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# An AQCQ-functional equation in matrix Banach spaces

Choonkil Park1, Jung Rye Lee2 and Dong Yun Shin3*

Author Affiliations

1 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea

2 Department of Mathematics, Daejin University, Pocheon, Kyeonggi, 487-711, Korea

3 Department of Mathematics, University of Seoul, Seoul, 130-743, Korea

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Advances in Difference Equations 2013, 2013:146  doi:10.1186/1687-1847-2013-146

 Received: 2 April 2013 Accepted: 8 May 2013 Published: 23 May 2013

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 method, we prove the Hyers-Ulam stability of an additive-quadratic-cubic-quartic functional equation in matrix normed spaces.

MSC: 47L25, 47H10, 39B82, 46L07, 39B52.

### 1 Introduction and preliminaries

The abstract characterization given for linear spaces of bounded Hilbert space operators in terms of matricially normed spaces[1] implies that quotients, mapping spaces and various tensor products of operator spaces may again be regarded as operator spaces. Owing in part to this result, the theory of operator spaces is having an increasingly significant effect on operator algebra theory (see [2]).

The proof given in [1] appealed to the theory of ordered operator spaces [3]. Effros and Ruan [4] showed that one can give a purely metric proof of this important theorem by using the technique of Pisier [5] and Haagerup [6] (as modified in [7]).

The stability problem of functional equations originated from a question of Ulam [8] concerning the stability of group homomorphisms.

The functional equation

is called the Cauchy additive functional equation. In particular, every solution of the Cauchy additive functional equation is said to be an additive mapping. Hyers [9] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [10] for additive mappings and by TM Rassias [11] for linear mappings by considering an unbounded Cauchy difference. A generalization of the TM Rassias theorem was obtained by Găvruta [12] by replacing the unbounded Cauchy difference by a general control function in the spirit of TM Rassias’ approach.

In 1990, TM Rassias [13] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for . In 1991, Gajda [14], following the same approach as in TM Rassias [11], gave an affirmative solution to this question for . It was shown by Gajda [14], as well as by TM Rassias and Šemrl [15], that one cannot prove a TM Rassias’ type theorem when (cf. the books of Czerwik [16], Hyers et al.[17]).

In 1982, JM Rassias [18] followed the innovative approach of the TM Rassias’ theorem [11] in which he replaced the factor by for with .

The functional equation

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [19] for mappings , where X is a normed space and Y is a Banach space. Cholewa [20] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [21] proved the Hyers-Ulam stability of the quadratic functional equation.

In [22], Jun and Kim considered the following cubic functional equation:

(1.1)

It is easy to show that the function satisfies functional equation (1.1), which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic mapping.

In [23], Lee et al. considered the following quartic functional equation:

(1.2)

It is easy to show that the function satisfies functional equation (1.2), which is called a quartic functional equation, and every solution of the quartic functional equation is said to be a quartic mapping. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [24-33]).

We will use the following notations:

;

is that -component is 1 and the other components are zero;

is that -component is x and the other components are zero;

For , ,

Note that is a matrix normed space if and only if is a normed space for each positive integer n and holds for , and , and that is a matrix Banach space if and only if X is a Banach space and is a matrix normed space.

Let E, F be vector spaces. For a given mapping and a given positive integer n, define by

for all .

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 1.1[34,35]

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, Isac and Rassias [36] 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 [37-43]).

In this paper, we prove the Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation:

(1.3)

in matrix normed spaces by using the fixed point method.

One can easily show that an odd mapping satisfies (1.3) if and only if the odd mapping is an additive-cubic mapping, i.e.,

It was shown in [[44], Lemma 2.2] that and are cubic and additive, respectively, and that .

One can easily show that an even mapping satisfies (1.3) if and only if the even mapping is a quadratic-quartic mapping, i.e.,

It was shown in [[45], Lemma 2.1] that and are quartic and quadratic, respectively, and that .

Throughout this paper, let be a matrix normed space and be a matrix Banach space.

### 2 Hyers-Ulam stability of AQCQ-functional equation (1.3) in matrix normed spaces: odd mapping case

In this section, we prove the Hyers-Ulam stability of AQCQ-functional equation (1.3) in matrix normed spaces for an odd mapping case.

Lemma 2.1Letbe a matrix normed space. Then:

(1) for.

(2) for.

(3) if and only iffor.

Proof (1) Since and , . Since , . So .

(2) Since and , . Since ,

(3) By

we get the result. □

For a mapping , define and by

for all and all .

Theorem 2.2Letbe 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 Let and except for in (2.2).

Putting in (2.2), we get

(2.4)

for all .

Replacing by in (2.2), we get

(2.5)

for all .

By (2.4) and (2.5),

(2.6)

for all . Replacing by and letting in (2.6), we get

for all . So

(2.7)

for all .

Consider the set

and introduce the generalized metric on S:

where, as usual, . It is easy to show that is complete (see [46,47]).

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 (2.7) that .

By Theorem 1.1, there exists a mapping satisfying the following:

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

(2.8)

for all . The mapping A is a unique fixed point of J in the set

This implies that A is a unique mapping satisfying (2.8) such that there exists a satisfying

for all ;

(2) as . This implies the equality

for all ;

(3) , which implies the inequality

So

(2.9)

for all .

It follows from (2.1) and (2.2) that

for all . Hence for all a, b. So is additive.

By Lemma 2.1 and (2.9),

for all . Thus is a unique additive mapping satisfying (2.3), as desired. □

Corollary 2.3Letr, θbe positive real numbers with. Letbe an odd mapping such that

(2.10)

for all. Then there exists a unique additive mappingsuch that

for all.

Proof The proof follows from Theorem 2.2 by taking for all . Then we can choose and we get the desired result. □

Theorem 2.4Letbe a function such that there exists anwith

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 2.2.

Now we consider the linear mapping such that

for all .

It follows from (2.7) that

for all . Thus . So

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 2.5Letr, θbe positive real numbers with. Letbe an odd mapping satisfying (2.10). Then there exists a unique additive mappingsuch that

for all.

Proof The proof follows from Theorem 2.4 by taking for all . Then we can choose and we get the desired result. □

Theorem 2.6Letbe a function such that there exists anwith

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

for all.

Proof Let be the generalized metric space defined in the proof of Theorem 2.2.

Replacing by and letting in (2.6), we get

for all . So

(2.11)

for all . Thus . So

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 2.7Letr, θbe positive real numbers with. Letbe an odd mapping satisfying (2.10). Then there exists a unique cubic mappingsuch that

for all.

Proof The proof follows from Theorem 2.6 by taking for all . Then we can choose and we get the desired result. □

Theorem 2.8Letbe a function such that there exists anwith

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

for all.

Proof Let be the generalized metric space defined in the proof of Theorem 2.2.

It follows from (2.11) that

for all . Thus . So

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 2.9Letr, θbe positive real numbers with. Letbe an odd mapping satisfying (2.10). Then there exists a unique cubic mappingsuch that

for all.

Proof The proof follows from Theorem 2.8 by taking for all . Then we can choose and we get the desired result. □

### 3 Hyers-Ulam stability of AQCQ-functional equation (1.3) in matrix normed spaces: even mapping case

In this section, we prove the Hyers-Ulam stability of AQCQ-functional equation (1.3) in matrix normed spaces for an even mapping case.

Theorem 3.1Letbe a function such that there exists anwith

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

for all.

Proof Let be the generalized metric space defined in the proof of Theorem 2.2.

Let and except for in (2.2).

Putting in (2.2), we get

(3.1)

for all .

Replacing by in (2.2), we get

(3.2)

for all .

By (3.1) and (3.2),

(3.3)

for all . Replacing by and letting in (3.3), we get

for all . So

(3.4)

for all . Thus . So

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 3.2Letr, θbe positive real numbers with. Letbe an even mapping satisfying (2.10). Then there exists a unique quadratic mappingsuch that

for all.

Proof The proof follows from Theorem 3.1 by taking for all . Then we can choose and we get the desired result. □

Theorem 3.3Letbe a function such that there exists anwith

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

for all.

Proof Let be the generalized metric space defined in the proof of Theorem 2.2.

It follows from (3.4) that

for all . Thus . So

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 3.4Letr, θbe positive real numbers with. Letbe an even mapping satisfying (2.10). Then there exists a unique quadratic mappingsuch that

for all.

Proof The proof follows from Theorem 3.3 by taking for all . Then we can choose and we get the desired result. □

Theorem 3.5Letbe a function such that there exists anwith

for all. Letbe an even mapping satisfyingand (2.2). Then there exists a unique quartic mappingsuch that

for all.

Proof Let be the generalized metric space defined in the proof of Theorem 2.2.

Replacing by and letting in (3.3), we get

for all . So

(3.5)

for all . Thus . So

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 3.6Letr, θbe positive real numbers with. Letbe an even mapping satisfying (2.10). Then there exists a unique quartic mappingsuch that

for all.

Proof The proof follows from Theorem 3.5 by taking for all . Then we can choose and we get the desired result. □

Theorem 3.7Letbe a function such that there exists anwith

for all. Letbe an even mapping satisfyingand (2.2). Then there exists a unique quartic mappingsuch that

for all.

Proof Let be the generalized metric space defined in the proof of Theorem 2.2.

It follows from (3.5) that

for all . Thus . So

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 3.8Letr, θbe positive real numbers with. Letbe an even mapping satisfying (2.10). Then there exists a unique quartic mappingsuch that

for all.

Proof The proof follows from Theorem 3.7 by taking for all . Then we can choose and we get the desired result. □

### 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.

### Acknowledgements

CP was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299), and DYS was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).

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