##### Journal of Inequalities and Applications
Research

C Park1, H Azadi Kenary2* and TM Rassias3

Author Affiliations

1 Department of Mathematics, Hanyang University, Seoul, South Korea

2 Department of Mathematics, College of Sciences, Yasouj University, Yasouj, 75914-353, Iran

3 Department of Mathematics, National Technical University of Athens, Zografou, Campus, Athens, 15780, Greece

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Journal of Inequalities and Applications 2012, 2012:174 doi:10.1186/1029-242X-2012-174

 Received: 29 September 2011 Accepted: 27 July 2012 Published: 6 August 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

Using the fixed point and direct methods, we prove the generalized Hyers-Ulam stability of the following additive-quadratic functional equation in non-Archimedean normed spaces

where r, s, γ are positive real numbers.

MSC: 39B55, 46S10.

##### Keywords:
Hyers-Ulam stability; fixed point method; non-Archimedean normed spaces

### 1 Introduction and preliminaries

A classical question in the theory of functional equations is the following: ‘When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation?’ If the problem accepts a solution, we say that the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [44] in 1940. In the next year, Hyers [23] gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces. In 1978, Rassias [39] proved a generalization of Hyers’ theorem for additive mappings. This new concept is known as generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. Furthermore, in 1994, a generalization of Rassias’ theorem was obtained by Gǎvruta [21] by replacing the bound by a general control function .

In 1983, a generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [43] for mappings , where X is a normed space and Y is a Banach space. In 1984, Cholewa [11] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group and, in 2002, Czerwik [13] proved the generalized Hyers-Ulam stability of the quadratic functional equation. The reader is referred to [1-42] and references therein for detailed information on stability of functional equations.

In 1897, Hensel [22] has introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications (see [16,25-27,33]).

Definition 1.1 By a non-Archimedean field, we mean a field equipped with a function (valuation) such that for all , the following conditions hold:

(1) if and only if ;

(2) ;

(3) .

Definition 1.2 Let X be a vector space over a scalar field with a non-Archimedean non-trivial valuation . A function is a non-Archimedean norm (valuation) if it satisfies the following conditions:

(1) if and only if ;

(2) (, );

(3) The strong triangle inequality (ultrametric); namely,

Then is called a non-Archimedean space.

Due to the fact that

Definition 1.3 A sequence is Cauchy 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.

Definition 1.4 Let X be a set. A function is called a generalized metric on X if d satisfies

(1) if and only if ;

(2) for all ;

(3) for all .

We recall a fundamental result in fixed point theory.

Theorem 1.5 ([13,17])

Letbe a complete generalized metric space and letbe a strictly contractive mapping with Lipschitz constant. Then for each given element, either

for all nonnegative integersnor there exists a positive integersuch that

(1) , ;

(2) the sequenceconverges to a fixed pointofJ;

(3) is the unique fixed point ofJin the set;

(4) for all.

In 1996, G. Isac and Th. M. Rassias [24] were the first to provide applications of stability theory of functional equations for the proof of new fixed-point theorems with applications. By using fixed-point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [14,15,35,36,40]).

This paper is organized as follows: In Section 2, using the fixed-point method, we prove the Hyers-Ulam stability of the following additive-quadratic functional equation:

(1.1)

where , in non-Archimedean normed space. In Section 3, using direct methods, we prove the Hyers-Ulam stability of the additive-quadratic functional equation (1.1) in non-Archimedean normed spaces.

It is easy to see that a mapping f with is a solution of equation (1.1) if and only if f is of the form for all .

### 2 Stability of functional equation (1.1): a fixed point method

In this section, we deal with the stability problem for the additive-quadratic functional equation (1.1). In the rest of the present article, let .

Theorem 2.1LetXis a non-Archimedean normed space and thatYbe a complete non-Archimedean space. Letbe a function such that there exists anwith

(2.1)

for all. Letbe an odd mapping satisfying

(2.2)

for all. Then there exists a unique additive mappingsuch that

(2.3)

for all.

Proof Putting in (2.2) and replacing x by 2x, we get

(2.4)

for all . Putting and in (2.2), we have

(2.5)

for all . By (2.4) and (2.5), we get

(2.6)

Consider the set and introduce the generalized metric on S:

where, as usual, . It is easy to show that is complete (see [30]). Now we consider the linear mapping such that for all . Let be given such that . Then

for all . Hence,

for all . So implies that . This means that for all .

It follows from (2.6) that . By Theorem 1.5, there exists a mapping satisfying the following:

(1) A is a fixed point of J, i.e.,

(2.7)

for all . The mapping A is a unique fixed point of J in the set . This implies that A is a unique mapping satisfying (2.7) such that there exists a satisfying for all ;

(2) as . This implies the equality

(3) , which implies the inequality . This implies that the inequalities (2.3) holds.

It follows from (2.1) and (2.2) that

for all . So

for all . Hence, satisfying (1.1). This completes the proof. □

Corollary 2.2Letθbe a positive real number andqis a real number with. Letbe an odd mapping satisfying

(2.8)

for all. Then there exists a unique additive mappingsuch that

for all.

Proof The proof follows from Theorem 2.1 by taking for all . Then we can choose and we get the desired result. □

Theorem 2.3LetXis a non-Archimedean normed space and thatYbe a complete non-Archimedean space. Letbe a function such that there exists anwith

(2.9)

for all. Letbe an odd mapping satisfying (2.2). Then there exists a unique additive mappingsuch that

for all.

Proof Let be the generalized metric space defined in the proof of Theorem 2.1.

Now we consider the linear mapping such that

for all .

Replacing x by in (2.6) and using (2.9), we have

(2.10)

So .

The rest of the proof is similar to the proof of Theorem 2.1. □

Corollary 2.4Letθbe a positive real number andqis a real number with. Letbe an odd mapping satisfying (2.8). Then there exists a unique additive mappingsuch that

for all.

Proof The proof follows from Theorem 2.3 by taking for all . Then we can choose and we get the desired result. □

Theorem 2.5LetXis a non-Archimedean normed space and thatYbe a complete non-Archimedean space. Letbe a function such that there exists anwith

(2.11)

for all. Letbe an even mapping withand satisfying (2.2). Then there exists a unique quadratic mappingsuch that

(2.12)

for all.

Proof Consider the set and the generalized metric in defined by

where, as usual, . It is easy to show that is complete (see [30]). Now we consider the linear mapping such that

for all .

Putting and in (2.2), we have

(2.13)

for all .

Substituting and then replacing x by 2x in (2.2), we obtain

(2.14)

By (2.13) and (2.14), we get

(2.15)

The rest of the proof is similar to the proof of Theorem 2.1. □

Corollary 2.6Letθbe a positive real number andqis a real number with. Letbe an even mapping withand satisfying (2.8). Then there exists a unique quadratic mappingsuch that

for all.

Proof The proof follows from Theorem 2.5 by taking for all . Then we can choose and we get the desired result. □

Theorem 2.7LetXis a non-Archimedean normed space and thatYbe a complete non-Archimedean space. Letbe a function such that there exists anwith

(2.16)

for all. Letbe an even mapping withand satisfying (2.2). Then there exists a unique quadratic mappingsuch that

(2.17)

for all.

Proof It follows from (2.15) that

The rest of the proof is similar to the proof of Theorems 2.1 and 2.5. □

Corollary 2.8Letθbe a positive real number andqis a real number with. Letbe an even mapping withand satisfying (2.8). Then there exists a unique quadratic mappingsuch that

for all.

Proof The proof follows from Theorem 2.7 by taking for all . Then we can choose and we get the desired result. □

Let be a mapping satisfying and (1.1). Let and . Then is an even mapping satisfying (1.1) and is an odd mapping satisfying (1.1) such that .

On the other hand

and

for all , where is the difference operator of the functional equation (1.1). So we obtain the following theorem.

Theorem 2.9LetXis a non-Archimedean normed space and thatYbe a complete non-Archimedean space. Letbe a function such that there exists anwith

for all. Letbe a mapping withand satisfying (2.2). Then there exist a unique additive mappingand a unique quadratic mappingsuch that

for all.

Theorem 2.10LetXis a non-Archimedean normed space and thatYbe a complete non-Archimedean space. Letbe a function such that there exists anwith

for all. Letbe a mapping withand satisfying (2.2). Then there exist a unique additive mappingand a unique quadratic mappingsuch that

for all.

### 3 Stability of functional equation (1.1): a direct method

In this section, using direct method, we prove the generalized Hyers-Ulam stability of the additive-quadratic functional equation (1.1) in non-Archimedean space.

Theorem 3.1LetGbe a vector space and thatXis a non-Archimedean Banach space. Assume thatbe a function such that

(3.1)

for all. Suppose that, for any, the limit

(3.2)

exists andbe an odd mapping satisfying

(3.3)

Then the limit

exists for alland defines an additive mappingsuch that

(3.4)

Moreover, if

thenAis the unique additive mapping satisfying (3.4).

Proof By (2.10), we know

(3.5)

for all . Replacing x by in (3.5), we obtain

(3.6)

Thus, it follows from (3.1) and (3.6) that the sequence is a Cauchy sequence. Since X is complete, it follows that is convergent. Set

By induction on n, one can show that

(3.7)

for all and . By taking in (3.7) and using (3.2), one obtains (3.4). By (3.1) and (3.3), we get

for all . Therefore, the mapping satisfies (1.1).

To prove the uniqueness property of A, let L be another mapping satisfying (3.4). Then we have

for all . Therefore, . This completes the proof. □

Corollary 3.2Letbe a function satisfying

for all. Assume thatandbe a mapping withsuch that

(3.8)

for all. Then there exists a unique additive mappingsuch that

Proof Defining by , then we have

for all . The last equality comes form the fact that . On the other hand, it follows that

exists for all . Also, we have

Thus, applying Theorem 3.1, we have the conclusion. This completes the proof. □

Theorem 3.3LetGbe a vector space and thatXis a non-Archimedean Banach space. Assume thatbe a function such that

(3.9)

for all. Suppose that, for any, the limit

(3.10)

exists andbe an odd mapping satisfying (3.3). Then the limitexists for alland

(3.11)

for all. Moreover, if

thenAis the unique mapping satisfying (3.11).

Proof By (2.6), we get

(3.12)

for all . Replacing x by in (3.12), we obtain

(3.13)

Thus, it follows from (3.9) and (3.13) that the sequence is convergent. Set

On the other hand, it follows from (3.13) that

for all and with . Letting , taking in the last inequality and using (3.10), we obtain (3.11).

The rest of the proof is similar to the proof of Theorem 3.1. This completes the proof. □

Theorem 3.4LetGbe a vector space and thatXis a non-Archimedean Banach space. Assume thatbe a function such that

(3.14)

for all. Suppose that, for any, the limit

(3.15)

exists andbe an even mapping withand satisfying (3.3). Then the limitexists for alland defines a quadratic mappingsuch that

(3.16)

Moreover, if

thenQis the unique additive mapping satisfying (3.16).

Proof It follows from (2.15) that

(3.17)

Replacing x by in (3.18), we have

(3.18)

It follows from (3.14) and (3.18) that the sequence is Cauchy sequence. The rest of the proof is similar to the proof of Theorem 3.1. □

Similarly, we can obtain the followings. We will omit the proof.

Theorem 3.5LetGbe a vector space and thatXis a non-Archimedean Banach space. Assume thatbe a function such that

(3.19)

for all. Suppose that, for any, the limit

(3.20)

exists andbe an even mapping withand satisfying (3.3). Then the limitexists for alland

(3.21)

for all. Moreover, if

thenQis the unique mapping satisfying (3.21).

Let be a mapping satisfying and (1.1). Let and . Then is an even mapping satisfying (1.1) and is an odd mapping satisfying (1.1) such that . On the other hand,

and

for all , where is the difference operator of the functional equation (1.1). So we obtain the following theorem.

Theorem 3.6LetGbe a vector space and thatXis a non-Archimedean Banach space. Assume thatbe a function such that

for all. Suppose that the limits

and

exist for allandbe a mapping withand satisfying (3.3). Then there exist an additive mappingand a quadratic mappingsuch that

(3.22)

for all. Moreover, if

and

thenA, Qare the unique mappings satisfying (3.22).

Theorem 3.7LetGbe a vector space and thatXis a non-Archimedean Banach space. Assume thatbe a function such that

for all. Suppose that the limits

and

exist for allandbe a mapping withand satisfying (3.3). Then there exist an additive mappingand a quadratic mappingsuch that

(3.23)

for all. Moreover, if

and

thenA, Qare the unique mappings satisfying (3.23).

### 4 Conclusion

We linked here two different disciplines, namely, the non-Archimedean normed spaces and functional equations. We established the generalized Hyers-Ulam stability of the functional equation (1.1) in non-Archimedean normed spaces.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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