Skip to content

Advertisement

  • Research Article
  • Open Access

Hyers-Ulam Stability of a Bi-Jensen Functional Equation on a Punctured Domain

Journal of Inequalities and Applications20102010:476249

https://doi.org/10.1155/2010/476249

  • Received: 16 November 2009
  • Accepted: 15 February 2010
  • Published:

Abstract

We obtain the Hyers-Ulam stability of a bi-Jensen functional equation: and simultaneously . And we get its stability on the punctured domain.

Keywords

  • Banach Space
  • Positive Integer
  • Functional Equation
  • Additive Mapping
  • Cauchy Sequence

1. Introduction

In 1940, Ulam [1] raised a question concerning the stability of homomorphisms: let be a group and let be a metric group with the metric . Given , does there exist a such that if a mapping satisfies the inequality

(1.1)

for all , then there is a homomorphism with

(1.2)

for all ? The case of approximately additive mappings was solved by Hyers [2] under the assumption that and are Banach spaces. In 1949, 1950, and 1978, Bourgin [3], Aoki [4], and Rassias [5] gave a generalization of it under the conditions bounded by variables. Since then, the further generalization has been extensively investigated by a number of mathematicians, such as Găvruta, Rassias, and so forth, [625].

Throughout this paper, let be a normed space and a Banach space. A mapping is called a Jensen mapping if satisfies the functional equation For a given mapping , we define

(1.3)

for all . A mapping is called a bi-Jensen mapping if satisfies the functional equations and .

In 2006, Bae and Park [26] obtained the generalized Hyers-Ulam stability of a bi-Jensen mapping. The following result is a special case of Theorem 6 in [26].

Theorem 1 A.

Let and let be a mapping such that
(1.4)
for all . Then there exist two bi-Jensen mappings such that
(1.5)

for all .

In Theorem A, they did not show that there exist a and a unique bi-Jensen mapping such that for all . In 2008, Jun et al. [7, 8] improved Bae and Park's results.

In Section 2, we show that there exists a unique bi-Jensen mapping such that for all . In Section 3, we investigate the Hyers-Ulam stability of a bi-Jensen functional equation on the punctured domain.

2. Stability of a Bi-Jensen Functional Equation

From Lemma 1 in [8], we get the following lemma.

Lemma.

Let be a bi-Jensen mapping. Then
(2.1)

for all and .

Now we will give the Hyers-Ulam stability for a bi-Jensen mapping.

Theorem 2.2.

Let and let be a mapping satisfying (1.4) for all . Then there exists a unique bi-Jensen mapping such that
(2.2)
for all with . In particular, the mapping is given by
(2.3)

for all .

Proof.

Let be the map defined by
(2.4)
for all and . By (1.4), we get
(2.5)
for all and . For given integers with , we obtain
(2.6)
for all . By the above inequality, the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Define by
(2.7)
for all . Putting and taking in (2.6), we obtain the inequality
(2.8)
for all . By (1.4) and the definition of , we get
(2.9)
for all . So is a bi-Jensen mapping satisfying (2.2). Now, let be another bi-Jensen mapping satisfying (2.2) with . By Lemma 2.1, we have
(2.10)

for all and . As , we may conclude that for all . Thus the bi-Jensen mapping is unique.

Example 2.3.

Let be the bi-Jensen mappings defined by
(2.11)

for all . Then satisfy (1.4) for all . In addition, satisfy (2.2) for all and also satisfy (2.2) for all . But we get . Hence the condition is necessary to show that the mapping is unique.

3. Stability of a Bi-Jensen Functional Equation on the Punctured Domain

Let be a subset of . and are punctured domain on the spaces and , respectively.

Throughout this paper, for a given mapping , let be the mappings defined by

(3.1)

for all .

Lemma.

Let be a subset of satisfying the following condition: for every , there exists a positive integer such that for all integer with , and such that for all integer with . Let be a mapping such that
(3.2)
for all . Then there exists a unique bi-Jensen mapping such that
(3.3)
for all . Moreover, the equality
(3.4)

holds for all .

Proof.

Note that , , , and for all . Let for any . From (3.2), we get the equality
(3.5)
for all , and we know that the equality
(3.6)
holds for all . From (3.2), we have
(3.7)
for all . From the above equalities, we obtain the equalities
(3.8)
(3.9)
(3.10)
(3.11)

for all and .

Let be the set defined by for each . From the above equalities, we can define by

(3.12)

From the definition of , we get the equalities

(3.13)
for all . By (3.10), we get the equality
(3.14)
for all and , where . And also we get the equality
(3.15)
for all and , where . Hence the equality
(3.16)
holds for all . From (3.8), (3.9), (3.10), and the definition of , we easily get
(3.17)
for all . And we obtain
(3.18)
for all with , where and . From this, we have
(3.19)
for all with , where . From the above equalities, we get
(3.20)
for all . By the similar method, we have
(3.21)
for all . Hence is a bi-Jensen mapping. Let be another bi-Jensen mapping satisfying
(3.22)
for all . Using the above equality, we show that the equalities
(3.23)

hold for all and as we desired, where .

Corollary 3.2.

Let be a mapping such that
(3.24)
for all . Then there exists a unique bi-Jensen mapping such that
(3.25)

for all .

Example 3.3.

Let be the mapping defined by
(3.26)

and let be the mapping defined by for all . Then the mappings satisfy the conditions of Corollary 3.2 with .

Now, we prove the Hyers-Ulam stability of a bi-Jensen functional equation on the punctured domain .

Theorem 3.4.

Let and . Let be a mapping such that
(3.27)
for all . Then there exists a unique bi-Jensen mapping such that
(3.28)
holds for all with . The mapping is given by
(3.29)

for all .

Proof.

By (3.27), we get
(3.30)
for all and . For given integers ( ), we have
(3.31)
(3.32)
(3.33)
(3.34)
(3.35)
for all . The sequences , , and are Cauchy sequences for all . Since is complete, the above sequences converge for all . From (3.34) and (3.35), we have
(3.36)
for all . Using the inequalities (3.31)–(3.35) and the above equality, we can define the mappings by
(3.37)
for all . By (3.27) and the definition of , we obtain
(3.38)
for all . Since and
(3.39)
for all with , where , we have
(3.40)
for all . Similarly, the equalities
(3.41)
hold for all . By Lemma 3.1, There exist bi-Jensen mappings such that
(3.42)
for all . Since the equalities
(3.43)
hold, are bi-Jensen mappings. Putting and taking in (3.31), (3.32), and (3.33), one can obtain the inequalities
(3.44)
for all . By (3.30) and the above equalities, we get
(3.45)
for all , where is given by
(3.46)
and . By (3.45), we get the inequalities
(3.47)
for all and , where , and the inequalities
(3.48)

for all and . Hence is a bi-Jensen mapping satisfying (3.28).

Now, let be another bi-Jensen mapping satisfying (3.28) with . By Lemma 2.1, we have

(3.49)

for all and . As , we may conclude that for all . By Lemma 3.1, as we desired.

Example 3.5.

Let be the mapping defined by
(3.50)

Let be the mapping defined by for all . Then satisfies the conditions in Theorem 3.4, and is a bi-Jensen mapping satisfying (3.28) but .

Corollary 3.6.

Let be a mapping satisfying (3.13) and (3.27) for all . Then there exists a bi-Jensen mapping such that
(3.51)

for all .

Proof.

Let be as in the proof of Theorem 3.4. By (3.30), we obtain
(3.52)
for . From the above inequalities and (3.45), we get the inequality
(3.53)

for all .

Authors’ Affiliations

(1)
Department of Mathematics, Kangnam University, Yongin, Gyeonggi, 446-702, South Korea
(2)
Department of Mathematics Education, Gongju National University of Education, Gongju, 314-711, South Korea

References

  1. Ulam SM: A Collection of Mathematical Problems. Interscience, New York, NY, USA; 1968.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 ArticleGoogle Scholar
  3. 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
  4. 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
  5. 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
  6. 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
  7. Jun K-W, Han M-H, Lee Y-H: On the Hyers-Ulam-Rassias stability of the bi-Jensen functional equation. Kyungpook Mathematical Journal 2008, 48(4):705–720.MATHMathSciNetView ArticleGoogle Scholar
  8. Jun K-W, Lee Y-H, Oh J-H: On the rassias stability of a bi-Jensen functional equation. Journal of Mathematical Inequalities 2008, 2(3):363–375.MATHMathSciNetView ArticleGoogle Scholar
  9. Jung S-M: Hyers-Ulam stability of linear differential equations of first order. Applied Mathematics Letters 2004, 17(10):1135–1140. 10.1016/j.aml.2003.11.004MATHMathSciNetView ArticleGoogle Scholar
  10. Jung S-M, Moslehian MS, Sahoo PK: Stability of generalized Jensen equation on restricted domains. to appear in Journal of Mathematical InequalitiesGoogle Scholar
  11. Kim GH: On the superstability of the pexider type trigonometric functional equation. Journal of Inequalities and Applications 2010, 2010:-14.Google Scholar
  12. Kim GH: On the superstability related with the trigonometric functional equation. Advances in Difference Equations 2009, 2009:-11.Google Scholar
  13. Kim GH: On the stability of trigonometric functional equations. Advances in Difference Equations 2007, 2007:-10.Google Scholar
  14. Kim GH: The stability of d'Alembert and Jensen type functional equations. Journal of Mathematical Analysis and Applications 2007, 325(1):237–248. 10.1016/j.jmaa.2006.01.062MATHMathSciNetView ArticleGoogle Scholar
  15. Kim GH, Dragomir SS: On the stability of generalized d'Alembert and Jensen functional equations. International Journal of Mathematics and Mathematical Sciences 2006, 2006:-12.Google Scholar
  16. Kim H-M: On the stability problem for a mixed type of quartic and quadratic functional equation. Journal of Mathematical Analysis and Applications 2006, 324(1):358–372. 10.1016/j.jmaa.2005.11.053MATHMathSciNetView ArticleGoogle Scholar
  17. 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
  18. Lee Y-S, Chung S-Y: Stability of a Jensen type functional equation. Banach Journal of Mathematical Analysis 2007, 1(1):91–100.MATHMathSciNetView ArticleGoogle Scholar
  19. Moslehian MS: The Jensen functional equation in non-Archimedean normed spaces. Journal of Function Spaces and Applications 2009, 7(1):13–24.MATHMathSciNetView ArticleGoogle Scholar
  20. Park C-G: Linear functional equations in banach modules over a -algebra. Acta Applicandae Mathematicae 2003, 77(2):125–161. 10.1023/A:1024014026789MATHMathSciNetView ArticleGoogle Scholar
  21. Park W-G, Bae J-H: On a Cauchy-Jensen functional equation and its stability. Journal of Mathematical Analysis and Applications 2006, 323(1):634–643. 10.1016/j.jmaa.2005.09.028MATHMathSciNetView ArticleGoogle Scholar
  22. 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
  23. Rassias JM: On a new approximation of approximately linear mappings by linear mappings. Discussiones Mathematicae 1985, 7: 193–196.MATHMathSciNetGoogle Scholar
  24. 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
  25. Rassias JM, Rassias MJ: On the ulam stability of Jensen and Jensen type mappings on restricted domains. Journal of Mathematical Analysis and Applications 2003, 281(2):516–524. 10.1016/S0022-247X(03)00136-7MATHMathSciNetView ArticleGoogle Scholar
  26. Bae J-H, Park W-G: On the solution of a bi-Jensen functional equation and its stability. Bulletin of the Korean Mathematical Society 2006, 43(3):499–507.MATHMathSciNetView ArticleGoogle Scholar

Copyright

© G. H. Kim and Y.-H. Lee. 2010

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.

Advertisement