Open Access

Stability of a Jensen Type Logarithmic Functional Equation on Restricted Domains and Its Asymptotic Behaviors

Advances in Difference Equations20102010:432796

https://doi.org/10.1155/2010/432796

Received: 28 June 2010

Accepted: 25 December 2010

Published: 30 December 2010

Abstract

Let be the set of positive real numbers, a Banach space, and , with . We prove the Hyers-Ulam stability of the Jensen type logarithmic functional inequality in restricted domains of the form for fixed with or and . As consequences of the results we obtain asymptotic behaviors of the inequality as .

1. Introduction

The stability problems of functional equations have been originated by Ulam in 1940 (see [1]). One of the first assertions to be obtained is the following result, essentially due to Hyers [2], that gives an answer for the question of Ulam.

Theorem 1.1.

Suppose that is an additive semigroup, is a Banach space, , and satisfies the inequality
(1.1)
for all . Then there exists a unique function satisfying
(1.2)
for which
(1.3)

for all .

In 1950-1951 this result was generalized by the authors Aoki [3] and Bourgin [4, 5]. Unfortunately, no results appeared until 1978 when Th. M. Rassias generalized the Hyers' result to a new approximately linear mappings [6]. Following the Rassias' result, a great number of the papers on the subject have been published concerning numerous functional equations in various directions [616]. For more precise descriptions of the Hyers-Ulam stability and related results, we refer the reader to the paper of Moszner [17]. Among the results, the stability problem in a restricted domain was investigated by Skof, who proved the stability problem of the inequality (1.1) in a restricted domain [16]. Developing this result, Jung considered the stability problems in restricted domains for the Jensen functional equation [11] and Jensen type functional equations [14]. The results can be summarized as follows: let and be a real normed space and a real Banach space, respectively. For fixed , if satisfies the functional inequalities (such as that of Cauchy, Jensen and Jensen type, etc.) for all with , the inequalities hold for all . We also refer the reader to [1826] for some interesting results on functional equations and their Hyers-Ulam stabilities in restricted conditions.

Throughout this paper, we denote by the set of positive real numbers, a Banach space, , and with . We prove the Hyers-Ulam stability of the Jensen type logarithmic functional inequality
(1.4)
in the restricted domains of the form for fixed with or , and . As a result, we prove that if the inequality (1.4) holds for all , there exists a unique function satisfying
(1.5)
for which
(1.6)
for all if ,
(1.7)
for all if , and
(1.8)
for all if . As a consequence of the result we obtain the stability of the inequality
(1.9)

in the restricted domains of the form for fixed with or , and . Also we obtain asymptotic behaviors of the inequalities (1.4) and (1.9) as and , respectively.

2. Hyers-Ulam Stability in Restricted Domains

We call the functions satisfying (1.5)logarithmic functions. As a direct consequence of Theorem 1.1, we obtain the stability of the logarithmic functional equation, viewing as a multiplicative group (see also the result of Forti [9]).

Theorem A.

Suppose that , , and
(2.1)
for all . Then there exists a unique logarithmic function satisfying
(2.2)

for all .

We first consider the usual logarithmic functional inequality (2.1) in the restricted domains .

Theorem 2.1.

Let , with or . Suppose that satisfies
(2.3)
for all , with . Then there exists a unique logarithmic function such that
(2.4)

for all .

Proof.

From the symmetry of the inequality we may assume that . For given , choose a such that , , and . Then we have
(2.5)

This completes the proof.

Now we consider the Hyers-Ulam stability of the Jensen type logarithmic functional inequality (1.4) in the restricted domains .

Theorem 2.2.

Let . Suppose that satisfies
(2.6)
for all , with . Then there exists a unique logarithmic function such that
(2.7)

for all .

Proof.

Replacing by , by in (2.6) we have
(2.8)

for all , with .

For given , choose a such that , , , and . Replacing by , by ; by , by ; by , by ; by , by in (2.8) we have
(2.9)
Now by Theorem A, there exists a unique logarithmic function such that
(2.10)

for all . This completes the proof.

As a matter of fact, we obtain that in Theorem 2.2 provided that and or is a rational number, or and or is a rational number.

Theorem 2.3.

Let , , . Suppose that and or is a rational number, or and or is a rational number, and satisfies
(2.11)
for all , with . Then one has
(2.12)

for all .

Proof.

We prove (2.12) only for the case that and or is a rational number since the other case is similarly proved. From (2.7) and (2.11), using the triangle inequality we have
(2.13)
for all , with , where . If , putting in (2.13) we have
(2.14)
for all , with . It is easy to see that for all and all rational numbers . Thus if is a rational number, it follows from (2.14) that
(2.15)
for all , with . If there exists such that , we can choose a rational number such that and (it is realized when is large if , and when is large if ). Now we have
(2.16)
Thus it follows that . If is a rational number, it follows from (2.14) that
(2.17)
for all , with , which implies
(2.18)
for all , with . Similarly, using (2.18) we can show that . If , choosing such that , putting in (2.13) and using the triangle inequality we have
(2.19)

for all . Similarly, using (2.19) we can show that . Thus the inequality (2.12) follows from (2.7). This completes the proof.

Theorem 2.4.

Let with or . Suppose that satisfies
(2.20)
for all , with . Then there exists a unique logarithmic function such that
(2.21)
for all if , and
(2.22)

for all if .

Proof.

Assume that . For given , choose a such that , , and . Replacing by , by ; by , by ; by , by ; by 1, by in (2.20) we have
(2.23)
Dividing (2.23) by and using Theorem A, we obtain that there exists a unique logarithmic function such that
(2.24)
for all . Assume that . For given , choose a such that and . Replacing by , by ; by , by ; by , by ; by 1, by in (2.20) we have
(2.25)
Dividing (2.25) by and using Theorem A, we obtain that there exists a unique logarithmic function such that
(2.26)

for all . This completes the proof.

From Theorem 2.4, using the same approach as in the proof of Theorem 2.3 we have the following.

Theorem 2.5.

Let , with or . Suppose that and or is a rational number, or and or is a rational number, and satisfies
(2.27)
for all , with . Then one has
(2.28)
for all if , and
(2.29)

for all if .

We call an additive function provided that
(2.30)

for all . Using Theorem 2.2 we have the following.

Corollary 2.6 (see [22]).

Let , with . Suppose that satisfies
(2.31)
for all , with . Then there exists a unique additive function such that
(2.32)

for all .

Proof.

Replacing by , by in (2.31) and setting we have
(2.33)
for all , with . Using Theorem 2.2, we have
(2.34)
for all , which implies
(2.35)

for all . Letting we get the result.

Using Theorem 2.3, we have the following.

Corollary 2.7.

Let , with . Suppose that and or is a rational number, or and or is a rational number, and satisfies
(2.36)
for all , with . Then one has
(2.37)

for all .

Using Theorem 2.4, we have the following.

Corollary 2.8.

Let , with or . Suppose that satisfies
(2.38)
for all , with . Then there exists a unique additive function such that
(2.39)
for all if , and
(2.40)

for all if .

Using Theorem 2.5, we have the following.

Corollary 2.9.

Let , with or . Suppose that and or is a rational number, or and or is a rational number, and satisfies
(2.41)
for all , with . Then one has
(2.42)
for all if , and
(2.43)

for all if .

3. Asymptotic Behavior of the Inequality

In this section, we consider asymptotic behaviors of the inequalities (1.4) and (2.1).

Theorem 3.1.

Let satisfy one of the conditions; , . Suppose that satisfies the asymptotic condition
(3.1)

as . Then is a logarithmic function.

Proof.

By the condition (3.1), for each , there exists such that
(3.2)
for all , with . By Theorem 2.1, there exists a unique logarithmic function such that
(3.3)
for all . From (3.4) we have
(3.4)
for all and all positive integers . Now, the inequality (3.4) implies . Indeed, for all and rational numbers we have
(3.5)

Letting in (3.5), we have . Thus, letting in (3.3), we get the result.

Theorem 3.2.

Let satisfy one of the conditions; , , . Suppose that satisfies the asymptotic condition
(3.6)
as . Then there exists a unique logarithmic function such that
(3.7)

for all .

Proof.

By the condition (3.6), for each , there exists such that
(3.8)
for all , with . By Theorems 2.2 and 2.4, there exists a unique logarithmic function such that
(3.9)
if ,
(3.10)
if , and
(3.11)
if . For all cases (3.9), (3.10), and (3.11), there exists such that
(3.12)

for all and all positive integers . Now as in the proof of Theorem 3.1, it follows from (3.12) that for all . Letting in (3.9), (3.10), and (3.11) we get the result.

Similarly using Theorems 2.3 and 2.5, we have the following.

Theorem 3.3.

Let satisfy one of the conditions; , , . Suppose that and or is a rational number, or and or is a rational number, and satisfies the asymptotic condition
(3.13)

as . Then is a constant function.

Using Corollaries 2.6 and 2.8 we have the following.

Corollary 3.4.

Let , satisfy one of the conditions , , or . Suppose that satisfies
(3.14)
as . Then there exists a unique additive function such that
(3.15)

for all .

Using Corollaries 2.7 and 2.9 we have the following.

Corollary 3.5.

Let , satisfy one of the conditions , , or . Suppose that and or is a rational number, or and or is a rational number, and satisfies
(3.16)

as . Then is a constant function.

Declarations

Acknowledgments

The author expresses his sincere gratitude to a referee of the paper for many useful comments and introducing the interesting related recent results including the papers [1726]. This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) (no. 2010-0016963).

Authors’ Affiliations

(1)
Department of Mathematics, Kunsan National University

References

  1. Ulam SM: A Collection of Mathematical Problems. Interscience, New York, NY, USA; 1960.MATHGoogle 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. 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/00210064MathSciNetView ArticleMATHGoogle Scholar
  4. Bourgin DG: Multiplicative transformations. Proceedings of the National Academy of Sciences of the United States of America 1950, 36: 564–570. 10.1073/pnas.36.10.564MathSciNetView ArticleMATHGoogle Scholar
  5. Bourgin DG: Classes of transformations and bordering transformations. Bulletin of the American Mathematical Society 1951, 57: 223–237. 10.1090/S0002-9904-1951-09511-7MathSciNetView ArticleMATHGoogle Scholar
  6. Rassias TM: 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
  7. Chung J: A distributional version of functional equations and their stabilities. Nonlinear Analysis: Theory, Methods & Applications 2005,62(6):1037–1051. 10.1016/j.na.2005.04.016MathSciNetView ArticleMATHGoogle Scholar
  8. Chung J: Stability of approximately quadratic Schwartz distributions. Nonlinear Analysis: Theory, Methods & Applications 2007,67(1):175–186. 10.1016/j.na.2006.05.005MathSciNetView ArticleMATHGoogle Scholar
  9. Forti GL: The stability of homomorphisms and amenability, with applications to functional equations. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1987, 57: 215–226. 10.1007/BF02941612MathSciNetView ArticleMATHGoogle Scholar
  10. Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and their Applications. Volume 34. Birkhäuser, Boston, Mass, USA; 1998:vi+313.Google Scholar
  11. 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
  12. Jun K-W, Kim H-M: Stability problem for Jensen-type functional equations of cubic mappings. Acta Mathematica Sinica 2006,22(6):1781–1788. 10.1007/s10114-005-0736-9MathSciNetView ArticleMATHGoogle Scholar
  13. Kim GH, Lee YW: Boundedness of approximate trigonometric functional equations. Applied Mathematics Letters 2009,31(4):439–443. 10.1016/j.aml.2008.06.013View ArticleGoogle Scholar
  14. Rassias JM: On the Ulam stability of mixed type mappings on restricted domains. Journal of Mathematical Analysis and Applications 2002,276(2):747–762. 10.1016/S0022-247X(02)00439-0MathSciNetView ArticleMATHGoogle Scholar
  15. 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)00136-7MathSciNetView ArticleMATHGoogle Scholar
  16. Skof F: Sull'approssimazione delle applicazioni localmente -additive. Atti della Reale Accademia delle Scienze di Torino. Classe di Scienze Fisiche, Matematiche e Naturali 1983, 117: 377–389.MathSciNetMATHGoogle Scholar
  17. Moszner Z: On the stability of functional equations. Aequationes Mathematicae 2009,77(1–2):33–88. 10.1007/s00010-008-2945-7MathSciNetView ArticleMATHGoogle Scholar
  18. Batko B: Stability of an alternative functional equation. Journal of Mathematical Analysis and Applications 2008,339(1):303–311. 10.1016/j.jmaa.2007.07.001MathSciNetView ArticleMATHGoogle Scholar
  19. Batko B: On approximation of approximate solutions of Dhombres' equation. Journal of Mathematical Analysis and Applications 2008,340(1):424–432. 10.1016/j.jmaa.2007.08.009MathSciNetView ArticleMATHGoogle Scholar
  20. Brzdęk J: On the quotient stability of a family of functional equations. Nonlinear Analysis: Theory, Methods & Applications 2009,71(10):4396–4404. 10.1016/j.na.2009.02.123MathSciNetView ArticleGoogle Scholar
  21. Brzdęk J: On a method of proving the Hyers-Ulam stability of functional equations on restricted domains. The Australian Journal of Mathematical Analysis and Applications 2009,6(1):1–10.MathSciNetGoogle Scholar
  22. Brzdęk J: On stability of a family of functional equations. Acta Mathematica Hungarica 2010,128(1–2):139–149. 10.1007/s10474-010-9169-8MathSciNetView ArticleMATHGoogle Scholar
  23. Brzdęk J, Sikorska J: A conditional exponential functional equation and its stability. Nonlinear Analysis: Theory, Methods & Applications 2010,72(6):2923–2934. 10.1016/j.na.2009.11.036MathSciNetView ArticleMATHGoogle Scholar
  24. Sikorska J: On two conditional Pexider functional equations and their stabilities. Nonlinear Analysis: Theory, Methods & Applications 2009,70(7):2673–2684. 10.1016/j.na.2008.03.054MathSciNetView ArticleMATHGoogle Scholar
  25. Sikorska J: On a Pexiderized conditional exponential functional equation. Acta Mathematica Hungarica 2009,125(3):287–299. 10.1007/s10474-009-9019-8MathSciNetView ArticleMATHGoogle Scholar
  26. Sikorska J: Exponential functional equation on spheres. Applied Mathematics Letters 2010,23(2):156–160. 10.1016/j.aml.2009.09.004MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Jae-Young Chung. 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.