Journal of Function Spaces

Volume 2014 (2014), Article ID 919646, 9 pages

http://dx.doi.org/10.1155/2014/919646

Research Article

Stability for Functional Equation of Mixed Type in Non-Archimedean Normed Spaces

Ick-Soon Chang¹ and Yang-Hi Lee²

¹Department of Mathematics, Chungnam National University, 79 Daehangno, Yuseong-gu, Daejeon 305-764, Republic of Korea

²Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Republic of Korea

Received 16 November 2013; Revised 8 January 2014; Accepted 10 January 2014; Published 25 February 2014

Academic Editor: Krzysztof Ciepliński

Copyright © 2014 Ick-Soon Chang and Yang-Hi Lee. 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.

Abstract.

We take into account the stability of the following functional equation in non-Archimedean normed spaces.

1. Introduction and Preliminaries

The study of stability problems has originally been formulated by Ulam [1]: under what condition does there exist a homomorphism near an approximate homomorphism? Hyers [2] had answered affirmatively the question of Ulam for Banach spaces. 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. The paper work of Rassias [4] has had a lot of influence in the development of what is called the generalized Hyers-Ulam stability of functional equations. Thereafter, many interesting results of the generalized Hyers-Ulam stability to a number of functional equations have been investigated by a number of mathematicians; see [5–13] and references therein.

Now we demonstrate some definitions used in this work.

Definition 1. A field , equipped with a function (valuation) from into , is called a non-Archimedean field if the function satisfies the following conditions:

(1) if and only if ;(2);(3)the strong triangle inequality, namely, for all .

Of course, it is easy to see that and for all nonzero integer .

Definition 2. Let be a vector space over the non-Archimedean field with a nontrivial non-Archimedean valuation . A function is said to be a non-Archimedean norm (valuation) if it satisfies the following conditions:

(1) if and only if ;(2) for all and all ;(3)the strong triangle inequality, namely,

In this case, is called a non-Archimedean space. Moreover, if every Cauchy sequence is convergent, then is said to be a complete non-Archimedean space.

It follows from the strong triangle inequality that

for all and all with . Therefore a sequence is a Cauchy sequence in non-Archimedean space if and only if the sequence converges to zero in the space.

On the other hand, Moslehian and Rassias [14] discussed the stability of the additive functional equation and the quadratic functional equation in non-Archimedean normed spaces. Quite recently, the new results on stability of functional equations in non-Archimedean metric spaces have been investigated (e.g., [6, 12, 15]).

Here and now, we consider a quadratic-additive type functional equation

whose solution is called a quadratic-additive mapping. Quite recently, the second author [16] investigated the Hyers-Ulam stability of the functional equation (3) on restricted domains for the case . In particular, hyperstability of the functional equation (3) in the case when was investigated in [17]. The purpose of this work establishes the generalized Hyers-Ulam stability of the functional equation (3) in non-Archimedean normed spaces.

2. Main Results

Throughout this section, we assume that is a non-Archimedean normed space and is a complete non-Archimedean space. For a given mapping with vector spaces and , we use the abbreviations

for all , where is a nonzero integer.

Theorem 3. Let and be vector spaces. A mapping satisfies equation for all if and only if there exist a quadratic mapping and an additive mapping such that

for all .

Proof. () We decompose into the even part and the odd part by putting

for all . Notice that . From the equalities

for all , we conclude that is a quadratic mapping and is an additive mapping.

() If there exist a quadratic mapping and an additive mapping such that

for all , then we find that

for all . We arrive at the desired conclusion.

Theorem 4. Let be a function such that

for all and, for each , let the limit

denoted by , exist. Suppose that is a mapping satisfying

for all . Then there exists a quadratic-additive mapping such that

for all , where the mapping is given by

for all .

Proof. For a given mapping and , let be a mapping defined by

for all . Note that and

for all and . It follows from (10) and (16) that the sequence is Cauchy. Since is complete, we conclude that is convergent. So we can define by

for all . An induction implies that

for all and all . By taking the limit as in (18) and using (10), we obtain inequality (13).

From (12), we get

for all . Take the limit as in the above inequality and then use (10) to have for all .

Corollary 5. Let be a real number and , . If a mapping satisfies the condition

for all , then there exists a unique quadratic-additive mapping such that

for all .

Proof. Let . Since and , we know that

for all . Therefore the conditions of Theorem 4 are fulfilled. In particular, it is easy to see that

Theorem 4 guarantees that there exists a quadratic-additive mapping with (21).

Now, to show uniqueness of the mapping , let us assume that is another quadratic-additive mapping satisfying (21). Then we have

for any , and thus we feel that

for all , which implies that is unique.

Theorem 6. Let be a function such that

for all and, for each , let the limit

denoted by , exist. Suppose that is a mapping satisfying

for all . Then there exists a quadratic-additive mapping such that

for all , where the mapping is given by

for all .

Proof. For a given mapping and , let be a mapping defined by

for all . Observe that and

for all and . It follows by (26) and (32) that the sequence is Cauchy. Since is complete, converges and so we can define by

for all . Applying the induction, we yield that

for all and all . Sending the limit as in (34) with (26), we arrive at inequality (29).

According to (28), we see that

for all . Taking the limit as and using (26), we get for all .

Corollary 7. Let be a real number and , . If a mapping satisfies the condition

for all , then there exists a unique quadratic-additive mapping such that

for all .

Proof. Let . Since and , the mapping satisfies equalities (26) for all . Moreover, we see that

Then it follows from Theorem 6 that there exists a quadratic-additive mapping satisfying (37).

In order to prove the uniqueness of , we assume that is another quadratic-additive mapping satisfying (37). Then

for any , whence we give that

for all , which means that is unique.

Theorem 8. Let be a function such that

for all and, for each , let the limit

denoted by , exist. Suppose that is a mapping satisfying

for all . Then there exists a quadratic-additive mapping such that

for all , where the mapping is given by

for all .

Proof. For a given mapping and , let be a mapping defined by

for all . Now we note that and

for all and . Hence it follows by (41) and (47) that the sequence is Cauchy. Since is complete, we conclude that is convergent. Thus we can define by

for all . Using the induction argument, we prove that

for all and all . Taking the limit as in (49) and using (41) lead to inequality (44).

By virtue of (43), we get

for all . Send the limit as and then use (41) to find for all .

Theorem 9. Let be a function such that

for all and, for each , let the limit

denoted by , exist. Suppose that is a mapping satisfying

for all . Then there exists a quadratic-additive mapping such that

for all , where the mapping is given by

for all .

Proof. For a given mapping and , let be a mapping defined by

for all . We remark that and

for all and . So it follows from (51) and (57) that the sequence is Cauchy. Due to the completeness of , this sequence is convergent. Let be a mapping defined by

for all . An induction implies that

for all and all . By passing the limit as in (59) with (51), we obtain the relation (54).

From (53), we find that

for all . Taking the limit as and using (51), we arrive at for all .

Corollary 10. Let and let be a real number such that . If a mapping satisfies the condition

for all , then there exists a unique quadratic-additive mapping such that

for all .

Proof. We consider . Based on the fact that and , the mapping satisfies conditions (51). In fact, it is easy to see that

On account of Theorem 9, there is a quadratic-additive mapping satisfying (62).

To show uniqueness of the mapping , we suppose that is another quadratic-additive mapping satisfying (62). Then

for any and so we figure out the following:

for all . This implies that is unique.

The problem is whether or not the previous corollaries hold the cases when or .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees for giving useful suggestions and for the improvement of this paper. The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2A10004419).

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