Journal of Applied MathematicsVolume 2011 (2011), Article ID 540274, 10 pagesdoi:10.1155/2011/540274

Research Article

Stability of the Pexiderized Lobacevski Equation

Gwang Hui Kim

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 is an exponential function.

Proof. Replacing with in (2.1), and then subtracting them and using triangle inequality, we have It follows from the inequality (2.5) that there exist constants such that for all . It follows from (2.6) and (2.7) that is bounded if and only if is bounded. If either of or is bounded, then we obtain (2.2) from (2.1).Now if is unbounded, then we can choose so that as . Letting in (2.1), dividing by , and letting , we have It follows from (2.1) and (2.8) that where , which implies for all .Letting in (2.10), we get for all , which implies that where (since is a nonzero and nonconstant function), and is an exponential. Exchanging the roles of and , by the same proceeding, we have where , and is an exponential.Putting (2.12) and (2.13) in (2.5), it implies Let in (2.14). Since and are exponentials, this implies that for all . Hence, from this and (2.14), we have which is for all .Since is unbounded from (2.2), we can choose so that as . Letting in (2.16), we get that . Let it be denoted by . Then (2.12) and (2.13) state nothing but (2.3). Putting (2.3) with in (2.1), we get the inequality (2.4). Finally, it is immediate that and in (2.3) satisfy (L), respectively.

Corollary 2.2. Suppose that satisfy the inequality for all . Then, either is bounded or satisfies (L). In particular, is represented by where is exponential.

Corollary 2.3. Suppose that satisfy the inequality for all . Then either there exist such that for all , or else each function and satisfies (L). In particular, and are represented by where is exponential.

Corollary 2.4. Suppose that satisfy the inequality for all . Then either is bounded or satisfies (L). In particular, is represented by where is exponential.

In Corollary 2.4, it is founded in papers [14, 15] that satisfies (L).

Theorem 2.5. Suppose that satisfy the inequality for all . Then, either is bounded or is an exponential by the multiplying of a scalar and satisfies (L).

Proof. Suppose that is unbounded. Then we can choose such that as . Letting in (2.24), dividing by , and letting , we have Thus, it follows from (2.24) and (2.25) that where , which implies for all .Letting in (2.27), we get for all . Namely, it means that is an exponential function by the multiplying of a scalar and satisfies (L).

Theorem 2.6. Suppose that satisfy the inequality for all . Then, either is bounded or is an exponential by the multiplying of a scalar and satisfies (L).

Proof. The proof runs along a slight change in the step-by-step procedure in Theorem 2.5.

Remark 2.7. (i) As Corollaries 2.2–2.4 of Theorem 2.1, by replacing and with in Theorems 2.5 and 2.6, we can obtain more corollaries for the following functional equations: in which the case of (2.30) is found in paper [15].(ii) For the results obtained from each equation of the above (i), by applying , we can obtain the same number of corollaries.

3. Extension to Banach Algebra

All obtained results can be extended to the stability on the Banach algebras. We will illustrate only for the case of Theorem 2.1 among them.

Theorem 3.1. Let be a semisimple commutative Banach algebra. Assume that satisfy the inequality for all . Then, for an arbitrary linear multiplicative functional , either there exist such that for all , or else each function and satisfies (L). Here and are represented by where is an exponential function.

Proof. Assume that (3.1) holds, and fix arbitrarily a linear multiplicative functional . As well known, we have , hence, for every , we have which states that the superpositions , , and satisfy the inequality (2.1) of Theorem 2.1. Due to the same processing as from (2.5) to (2.7), to fix arbitrarily a linear multiplicative functional , indeed, we have It follows from the inequality (3.6) that there exist constants such that for all . Since is an arbitrarily linear multiplicative functional, it follows from (3.7) that is bounded if and only if is bounded. Assume that one of or is bounded. From (3.1), we arrive at (3.2).By the assumption (3.2), an appeal to Theorem 2.1 shows that 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.

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