Journal of Applied Mathematics

Volume 2011, Article ID 540274, 10 pages

http://dx.doi.org/10.1155/2011/540274

## Stability of the Pexiderized Lobacevski Equation

Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Republic of Korea

Received 16 April 2011; Accepted 11 June 2011

Academic Editor: Junjie Wei

Copyright © 2011 Gwang Hui Kim. 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.

#### Abstract

The aim of this paper is to investigate the solution and the superstability of the Pexiderized Lobacevski equation , where , , : are unknown functions on an Abelian semigroup . The obtained result is a generalization of Gǎvruţa's result in 1994 and Kim's result in 2010.

#### 1. Introduction

The stability problem of the functional equation was conjectured by Ulam [1] during the conference in the University of Wisconsin in 1940. In the next year, it was solved by Hyers [2] in the case of additive mapping, which is called the Hyers-Ulam stability. Thereafter, this problem was improved by Bourgin [3], Aoki [4], Rassias [5], Ger [6], and Gǎvruţa et al. [7, 8] in which Rassias’ result is called the Hyers-Ulam-Rassias stability.

In 1979, Baker et al. [9] developed the superstability, which is that if is a function from a vector space to satisfying for some fixed , then either is bounded or satisfies the exponential functional equation

In 1983, the superstability bounded by a constant for the sine functional equation was investigated by Cholewa [10] and was improved by Badora and Ger [11]. Recently, the superstability bounded by some function for the Pexider type sine functional equation has been investigated by Kim [12, 13].

In 1994, Gǎvruţa [14] proved the superstability of the Lobacevski equation under the condition bounded by a constant.

Kim [15] improved his result under the condition bounded by an unknown function. In there, author conjectured through an example that the Lobacevski equation (L) will have a solution as an exponential function. Namely, for a simple example of this equation, we can find the functional equation .

The aim of this paper is to investigate the solution and the superstability of the Pexiderized Lobacevski equation under the condition bounded by a function. Namely, this has improved in the Pexider type for the results of Gǎvruţa and Kim.

Furthermore, the range of the function in all results is expanded to the Banach space.

The solution of (PL) will be represented as an exponential, namely, for a simple example of this equation, it will be considered as a geometric mean

In this paper, let be a uniquely 2-divisible Abelian semigroup (i.e., for each , there exists a unique such that : such will be denoted by ), is the field of complex numbers, the field of real numbers, and the set of positive reals. We assume that are nonzero and nonconstant functions, is a nonnegative real constant, and is a mapping.

#### 2. Stability of the Pexiderized Lobacevski Equation (PL)

We will investigate the solution and the superstability of the Pexiderized Lobacevski equation (PL).

Theorem 2.1. *Suppose that satisfy the inequality
**
for all .**Then, either there exist such that
**
for all , or else each function and satisfies (L). Here and are represented by
**
where are exponentials. In other words, bearing the linear multiplicativity of in mind, for all , each difference derived from (3.8) and (3.9)
**
falls into the kernel of . Therefore, in view of the unrestricted choice of , we infer that
**
for all . Since the algebra has been assumed to be semisimple, the last term of the previous formula coincides with the singleton , that is,
**Putting (3.13) in (3.6), following the same proceeding as after (2.13) in Theorem 2.1, then we arrive that . Indeed, we have
**
for all . This implies that
**Letting in (3.15), it implies for all . Thus, from this and (3.15), we have
**
which is
**
for all .**Since is unbounded from (3.2), we can choose so that as . Letting in (3.17), which arrive that
**Using the same logic as before, that is, bearing the linear multiplicativity of in mind, the difference derived from (3.18), , falls into the kernel of . Then, the semisimplicity of implies that . Let it be denoted by , which arrive the claimed (3.3) and (3.4).**Since is exponential, it is immediate from (3.3) that each function and satisfies (L).*

*Remark 3.2. *All results of Section 2 containing Remark 2.7 can be extended to the Banach space as Theorem 3.1.

#### Acknowledgment

This work was supported by Kangnam University Research Grant in 2010.

#### References

- S. M. Ulam,
*Problems in Modern Mathematics*, chapter 6, John Wiley & Sons, New York, NY, USA, 1964. - D. H. Hyers, “On the stability of the linear functional equation,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 27, pp. 222–224, 1941. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. G. Bourgin, “Multiplicative transformations,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 36, pp. 564–570, 1950. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Aoki, “On the stability of the linear transformation in Banach spaces,”
*Journal of the Mathematical Society of Japan*, vol. 2, pp. 64–66, 1950. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,”
*Proceedings of the American Mathematical Society*, vol. 72, no. 2, pp. 297–300, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Ger, “Superstability is not natural,”
*Rocznik Naukowo-Dydaktyczny*, vol. 159, no. 13, pp. 109–123, 1993. View at Google Scholar · View at Zentralblatt MATH - P. Gǎvruţa, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 184, no. 3, pp. 431–436, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Frank, P. Gǎvruţa, and M. S. Moslehian, “Superstability of adjointable mappings on Hilbert ${C}^{*}$-modules,”
*Applicable Analysis and Discrete Mathematics*, vol. 3, no. 1, pp. 39–45, 2009. View at Publisher · View at Google Scholar - J. Baker, J. Lawrence, and F. Zorzitto, “The stability of the equation $f(x+y)=f(x)f(y)$,”
*Proceedings of the American Mathematical Society*, vol. 74, no. 2, pp. 242–246, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. W. Cholewa, “The stability of the sine equation,”
*Proceedings of the American Mathematical Society*, vol. 88, no. 4, pp. 631–634, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Badora and R. Ger, “On some trigonometric functional inequalities,” in
*Functional Equations—Results and Advances*, vol. 3, pp. 3–15, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002. View at Google Scholar · View at Zentralblatt MATH - G. H. Kim, “A stability of the generalized sine functional equations,”
*Journal of Mathematical Analysis and Applications*, vol. 331, no. 2, pp. 886–894, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. H. Kim, “On the stability of the generalized sine functional equations,”
*Acta Mathematica Sinica*, vol. 25, no. 1, pp. 29–38, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Gǎvruţa, “On the stability of some functional equations,” in
*Stability of Mappings of Hyers-Ulam Type*, pp. 93–98, Hadronic Press, Palm Harbor, Fla, USA, 1994. View at Google Scholar · View at Zentralblatt MATH - G. H. Kim, “Stability of the Lobacevski equation,”
*Journal of Nonlinear Science and its Applications*, vol. 4, no. 1, pp. 11–18, 2011. View at Google Scholar · View at Zentralblatt MATH