- MEshaghi Gordji
^{1, 2}, - Z Alizadeh
^{1, 2}, - YJ Cho
^{3}Email author and - H Khodaei
^{1, 2}

**2011**:454093

https://doi.org/10.1155/2011/454093

© M. Eshaghi Gordji et al. 2011

**Received: **21 November 2010

**Accepted: **6 March 2011

**Published: **14 March 2011

## Abstract

## 1. Introduction

Following the terminology of [1], a nonempty set
with a ternary operation
is called a *ternary groupoid*, which is denoted by
. The ternary groupoid
is said to be *commutative* if
for all
and all permutations
of
. If a binary operation
is defined on
such that
for all
, then we say that
is derived from
. We say that
is a *ternary semigroup* if the operation
is associative, that is, if
holds for all
(see [2]). Since it is extensively discussed in [3], the full description of a physical system
implies the knowledge of three basis ingredients: the set of the observables, the set of the states, and the dynamics that describes the time evolution of the system by means of the time dependence of the expectation value of a given observable on a given statue. Originally, the set of the observable was considered to be a
-algebra [4]. In many applications, however, it was shown not to be the most convenient choice and the
-algebra was replaced by a von Neumann algebra because the role of the representation turns out to be crucial mainly when long-range interactions are involved (see [5] and references therein). Here we used a different algebraic structure.

A
-*ternary ring* is a complex Banach space
, equipped with a ternary product
of
into
, which is
-linear in the outer variables, conjugate
-linear in the middle variable and associative in the sense that
and satisfies
and
.

If a -ternary ring has an identity, that is, an element such that for all , then it is routine to verify that , endowed with and , is a unital -algebra. Conversely, if is a unital -algebra, then makes into a -ternary algebra.

Consider the functional equation
in a certain general setting. A function
is an approximate solution of
if
and
are close in some sense. The Ulam stability problem asks whether or not there exists a true solution of
near
. A functional equation is said to be *superstable* if every approximate solution of the equation is an exact solution of the functional equation. The problem of stability of functional equations originated from a question of Ulam [6] concerning the stability of group homomorphisms.

for all , then there exists a homomorphism with for all ?

If the answer is affirmative, we say that the equation of homomorphism
is *stable*. The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus the stability question of functional equations is that how do the solutions of the inequality differ from those of the given functional equation?

In 1941, Hyers [7] gave a first affirmative answer to the question of Ulam for Banach spaces.

for all and some . Then there exists a unique additive mapping such that for all .

for all , where and . This result was later extended to all and generalized by Gajda [11], Th. M. Rassias and Šemrl [12], and Isac and Th. M. Rassias [13].

In 2000, Lee and Jun [14] have improved the stability problem for approximately additive mappings. The problem when is not true. Counter examples for the corresponding assertion in the case were constructed by Gadja [11], Th. M. Rassias and Šemrl [12].

and by a general control function . In 1949 and 1951, Bourgin [19, 20] is the first mathematician dealing with stability of (ring) homomorphism . The topic of approximation of functional equations on Banach algebras was studied by a number of mathematicians (see [21–33]).

is related to a symmetric biadditive mapping [34, 35]. It is natural that this equation is called a *quadratic functional equation*. For more details about various results concerning such problems, the readers refer to [36–43].

*cubic functional equation*. In 2005, Lee et al. [45] considered the following functional equation

It is easy to see that the mapping
is a solution of the functional equation (1.7), which is called the *quartic functional equation*.

## 2. Preliminaries

*ternary quadratic mapping*if is a quadratic mapping satisfies

for all . Let be a -ternary ring derived from a unital commutative -algebra and let satisfy for all . It is easy to show that the mapping is a -ternary quadratic mapping.

and they have obtained the stability of the functional equations (2.2) and (2.3).

Obviously, the monomial is a solution of the functional equation (2.4) for each .

For , Bae and Park [47, 48] showed that the functional equation (2.4) is equivalent to the additive equation and quadratic equation, respectively.

If , the functional equation (2.4) is equivalent to the cubic equation [44]. Moreover, Lee et al. [45] solved the solution of the functional equation (2.4) for .

In this paper, using the idea of Park and Cui [46], we study the further generalized stability of -ternary additive, quadratic, cubic, and quartic mappings over -ternary algebra via fixed point method for the functional equation (2.4). Moreover, we establish the superstability of this functional equation by suitable control functions.

Definition 2.1.

Let and be two -ternary algebras.

(1)A mapping
is called a
*-ternary additive homomorphism*
briefly,
-ternary 1-homomorphism
if
is an additive mapping satisfying (2.1) for all
.

(2)A mapping
is called a
*-ternary quadratic mapping*
briefly,
-ternary 2-homomorphism
if
is a quadratic mapping satisfying (2.1) for all
.

(3)A mapping
is called a
*-ternary cubic mapping*
briefly,
-ternary 3-homomorphism
if
is a cubic mapping satisfying (2.1) for all
.

(4)A mapping
is called a
*-ternary quartic homomorphism*
briefly,
-ternary 4-homomorphism
if
is a quartic mapping satisfying (2.1) for all
.

Now, we state the following notion of fixed point theorem. For the proof, refer to [49] (see also Chapter 5 in [50] and [51, 52]). In 2003, Radu [53] proposed a new method for obtaining the existence of exact solutions and error estimations, based on the fixed point alternative (see also [54–57]).

Let
be a generalized metric space. We say that a mapping
satisfies a Lipschitz condition if there exists a constant
such that
for all
, where the number
is called the Lipschitz constant. If the Lipschitz constant
is less than 1, then the mapping
is called a *strictly contractive mapping*. Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity.

The following theorem was proved by Diaz and Margolis [49] and Radu [53].

Theorem 2.2.

or there exists a natural number such that

(2)the sequence is convergent to a fixed point of ;

## 3. Approximation of -Ternary -Homomorphisms between -Ternary Algebras

In this section, we investigate the generalized stability of -ternary -homomorphism between -ternary algebras for the functional equation (2.4).

From now on, let be a positive integer less than 5.

Theorem 3.1.

Proof.

for all . By (3.6), and so . Letting in (3.2), we get and so .

It is easy to show that is a generalized complete metric space [55].

for all . This completes the proof.

Corollary 3.2.

Proof.

for all . Then we can choose and so the desired conclusion follows.

Remark 3.3.

In the following, we formulate and prove a theorem in superstability of -ternary -homomorphism in -ternary rings for the functional equation (2.4).

Theorem 3.4.

for all , then is a -ternary -homomorphism.

Proof.

for all , and . Hence, letting in (3.34) and using (3.31), we have for all .

for all and . Thus, letting in (3.35) and using (3.30), we have for all . Therefore, is a -ternary -homomorphism. This completes the proof.

Corollary 3.5.

for all , then is a -ternary -homomorphism.

Remark 3.6.

for all , then is a -ternary -homomorphism.

In the rest of this section, assume that is a unital -ternary algebra with the unit and is a -ternary algebra with the unit .

Theorem 3.7.

Let , , be positive real numbers with , and . Suppose that is a mapping satisfying (3.19) and (3.20). If there exist a real number and such that , then the mapping is a -ternary -homomorphism.

Proof.

for all and . Therefore, by the assumption, we get that .

for all and so for all . Letting in the last equality, we get for all .

Similarly, one can show that for all when and . Therefore, the mapping is a -ternary -homomorphism. This completes the proof.

Theorem 3.8.

for all . If there exist a real number and such that , then the mapping is a -ternary -homomorphism.

Proof.

for all and . Therefore, by the assumption, we get that .

for all and so for all . Letting in the last equality, we get for all .

Similarly, one can show that for all when and . Therefore, the mapping is a -ternary -homomorphism. This completes the proof.

## Declarations

### Acknowledgment

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).

## Authors’ Affiliations

## References

- Duplij S:
**Ternary Hopf algebras.**In*Symmetry in Nonlinear Mathematical Physics, Natsīonal'noï Akademīï Nauk Ukraïni Mat. Zastos., 43, Part 1, 2; Natsīonal'noï Akademīï Nauk Ukraïni, Institute of Mathematics, Kyiv*.*Volume 2*. Natsīonal'noï Akademīï Nauk Ukraïni, Kiev, Ukraine; 2002:439–448.Google Scholar - Bazunova N, Borowiec A, Kerner R:
**Universal differential calculus on ternary algebras.***Letters in Mathematical Physics*2004,**67**(3):195–206.MATHMathSciNetView ArticleGoogle Scholar - Sewell GL:
*Quantum Mechanics and Its Emergent Macrophysics*. Princeton University Press, Princeton, NJ, USA; 2002:xii+292.MATHGoogle Scholar - Haag R, Kastler D:
**An algebraic approach to quantum field theory.***Journal of Mathematical Physics*1964,**5:**848–861. 10.1063/1.1704187MATHMathSciNetView ArticleGoogle Scholar - Bagarello F, Morchio G:
**Dynamics of mean-field spin models from basic results in abstract differential equations.***Journal of Statistical Physics*1992,**66**(3–4):849–866. 10.1007/BF01055705MATHMathSciNetView ArticleGoogle Scholar - Ulam SM:
*Problems in Modern Mathematics*. John Wiley & Sons, New York, NY, USA; 1964:xvii+150.MATHGoogle Scholar - Hyers DH:
**On the stability of the linear functional equation.***Proceedings of the National Academy of Sciences of the United States of America*1941,**27:**222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar - Aoki T:
**On the stability of the linear transformation in Banach spaces.***Journal of the Mathematical Society of Japan*1950,**2:**64–66. 10.2969/jmsj/00210064MATHMathSciNetView ArticleGoogle Scholar - Bourgin DG:
**Classes of transformations and bordering transformations.***Bulletin of the American Mathematical Society*1951,**57:**223–237. 10.1090/S0002-9904-1951-09511-7MATHMathSciNetView ArticleGoogle Scholar - Rassias ThM:
**On the stability of the linear mapping in Banach spaces.***Proceedings of the American Mathematical Society*1978,**72**(2):297–300. 10.1090/S0002-9939-1978-0507327-1MATHMathSciNetView ArticleGoogle Scholar - Gajda Z:
**On stability of additive mappings.***International Journal of Mathematics and Mathematical Sciences*1991,**14**(3):431–434. 10.1155/S016117129100056XMATHMathSciNetView ArticleGoogle Scholar - Rassias ThM, Šemrl P:
**On the behavior of mappings which do not satisfy Hyers-Ulam stability.***Proceedings of the American Mathematical Society*1992,**114**(4):989–993. 10.1090/S0002-9939-1992-1059634-1MATHMathSciNetView ArticleGoogle Scholar - Isac G, Rassias ThM:
**On the Hyers-Ulam stability of -additive mappings.***Journal of Approximation Theory*1993,**72**(2):131–137. 10.1006/jath.1993.1010MATHMathSciNetView ArticleGoogle Scholar - Lee Y-H, Jun K-W:
**On the stability of approximately additive mappings.***Proceedings of the American Mathematical Society*2000,**128**(5):1361–1369. 10.1090/S0002-9939-99-05156-4MATHMathSciNetView ArticleGoogle Scholar - Rassias JM:
**On approximation of approximately linear mappings by linear mappings.***Journal of Functional Analysis*1982,**46**(1):126–130. 10.1016/0022-1236(82)90048-9MATHMathSciNetView ArticleGoogle Scholar - Rassias JM:
**On approximation of approximately linear mappings by linear mappings.***Bulletin des Sciences Mathématiques*1984,**108**(4):445–446.MATHMathSciNetGoogle Scholar - Rassias JM:
**Solution of a problem of Ulam.***Journal of Approximation Theory*1989,**57**(3):268–273. 10.1016/0021-9045(89)90041-5MATHMathSciNetView ArticleGoogle Scholar - Găvruţa P:
**A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings.***Journal of Mathematical Analysis and Applications*1994,**184**(3):431–436. 10.1006/jmaa.1994.1211MATHMathSciNetView ArticleGoogle Scholar - Bourgin DG:
**Classes of transformations and bordering transformations.***Bulletin of the American Mathematical Society*1951,**57:**223–237. 10.1090/S0002-9904-1951-09511-7MATHMathSciNetView ArticleGoogle Scholar - Bourgin DG:
**Approximately isometric and multiplicative transformations on continuous function rings.***Duke Mathematical Journal*1949,**16:**385–397. 10.1215/S0012-7094-49-01639-7MATHMathSciNetView ArticleGoogle Scholar - Badora R:
**On approximate ring homomorphisms.***Journal of Mathematical Analysis and Applications*2002,**276**(2):589–597. 10.1016/S0022-247X(02)00293-7MATHMathSciNetView ArticleGoogle Scholar - Baker J, Lawrence J, Zorzitto F:
**The stability of the equation .***Proceedings of the American Mathematical Society*1979,**74:**242–246.MATHMathSciNetGoogle Scholar - Savadkouhi MB, Gordji ME, Rassias JM, Ghobadipour N: Approximate ternary Jordan derivations on Banach ternary algebras. Journal of Mathematical Physics 2009,50(4, article 042303):-9.Google Scholar
- Eshaghi Gordji M, Najati A:
**Approximately -homomorphisms: a fixed point approach.***Journal of Geometry and Physics*2010,**60**(5):809–814. 10.1016/j.geomphys.2010.01.012MATHMathSciNetView ArticleGoogle Scholar - Park C-G:
**Homomorphisms between Lie -algebras and Cauchy-Rassias stability of Lie -algebra derivations.***Journal of Lie Theory*2005,**15**(2):393–414.MATHMathSciNetGoogle Scholar - Eshaghi Gordji M, Ghaemi MB, Kaboli Gharetapeh S, Shams S, Ebadian A:
**On the stability of -derivations.***Journal of Geometry and Physics*2010,**60**(3):454–459. 10.1016/j.geomphys.2009.11.004MATHMathSciNetView ArticleGoogle Scholar - Eshaghi Gordji M, Kaboli Gharetapeh S, Rashidi E, Karimi T, Aghaei M:
**Ternary Jordan -derivations in -ternary algebras.***Journal of Computational Analysis and Applications*2010,**12**(2):463–470.MATHMathSciNetGoogle Scholar - Park C, Gordji ME:
**Comment on "Approximate ternary Jordan derivations on Banach ternary algebras" [Bavand Savadkouhi et al., Journal of Mathematical Physics, vol. 50, article 042303, 2009].***Journal of Mathematical Physics*2010,**51**(4, article 044102):7.MathSciNetView ArticleMATHGoogle Scholar - Park C, Najati A:
**Generalized additive functional inequalities in Banach algebras.***International Journal of Nonlinear Analysis and Applications*2010,**1:**54–62.MATHGoogle Scholar - Park C, Rassias ThM:
**Isomorphisms in unital -algebras.***International Journal of Nonlinear Analysis and Applications*2010,**1:**1–10.MATHMathSciNetGoogle Scholar - Eshaghi Gordji M, Karimi T, Kaboli Gharetapeh S:
**Approximately -Jordan homomorphisms on Banach algebras.***Journal of Inequalities and Applications*2009,**2009:**-8.MathSciNetView ArticleMATHGoogle Scholar - Eshaghi Gordji M, Rassias JM, Ghobadipour N:
**Generalized Hyers-Ulam stability of generalized -derivations.***Abstract and Applied Analysis*2009,**2009:**-8.MathSciNetMATHGoogle Scholar - Eshaghi Gordji M, Savadkouhi MB:
**Approximation of generalized homomorphisms in quasi-Banach algebras.***Analele Stiintifice ale Universitatii Ovidius Constanta*2009,**17**(2):203–213.MATHMathSciNetGoogle Scholar - Aczél J, Dhombres J:
*Functional Equations in Several Variables*.*Volume 31*. Cambridge University Press, Cambridge, UK; 1989:xiv+462.MATHView ArticleGoogle Scholar - Kannappan P:
**Quadratic functional equation and inner product spaces.***Results in Mathematics. Resultate der Mathematik*1995,**27**(3–4):368–372.MATHMathSciNetView ArticleGoogle Scholar - Czerwik S:
**On the stability of the quadratic mapping in normed spaces.***Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg*1992,**62:**59–64. 10.1007/BF02941618MATHMathSciNetView ArticleGoogle Scholar - Eshaghi Gordji M, Khodaei H:
**Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces.***Nonlinear Analysis. Theory, Methods & Applications*2009,**71**(11):5629–5643. 10.1016/j.na.2009.04.052MATHMathSciNetView ArticleGoogle Scholar - Eshaghi Gordji M, Khodaei H:
**On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations.***Abstract and Applied Analysis*2009,**2009:**-11.MathSciNetMATHGoogle Scholar - Forti GL:
**An existence and stability theorem for a class of functional equations.***Stochastica*1980,**4**(1):23–30. 10.1080/17442508008833155MATHMathSciNetView ArticleGoogle Scholar - Forti G-L:
**Elementary remarks on Ulam-Hyers stability of linear functional equations.***Journal of Mathematical Analysis and Applications*2007,**328**(1):109–118. 10.1016/j.jmaa.2006.04.079MATHMathSciNetView ArticleGoogle Scholar - Jung S-M:
**Hyers-Ulam-Rassias stability of Jensen's equation and its application.***Proceedings of the American Mathematical Society*1998,**126**(11):3137–3143. 10.1090/S0002-9939-98-04680-2MATHMathSciNetView ArticleGoogle Scholar - Jung S-M:
*Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis*. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar - Khodaei H, Rassias ThM:
**Approximately generalized additive functions in several variables.***International Journal of Nonlinear Analysis*2010,**1:**22–41.MATHGoogle Scholar - Jun K-W, Kim H-M:
**The generalized Hyers-Ulam-Rassias stability of a cubic functional equation.***Journal of Mathematical Analysis and Applications*2002,**274**(2):867–878. 10.1016/S0022-247X(02)00415-8MATHMathSciNetView ArticleGoogle Scholar - Lee SH, Im SM, Hwang IS:
**Quartic functional equations.***Journal of Mathematical Analysis and Applications*2005,**307**(2):387–394. 10.1016/j.jmaa.2004.12.062MATHMathSciNetView ArticleGoogle Scholar - Park C, Cui J:
**Generalized stability of -ternary quadratic mappings.***Abstract and Applied Analysis*2007,**2007:**-6.MathSciNetMATHGoogle Scholar - Bae J-H, Park W-G:
**A functional equation having monomials as solutions.***Applied Mathematics and Computation*2010,**216**(1):87–94. 10.1016/j.amc.2010.01.006MATHMathSciNetView ArticleGoogle Scholar - Bae J-H, Park W-G:
**On a cubic equation and a Jensen-quadratic equation.***Abstract and Applied Analysis*2007,**2007:**-10.MathSciNetView ArticleMATHGoogle Scholar - Diaz JB, Margolis B:
**A fixed point theorem of the alternative, for contractions on a generalized complete metric space.***Bulletin of the American Mathematical Society*1968,**74:**305–309. 10.1090/S0002-9904-1968-11933-0MATHMathSciNetView ArticleGoogle Scholar - Rus IA:
*Principles and Applications of Fixed Point Theory*. , Cluj-Napoca; 1979.Google Scholar - Hyers DH, Isac G, Rassias ThM:
*Stability of Functional Equations in Several Variables*. Birkhäuser Boston Inc., Boston, Mass, USA; 1998:vi+313.MATHView ArticleGoogle Scholar - Zeidler E:
**Nonlinear functional analysis and its applications.**In*Fixed-Point Theorems. 2*.*Volume 1*. Springer; 1993:xxiii +909.Google Scholar - Radu V:
**The fixed point alternative and the stability of functional equations.***Fixed Point Theory*2003,**4**(1):91–96.MATHMathSciNetGoogle Scholar - Cădariu L, Radu V:
**Fixed points and the stability of Jensen's functional equation.***Journal of Inequalities in Pure and Applied Mathematics*2003,**4**(1, article 4):7.MATHGoogle Scholar - Cădariu L, Radu V:
**On the stability of the Cauchy functional equation: a fixed point approach.**In*Iteration Theory, Grazer Mathematische Berichte*.*Volume 346*. Karl-Franzens-Universitaet Graz, Graz, Austria; 2004:43–52.Google Scholar - Ebadian A, Ghobadipour N, Gordji ME:
**A fixed point method for perturbation of bimultipliers and Jordan bimultipliers in -ternary algebras.***Journal of Mathematical Physics*2010,**51:**-10.View ArticleMathSciNetMATHGoogle Scholar - Eshaghi Gordji M, Khodaei H:
**The fixed point method for fuzzy approximation of a functional equation associated with inner product spaces.***Discrete Dynamics in Nature and Society*2010,**2010:**-15.MathSciNetMATHGoogle Scholar

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