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

Advertisement