 3. Finally, in Section 4, we show the generalized Hyers-Ulam stability of the AQCQ-functional equation (1.14) in non-Archimedean normed spaces.

Throughout this paper, assume that is an additive group, is a complete non-Archimedean spaces, and , are vector spaces. Before taking up the main subject, given , we define the difference operator where and for all .

2. Stability of the AQCQ-Functional Equation (1.14): For an Odd Case

In this section, we prove the generalized Hyers-Ulam stability of the functional equation in complete non-Archimedean spaces: an odd case.

Lemma 2.1 (see [49, 59, 63]). If an odd function satisfies (1.14), then the function defined by is additive.

Theorem 2.2. Let be fixed, and let be a function such that for all . Suppose that an odd function satisfies the inequality for all . Then there exists a unique additive function such that for all , where for all .

Proof. Let . It follows from (2.2) and using oddness of that for all . Putting in (2.8), we have for all . It follows from (2.9) that for all . Replacing and by and in (2.8), respectively, we get for all . Setting in (2.8), one obtains for all . Putting in (2.8), we obtain for all . Replacing and by and in (2.8), respectively, we get for all . Replacing and by and in (2.8), respectively, one gets for all . Replacing and by and in (2.8), respectively, we obtain for all . Replacing and by and in (2.8), respectively, we have for all . It follows from (2.9), (2.11), (2.12), (2.14), and (2.15) that for all . Also, from (2.9), (2.10), (2.12), (2.13), (2.16), and (2.17), we conclude that for all . Finally, by using (2.18) and (2.19), we obtain that for all . Let be a function defined by for all . From (2.20), we conclude that for all . If we replace in (2.21) by , we get for all . It follows from (2.1) and (2.22) that the sequence is Cauchy. Since is complete, we conclude that is convergent. So one can define the function by for all . By using induction, it follows from (2.21) and (2.22) that for all and all . By taking to approach infinity in (2.24) and using (2.4) one gets (2.3). Now we show that is additive. It follows from (2.1), (2.22), and (2.23) that for all . So for all . On the other hand it follows from (2.1), (2.2), and (2.23) that for all . Hence the function satisfies (1.14). Thus by Lemma 2.1, the function is cubic-additive. Therefore (2.26) implies that the function is additive. If is another additive function satisfying (2.3), by using (2.1), we have for all . Therefore . For , we can prove the theorem by a similar technique.

Lemma 2.3 (see [49, 59, 63]). If an odd function satisfies (1.14), then the function defined by is cubic.

Theorem 2.4. Let be fixed and let be a function such that for all . Suppose that an odd function satisfies inequality (2.2) for all . Then there exists a unique cubic function such that for all , where and is defined as in (2.5) for all .

Proof. Let . Similar to the proof of Theorem 2.2, we have for all , where is defined as in (2.5) for all . Let be a function defined by for all . From (2.32), we conclude that for all . If we replace in (2.33) by , we get for all . It follows from (2.29) and (2.34) that the sequence is Cauchy. Since is complete, we conclude that is convergent. So one can define the function by for all . It follows from (2.33) and (2.34) by using induction that for all and all . By taking to approach infinity in (2.36) and using (2.29), one gets (2.30). Now we show that is cubic. It follows from (2.29), (2.34), and (2.35) that for all . So for all . On the other hand it follows from (2.2), (2.29), and (2.35) that for all . Hence the function satisfies (1.14). Thus by Lemma 2.1, the function is cubic-additive. Therefore (2.38) implies that the function is cubic. The rest of the proof is similar to the proof of Theorem 2.2. For , we can prove the theorem by a similar technique.

Lemma 2.5 (see [49, 63]). If an odd function satisfies (1.14), then is cubic-additive function.

Theorem 2.6. Let be fixed, and let be a function such that for all . Suppose that an odd function satisfies inequality (2.2) for all . Then there exist a unique additive function and a unique cubic function such that for all , where and are defined as in Theorems 2.2 and 2.4.

Proof. Let . By Theorems 2.2 and 2.4, there exists a additive function and a cubic function such that for all . So we obtain (2.41) by letting and for all .
To prove the uniqueness property of and , let be other additive and cubic functions satisfying (2.41). Let and . Hence for all . Since for all , for all . Therefore, we get and then , and the proof is complete. For , we can prove the theorem by a similar technique.

Theorem 2.7. Let be a function such that for all . Suppose that an odd function satisfies inequality (2.2) for all . Then there exist a unique additive function and a unique cubic function such that for all , where and are defined as in Theorems 2.2 and 2.4.

Proof. The proof is similar to the proof of Theorem 2.6, and the result follows from Theorems 2.2 and 2.4.

3. Stability of the AQCQ-Functional Equation (1.14): For an Even Case

In this section, we prove the generalized Hyers-Ulam stability of the functional equation in complete non-Archimedean spaces: an even case.

Lemma 3.1 (see [63]). If an even function satisfies (1.14), then the function defined by is quadratic.

Theorem 3.2. Let be fixed, and let be a function such that for all . Suppose that an even function with satisfies inequality (2.2) for all . Then there exists a unique quadratic function such that for all , where exists for all .

Proof. Let . It follows from (2.2) and using the evenness of that for all . Interchanging with in (3.5), we get by the evenness of : for all . Setting in (3.6), we have for all . Putting in (3.6), we obtain for all . Replacing and by and 0 in (3.6), respectively, we see that for all . Setting in (3.6) and using the evenness of , we get for all . It follows from (3.7), (3.8), (3.9), and (3.10) that for all . Let be a function defined by for all . From (3.11), we conclude that for all . Replacing by in (3.12), we have for all . It follows from (3.1) and (3.13) that the sequence is Cauchy. Since is complete, we conclude that is convergent. So one can define the function by for all . It follows from (3.12) and (3.13) by using induction that for all and all . By taking to approach infinity in (3.15) and using (3.3), one gets (3.2). Now we show that is quadratic. It follows from (3.1), (3.13), and (3.14) that for all . So for all . On the other hand it follows from (2.2), (3.1), and (3.14) that for all . Hence the function satisfies (1.14). Thus by Lemma 3.1, the function is quartic-quadratic. Therefore (3.17) implies that the function is quadratic. The rest of the proof is similar to the proof of Theorem 2.2. For , we can prove the theorem by a similar technique.

Lemma 3.3 (see [63]). If an even function satisfies (1.14), then the function defined by is quartic.

Theorem 3.4. Let be fixed, and let be a function such that for all . Suppose that an even function with satisfies inequality (2.2) for all . Then there exists a unique quartic function such that for all , where and is defined as in (3.4) for all .

Lemma 3.5 (see [63]). If an even function satisfies (1.14), then is quartic-quadratic function.

Theorem 3.6. Let be fixed, and let be a function such that for all . Suppose that an even function with satisfies inequality (2.2) for all . Then there exist a unique quadratic function and a unique quartic function such that for all , where and are defined as in Theorems 3.2 and 3.4.

Proof. The proof is similar to the proof of Theorem 2.6 and the result follows from Theorems 3.2 and 3.4.

Theorem 3.7. Let be a function such that for all . Suppose that an even function with satisfies inequality (2.2) for all . Then there exist a unique quadratic function and a unique quartic function such that for all , where and are defined as in Theorems 3.2 and 3.4.

4. AQCQ-Functional Equation in Non-Archimedean Normed Spaces

Now, we are ready to prove the main theorems concerning the generalized Hyers-Ulam stability problem for (1.14) in non-Archimedean spaces.

Lemma 4.1 (see [63]). A function satisfies (1.14) for all if and only if there exist a unique symmetric biquadratic function , a unique function , a unique symmetric biadditive function , and a unique additive function , such that for all , where the function is symmetric for each fixed variable and is additive for fixed two variables.

Theorem 4.2. Let be fixed, and let be a function satisfying (2.41) and (3.22) for all . Then exist for all , where and are defined as in (2.3) and (3.3) for all . Suppose that a function with satisfies inequality (2.2) for all . Then there exist a unique additive function , a unique quadratic function , a unique cubic function , and a unique quartic function such that for all , where for all , and , , and are defined as in Theorems 2.2, 2.4, 3.2, and 3.4.

Proof. Let and for all . Then for all . From Theorem 2.6, it follows that there exist a unique additive function and a unique cubic function satisfying for all . Also, let for all . Then for all . From Theorem 3.6, it follows that there exist a quadratic function and a quartic function satisfying for all . Hence, (4.2) follows from (4.7) and (4.9). To prove the uniqueness property of , and , let be other additive, quadratic, cubic, and quartic functions satisfying (4.2). Let , and . So for all . Since for all , if we replace in (4.10) by and multiply both sides of (4.10) by , we get for all . Therefore . Putting and in (4.10), we obtain for all . Therefore . Also by putting and in (4.10), we have for all . Therefore , and then .
For , we can prove the theorem by a similar technique.

Theorem 4.3. Let be a function satisfying (2.47) and (3.24) for all . Suppose that a function with satisfies inequality (2.2) for all . Then there exist a unique additive function , a unique quadratic function , a unique cubic function , and a unique quartic function such that for all , where is defined as in Theorem 4.2.

Proof. The proof is similar to the proof of Theorem 4.2, and the result follows from Theorems 2.7 and 3.7. To prove the uniqueness property of , and , let be other additive, quadratic, cubic and quartic functions satisfying (4.15). Let , and . So for all . Since for all , if we replace in (4.16) by and divide both sides of (4.16) by , we get for all . Therefore . It follows that for all . Therefore . Also by putting and in (4.16), we have for all . Therefore , and then .

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2010-0007143).

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Copyright © 2010 M. Eshaghi Gordji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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