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# Generalized Hyers-Ulam Stability of the Pexiderized Cauchy Functional Equation in Non-Archimedean Spaces

*Fixed Point Theory and Applications*
**volume 2011**, Article number: 309026 (2011)

## Abstract

We prove the generalized Hyers-Ulam stability of the Pexiderized Cauchy functional equation in non-Archimedean spaces.

## 1. Introduction

The stability problem of functional equations was originated from a question of Ulam [1] concerning the stability of group homomorphisms.

Let be a group and let be a metric group with the metric . Given , does there exist a such that, if a function satisfies the inequality for all , then there exists a homomorphism with for all

In other words, we are looking for situations when the homomorphisms are stable, that is, if a mapping is almost a homomorphism, then there exists a true homomorphism near it. If we turn our attention to the case of functional equations, we can ask the following question.

When the solutions of an equation differing slightly from a given one must be close to the true solution of the given equation.

For Banach spaces, the Ulam problem was first solved by Hyers [2] in 1941, which states that, if and is a mapping, where are Banach spaces, such that

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

for all . Rassias [3] succeeded in extending the result of Hyers by weakening the condition for the Cauchy difference to be unbounded. A number of mathematicians were attracted to this result of Rassias and stimulated to investigate the stability problems of functional equations. The stability phenomenon that was introduced and proved by Rassias is called the *generalized Hyers-Ulam stability.* Forti [4] and Găvruţa [5] have generalized the result of Rassias, which permitted the Cauchy difference to become arbitrary unbounded. 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. A large list of references can be found, for example, in [3, 6–30].

Definition 1.1.

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) for all .

Clearly, and for all .

Definition 1.2.

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

if and only if ;

;

the strong triangle inequality, namely,

for all and .

The pair is called a *non-Archimedean space* if is non-Archimedean norm on .

It follows from that

for all , where with . Therefore, a sequence is a Cauchy sequence in non-Archimedean space if and only if the sequence converges to zero in . In a complete non-Archimedean space, every Cauchy sequence is convergent.

In 1897, Hensel [31] discovered the -adic number as a number theoretical analogue of power series in complex analysis. Fix a prime number . For any nonzero rational number , there exists a unique integer such that , where and are integers not divisible by . Then defines a non-Archimedean norm on . The completion of with respect to metric , which is denoted by , is called *-adic number field*. In fact, is the set of all formal series , where are integers. The addition and multiplication between any two elements of are defined naturally. The norm is a non-Archimedean norm on , and it makes a locally compact field (see [32, 33]).

In [34], Arriola and Beyer showed that, if is a continuous mapping for which there exists a fixed such that for all , then there exists a unique additive mapping such that for all . The stability problem of the Cauchy functional equation and quadratic functional equation has been investigated by Moslehian and Rassias [19] in non-Archimedean spaces.

According to Theorem 6 in [16], a mapping satisfying is a solution of the Jensen functional equation

for all if and only if it satisfies the additive Cauchy functional equation .

In this paper, by using the idea of Găvruţa [5], we prove the stability of the Jensen functional equation and the Pexiderized Cauchy functional equation:

## 2. Generalized Hyers-Ulam Stability of the Jensen Functional Equation

Throughout this section, let be a normed space with norm and a complete non-Archimedean space with norm .

Theorem 2.1.

Let be a function such that

for all and the limit

for all , which is denoted by , exist. Suppose that a mapping with satisfies the inequality

for all . Then the limit

exists for all and is an additive mapping satisfying

for all . Moreover, if

for all , then is a unique additive mapping satisfying (2.5).

Proof.

Letting in (2.3), we get

for all . If we replace in (2.7) by and multiply both sides of (2.7) to , then we have

for all and all nonnegative integers . It follows from (2.1) and (2.8) that the sequence is a Cauchy sequence in for all . Since is complete, the sequence converges for all . Hence one can define the mapping by (2.4).

By induction on , one can conclude that

for all and . By passing the limit in (2.9) and using (2.2), we obtain (2.5).

Now, we show that is additive. It follows from (2.1), (2.3), and (2.4) that

for all . Therefore, the mapping is additive.

To prove the uniqueness of , let be another additive mapping satisfying (2.5). Since

for all , it follows from (2.6) that

for all . So . This completes the proof.

The following theorem is an alternative result of Theorem 2.1, and its proof is similar to the proof of Theorem 2.1.

Theorem 2.2.

Let be a function such that

for all and the limit

for all , denoted by , exist. Suppose that a mapping with satisfies the inequality

for all . Then the limit

exists for all , and is an additive mapping satisfying

for all . Moreover, if

for all , then is a unique additive mapping satisfying (2.17).

## 3. Generalized Hyers-Ulam Stability of the Pexiderized Cauchy Functional Equation

Throughout this section, let be a normed space with norm and a complete non-Archimedean space with norm .

Theorem 3.1.

Let be a function such that

for all and the limits

exist for all . Suppose that mappings with satisfy the inequality

for all . Then the limits

exist for all and is an additive mapping satisfying

for all . Moreover, if

for all , then is a unique additive mapping satisfying (3.7), (3.8), and (3.9).

Proof.

It follows from (3.5) that

for all . Let

for all . It follows from (3.1) and (3.2) that

for all . By Theorem 2.1, there exists an additive mapping satisfying (3.7) and

for all . From (3.5), we get

for all . Let

for all . By (3.1) and (3.3), we have

for all . By Theorem 2.1, there exists an additive mapping satisfying (3.8) and

for all . Similarly, (3.5) implies that

for all . Let

for all . By (3.1) and (3.4), we have

for all . By Theorem 2.1, there exists an additive mapping satisfying (3.9) and

for all . The uniqueness of , and follows from (3.10).

Now, we show that . Replacing and by and 0 in (3.5), respectively, and dividing both sides of (3.5) by , we get

for all . By passing the limit in (3.23), we conclude that

for all . Similarly, we get for all . Therefore, (3.6) follows from (3.14), (3.18), and (3.22). This completes the proof.

The next theorem is an alternative result of Theorem 3.1.

Theorem 3.2.

Let be a function such that

for all and the limits

exist for all . Suppose that mappings with satisfy the inequality

for all . Then the limits

exist for all and is an additive mapping satisfying

for all . Moreover, if

for all , then is a unique additive mapping satisfying the above inequalities.

## References

- [1]
Ulam SM:

*A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8*. Interscience, New York, NY, USA; 1960:xiii+150. - [2]
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.27.4.222 - [3]
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/S0002-9939-1978-0507327-1 - [4]
Forti GL:

**An existence and stability theorem for a class of functional equations.***Stochastica*1980,**4**(1):23–30. 10.1080/17442508008833155 - [5]
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.1994.1211 - [6]
Jun K-W, Bae J-H, Lee Y-H:

**On the Hyers-Ulam-Rassias stability of an -dimensional Pexiderized quadratic equation.***Mathematical Inequalities & Applications*2004,**7**(1):63–77. - [7]
Faĭziev VA, Rassias ThM, Sahoo PK:

**The space of -additive mappings on semigroups.***Transactions of the American Mathematical Society*2002,**354**(11):4455–4472. 10.1090/S0002-9947-02-03036-2 - [8]
Forti GL:

**Hyers-Ulam stability of functional equations in several variables.***Aequationes Mathematicae*1995,**50**(1–2):143–190. 10.1007/BF01831117 - [9]
Forti G-L:

**Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations.***Journal of Mathematical Analysis and Applications*2004,**295**(1):127–133. 10.1016/j.jmaa.2004.03.011 - [10]
Haruki H, Rassias ThM:

**A new functional equation of Pexider type related to the complex exponential function.***Transactions of the American Mathematical Society*1995,**347**(8):3111–3119. 10.2307/2154775 - [11]
Hyers DH, Isac G, Rassias ThM:

*Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications, 34*. Birkhäuser, Boston, Mass, USA; 1998:vi+313. - [12]
Hyers DH, Isac G, Rassias ThM:

**On the asymptoticity aspect of Hyers-Ulam stability of mappings.***Proceedings of the American Mathematical Society*1998,**126**(2):425–430. 10.1090/S0002-9939-98-04060-X - [13]
Jun K-W, Kim H-M:

**On the Hyers-Ulam stability of a generalized quadratic and additive functional equation.***Bulletin of the Korean Mathematical Society*2005,**42**(1):133–148. - [14]
Jun K-W, Kim H-M:

**Ulam stability problem for generalized -quadratic mappings.***Journal of Mathematical Analysis and Applications*2005,**305**(2):466–476. 10.1016/j.jmaa.2004.10.058 - [15]
Jun K-W, Lee Y-H:

**On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality.***Mathematical Inequalities & Applications*2001,**4**(1):93–118. - [16]
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/S0002-9939-98-04680-2 - [17]
Khrennikov A:

*Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Mathematics and Its Applications*.*Volume 427*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:xviii+371. - [18]
Lee Y-H, Jun K-W:

**A note on the Hyers-Ulam-Rassias stability of Pexider equation.***Journal of the Korean Mathematical Society*2000,**37**(1):111–124. - [19]
Moslehian MS, Rassias ThM:

**Stability of functional equations in non-Archimedean spaces.***Applicable Analysis and Discrete Mathematics*2007,**1**(2):325–334. 10.2298/AADM0702325M - [20]
Najati A:

**On the stability of a quartic functional equation.***Journal of Mathematical Analysis and Applications*2008,**340**(1):569–574. 10.1016/j.jmaa.2007.08.048 - [21]
Najati A, Moghimi MB:

**Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces.***Journal of Mathematical Analysis and Applications*2008,**337**(1):399–415. 10.1016/j.jmaa.2007.03.104 - [22]
Najati A, Park C:

**Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation.***Journal of Mathematical Analysis and Applications*2007,**335**(2):763–778. 10.1016/j.jmaa.2007.02.009 - [23]
Park C-G:

**On the stability of the quadratic mapping in Banach modules.***Journal of Mathematical Analysis and Applications*2002,**276**(1):135–144. 10.1016/S0022-247X(02)00387-6 - [24]
Park C-G:

**On the Hyers-Ulam-Rassias stability of generalized quadratic mappings in Banach modules.***Journal of Mathematical Analysis and Applications*2004,**291**(1):214–223. 10.1016/j.jmaa.2003.10.027 - [25]
Rassias ThM, Tabo J (Eds):

*Stability of Mappings of Hyers-Ulam Type, Hadronic Press Collection of Original Articles*. Hadronic Press, Palm Harbor, Fla, USA; 1994:vi+173. - [26]
Rassias ThM:

**On the stability of the quadratic functional equation and its applications.***Studia Mathematica. Universitatis Babeş-Bolyai*1998,**43**(3):89–124. - [27]
Rassias ThM:

**On the stability of functional equations and a problem of Ulam.***Acta Applicandae Mathematicae*2000,**62**(1):23–130. 10.1023/A:1006499223572 - [28]
Rassias ThM (Ed):

*Functional Equations and Inequalities, Mathematics and Its Applications*.*Volume 518*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:xii+336. - [29]
Rassias ThM:

**On the stability of functional equations in Banach spaces.***Journal of Mathematical Analysis and Applications*2000,**251**(1):264–284. 10.1006/jmaa.2000.7046 - [30]
Šemrl P:

**On quadratic functionals.***Bulletin of the Australian Mathematical Society*1988,**37**(1):27–28. 10.1017/S0004972700004111 - [31]
Hensel K:

**Über eine neue Begrunduring der Theorie der algebraischen Zahlen.***Jahresbericht der Deutschen Mathematiker-Vereinigung*1987,**6:**83–88. - [32]
Gouvêa FQ:

*p-Adic Numbers, Universitext*. 2nd edition. Springer, Berlin, Germany; 1997:vi+298. - [33]
Robert AM:

*A Course in p-Adic Analysis, Graduate Texts in Mathematics*.*Volume 198*. Springer, New York, NY, USA; 2000:xvi+437. - [34]
Arriola LM, Beyer WA:

**Stability of the Cauchy functional equation over -adic fields.***Real Analysis Exchange*2006,**31**(1):125–132.

## Acknowledgment

Y. J. Cho was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).

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Najati, A., Cho, Y. Generalized Hyers-Ulam Stability of the Pexiderized Cauchy Functional Equation in Non-Archimedean Spaces.
*Fixed Point Theory Appl* **2011, **309026 (2011). https://doi.org/10.1155/2011/309026

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### Keywords

- Functional Equation
- Stability Problem
- Additive Mapping
- Alternative Result
- Quadratic Functional Equation