Stability of the Pexiderized Lobacevski Equation

Abstract

The aim of this paper is to investigate the solution and the superstability of the Pexiderized Lobacevski equation 𝑓((𝑥+𝑦)/2)²=𝑔(𝑥)ℎ(𝑦), where 𝑓, 𝑔, ℎ : 𝐺²→ℂ are unknown functions on an Abelian semigroup (𝐺,+). 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 𝑓 is a function from a vector space to ℝ satisfying 𝑓𝑥+𝑦−𝑓(𝑥)𝑓𝑦≤𝜀 (1.1) for some fixed 𝜀>0, then either 𝑓 is bounded or satisfies the exponential functional equation 𝑓𝑥+𝑦=𝑓(𝑥)𝑓𝑦. (E)

In 1983, the superstability bounded by a constant for the sine functional equation 𝑓𝑥+𝑦2²−𝑓𝑥−𝑦2²=𝑓(𝑥)𝑓𝑦 (S) 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 𝑓𝑥+𝑦2²−𝑓𝑥−𝑦2²=𝑔(𝑥)ℎ𝑦 (1.2) has been investigated by Kim [12, 13].

In 1994, Gǎvruţa [14] proved the superstability of the Lobacevski equation 𝑓𝑥+𝑦2²=𝑓(𝑥)𝑓𝑦 (L) 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 (𝑒^{(𝑥+𝑦)/2})²=𝑒^𝑥𝑒^𝑦.

The aim of this paper is to investigate the solution and the superstability of the Pexiderized Lobacevski equation 𝑓𝑥+𝑦2²=𝑔(𝑥)ℎ𝑦 (PL) 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 𝑓(𝑥)=𝛼𝛽𝑒^𝑥=(𝛼𝑒^𝑥)𝛽𝑒^𝑥=𝑔(𝑥)ℎ(𝑥),where𝛼,𝛽>0. (1.3)

In this paper, let (𝐺,+) be a uniquely 2-divisible Abelian semigroup (i.e., for each 𝑥∈𝐺, there exists a unique 𝑦∈𝐺 such that 𝑦+𝑦=𝑥: such 𝑦 will be denoted by 𝑥/2), ℂ is the field of complex numbers, ℝ the field of real numbers, and ℝ₊ the set of positive reals. We assume that 𝑓,𝑔,ℎ∶𝐺→ℂ are nonzero and nonconstant functions, 𝜀 is a nonnegative real constant, and 𝜑∶𝐺→ℝ₊ 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 𝑓,𝑔,ℎ∶𝐺→ℝ satisfy the inequality 𝑓𝑥+𝑦2²−𝑔(𝑥)ℎ𝑦≤𝜀 (2.1) for all 𝑥,𝑦∈𝐺.
Then, either there exist 𝐶₁,𝐶₂,𝐶₃>0 such that 𝑔(𝑥)≤𝐶₁,|ℎ(𝑥)|≤𝐶₂,𝑓(𝑥)≤𝐶₃, (2.2) for all 𝑥∈𝐺, or else each function 𝑔 and ℎ satisfies (L). Here 𝑔 and ℎ are represented by 𝑔(𝑥)(0)𝑒(𝑥)(𝑥)(0)𝑒(𝑥),𝑓(𝑥)=𝑔,ℎ=ℎ₂−𝑔(0)ℎ(0)⋅𝑒(𝑥)²≤𝜀, (2.3)(2.4)𝑥^∗∘𝑓(𝑥)−𝑥^∗∘𝑔(0)ℎ(0)𝑒₃(𝑥)≤𝜀, (3.10) where 𝑒₁,𝑒₂,𝑒₃∶𝐺→ℝ are exponentials. In other words, bearing the linear multiplicativity of 𝑥^∗ in mind, for all 𝑥∈𝐺, each difference derived from (3.8) and (3.9) 𝐷(3.8)(𝑥)∶=𝑔(𝑥)−𝑔(0)𝑒₁(𝑥),𝐷(3.9)(𝑥)∶=ℎ(𝑥)−ℎ(0)𝑒₂(𝑥), (3.11) falls into the kernel of 𝑥^∗. Therefore, in view of the unrestricted choice of 𝑥^∗, we infer that 𝐷(3.8)(𝑥),𝐷(3.9)(𝑥)∈ker𝑥^∗∶𝑥^∗isamultiplicativememberof𝐸^∗ (3.12) for all 𝑥∈𝐺. Since the algebra 𝐸 has been assumed to be semisimple, the last term of the previous formula coincides with the singleton {0}, that is, 𝑔(𝑥)−𝑔(0)𝑒₁(𝑥)=0,ℎ(𝑥)−ℎ(0)𝑒₂(𝑥)=0,𝑥∈𝐺. (3.13)
Putting (3.13) in (3.6), following the same proceeding as after (2.13) in Theorem 2.1, then we arrive that 𝑒₁(𝑥)=𝑒₂(𝑥). Indeed, we have 𝑥^∗∘𝑔(0)𝑒₁(𝑥)𝑥^∗∘ℎ(0)𝑒₂𝑦−𝑥^∗∘𝑔(0)𝑒₁𝑦𝑥^∗∘ℎ(0)𝑒₂(𝑥)≤2𝜀 (3.14) for all 𝑥,𝑦∈𝐺. This implies that 𝑥^∗∘𝑒₁(𝑥)𝑥^∗∘𝑒₂𝑦−𝑥^∗∘𝑒₁𝑦𝑥^∗∘𝑒₂(𝑥)≤2𝜀𝑔(0)ℎ(0)=𝑀∀𝑥,𝑦∈𝐺. (3.15)
Letting 𝑥=0 in (3.15), it implies |(𝑥^∗∘𝑒₂)(𝑦)−(𝑥^∗∘𝑒₁)(𝑦)|≤𝑀/𝑥^∗(1)=𝑀′ for all 𝑦∈𝐺. Thus, from this and (3.15), we have 𝑥^∗∘𝑒₁𝑦𝑥^∗∘𝑒₁(𝑥)−𝑥^∗∘𝑒₂𝑥=𝑥()_∗∘𝑒₁(𝑥)𝑥^∗∘𝑒₁𝑦−𝑥^∗∘𝑒₂𝑦+𝑥^∗∘𝑒₁(𝑥)𝑥^∗∘𝑒₂𝑦−𝑥^∗∘𝑒₁𝑦𝑥^∗∘𝑒₂(𝑥≤𝑥_∗)∘𝑒₁(𝑥)𝑀′+𝑀, (3.16) which is 𝑥^∗∘𝑒₁(𝑥)−𝑥^∗∘𝑒₂(𝑥≤𝑥)^∗∘𝑒₁(𝑥)𝑀′+𝑀𝑥^∗∘𝑒₁𝑦, (3.17) for all 𝑥,𝑦∈𝐺.
Since 𝑥^∗∘𝑔 is unbounded from (3.2), we can choose (𝑦_𝑛)∈𝐺 so that |(𝑥^∗∘𝑔)(𝑦_𝑛)|=|𝑔(0)(𝑥^∗∘𝑒₁)(𝑦_𝑛)|→∞ as 𝑛→∞. Letting 𝑦=𝑦_𝑛 in (3.17), which arrive that 𝑥^∗∘𝑒₁(𝑥)=𝑥^∗∘𝑒₂(𝑥). (3.18)
Using the same logic as before, that is, bearing the linear multiplicativity of 𝑥^∗ in mind, the difference derived from (3.18), 𝐷(3.18)(𝑥)∶=𝑒₁(𝑥)−𝑒₂(𝑥), falls into the kernel of 𝑥^∗. Then, the semisimplicity of 𝐸 implies that 𝑒₁(𝑥)=𝑒₂(𝑥). Let it be denoted by 𝑒(𝑥), which arrive the claimed (3.3) and (3.4).
Since 𝑒(𝑥)∶𝐺→ℝ is exponential, it is immediate from (3.3) that each function 𝑔 and ℎ 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.