Open Access

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

Fixed Point Theory and Applications20112011:309026

https://doi.org/10.1155/2011/309026

Received: 22 October 2010

Accepted: 8 March 2011

Published: 14 March 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
(1.1)
for all , then there exists a unique additive mapping such that
(1.2)

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, 630].

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,

(1.3)

for all and .

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

It follows from that
(1.4)

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
(1.5)

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:
(1.6)

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
(2.1)
for all and the limit
(2.2)
for all , which is denoted by , exist. Suppose that a mapping with satisfies the inequality
(2.3)
for all . Then the limit
(2.4)
exists for all and is an additive mapping satisfying
(2.5)
for all . Moreover, if
(2.6)

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

Proof.

Letting in (2.3), we get
(2.7)
for all . If we replace in (2.7) by and multiply both sides of (2.7) to , then we have
(2.8)

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
(2.9)

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
(2.10)

for all . Therefore, the mapping is additive.

To prove the uniqueness of , let be another additive mapping satisfying (2.5). Since
(2.11)
for all , it follows from (2.6) that
(2.12)

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
(2.13)
for all and the limit
(2.14)
for all , denoted by , exist. Suppose that a mapping with satisfies the inequality
(2.15)
for all . Then the limit
(2.16)
exists for all , and is an additive mapping satisfying
(2.17)
for all . Moreover, if
(2.18)

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
(3.1)
for all and the limits
(3.2)
(3.3)
(3.4)
exist for all . Suppose that mappings with satisfy the inequality
(3.5)
for all . Then the limits
(3.6)
exist for all and is an additive mapping satisfying
(3.7)
(3.8)
(3.9)
for all . Moreover, if
(3.10)

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

Proof.

It follows from (3.5) that
(3.11)
for all . Let
(3.12)
for all . It follows from (3.1) and (3.2) that
(3.13)
for all . By Theorem 2.1, there exists an additive mapping satisfying (3.7) and
(3.14)
for all . From (3.5), we get
(3.15)
for all . Let
(3.16)
for all . By (3.1) and (3.3), we have
(3.17)
for all . By Theorem 2.1, there exists an additive mapping satisfying (3.8) and
(3.18)
for all . Similarly, (3.5) implies that
(3.19)
for all . Let
(3.20)
for all . By (3.1) and (3.4), we have
(3.21)
for all . By Theorem 2.1, there exists an additive mapping satisfying (3.9) and
(3.22)

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
(3.23)
for all . By passing the limit in (3.23), we conclude that
(3.24)

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
(3.25)
for all and the limits
(3.26)
exist for all . Suppose that mappings with satisfy the inequality
(3.27)
for all . Then the limits
(3.28)
exist for all and is an additive mapping satisfying
(3.29)
for all . Moreover, if
(3.30)

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

Declarations

Acknowledgment

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

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili
(2)
Department of Mathematics Education and the RINS, Gyeongsang National University

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.Google Scholar
  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.27.4.222MathSciNetView ArticleMATHGoogle Scholar
  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/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar
  4. Forti GL: An existence and stability theorem for a class of functional equations. Stochastica 1980,4(1):23–30. 10.1080/17442508008833155MathSciNetView ArticleMATHGoogle Scholar
  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.1994.1211MathSciNetView ArticleMATHGoogle Scholar
  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.MathSciNetView ArticleMATHGoogle Scholar
  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/S0002-9947-02-03036-2MathSciNetView ArticleMATHGoogle Scholar
  8. Forti GL: Hyers-Ulam stability of functional equations in several variables. Aequationes Mathematicae 1995,50(1–2):143–190. 10.1007/BF01831117MathSciNetView ArticleMATHGoogle Scholar
  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.2004.03.011MathSciNetView ArticleMATHGoogle Scholar
  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. 10.2307/2154775MathSciNetView ArticleMATHGoogle Scholar
  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.View ArticleMATHGoogle Scholar
  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-XMathSciNetView ArticleMATHGoogle Scholar
  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.MathSciNetView ArticleMATHGoogle Scholar
  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.2004.10.058MathSciNetView ArticleMATHGoogle Scholar
  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.MathSciNetView ArticleMATHGoogle Scholar
  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/S0002-9939-98-04680-2MathSciNetView ArticleMATHGoogle Scholar
  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.MATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  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/AADM0702325MMathSciNetView ArticleMATHGoogle Scholar
  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.2007.08.048MathSciNetView ArticleMATHGoogle Scholar
  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.2007.03.104MathSciNetView ArticleMATHGoogle Scholar
  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.2007.02.009MathSciNetView ArticleMATHGoogle Scholar
  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)00387-6MathSciNetView ArticleMATHGoogle Scholar
  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.2003.10.027MathSciNetView ArticleMATHGoogle Scholar
  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.Google Scholar
  26. Rassias ThM: On the stability of the quadratic functional equation and its applications. Studia Mathematica. Universitatis Babeş-Bolyai 1998,43(3):89–124.MathSciNetMATHGoogle Scholar
  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:1006499223572MathSciNetView ArticleMATHGoogle Scholar
  28. Rassias ThM (Ed): Functional Equations and Inequalities, Mathematics and Its Applications. Volume 518. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:xii+336.Google Scholar
  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.2000.7046MathSciNetView ArticleMATHGoogle Scholar
  30. Šemrl P: On quadratic functionals. Bulletin of the Australian Mathematical Society 1988,37(1):27–28. 10.1017/S0004972700004111MathSciNetView ArticleMATHGoogle Scholar
  31. Hensel K: Über eine neue Begrunduring der Theorie der algebraischen Zahlen. Jahresbericht der Deutschen Mathematiker-Vereinigung 1987, 6: 83–88.MATHGoogle Scholar
  32. Gouvêa FQ: p-Adic Numbers, Universitext. 2nd edition. Springer, Berlin, Germany; 1997:vi+298.View ArticleMATHGoogle Scholar
  33. Robert AM: A Course in p-Adic Analysis, Graduate Texts in Mathematics. Volume 198. Springer, New York, NY, USA; 2000:xvi+437.View ArticleGoogle Scholar
  34. Arriola LM, Beyer WA: Stability of the Cauchy functional equation over -adic fields. Real Analysis Exchange 2006,31(1):125–132.MathSciNetMATHGoogle Scholar

Copyright

© Abbas Najati and Yeol Je Cho. 2011

This article is published under license to BioMed Central Ltd. 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.