Stability of the Pexiderized Lobacevski Equation
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.
The stability problem of the functional equation was conjectured by Ulam  during the conference in the University of Wisconsin in 1940. In the next year, it was solved by Hyers  in the case of additive mapping, which is called the Hyers-Ulam stability. Thereafter, this problem was improved by Bourgin , Aoki , Rassias , Ger , and Gǎvruţa et al. [7, 8] in which Rassias’ result is called the Hyers-Ulam-Rassias stability.
In 1979, Baker et al.  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  and was improved by Badora and Ger . Recently, the superstability bounded by some function for the Pexider type sine functional equation has been investigated by Kim [12, 13].
In 1994, Gǎvruţa  proved the superstability of the Lobacevski equation under the condition bounded by a constant.
Kim  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).
This work was supported by Kangnam University Research Grant in 2010.
S. M. Ulam, Problems in Modern Mathematics, chapter 6, John Wiley & Sons, New York, NY, USA, 1964.