1. Introduction

Baker et al. in [1] stated the following: if satisfies the inequality , then either is bounded or . This is frequently referred to as superstability.

The superstability of the cosine functional equation (also called the d'Alembert equation)

and the sine functional equation

were investigated by Baker [2] and Cholewa [3], respectively. Their results were improved by Badora [4], Badora and Ger [5], Gvruţa [6], and Kim (see [7, 8]).

The superstability of the Wilson equation

was investigated by Kannappan and Kim [9].

The superstability of the trigonometric functional equation concerned with the sine and the cosine equations

was investigated by Kim [10, 11], Kim and Lee [12].

The hyperbolic cosine function, hyperbolic sine function, hyperbolic trigonometric function, and some exponential functions also satisfy the above mentioned equations; thus they can be called by the hyperbolic cosine (sine, trigonometric, exponential) functional equations, respectively.

For example, where and are constants.

The aim of this paper is to investigate the superstability of the (hyperbolic) sine functional equation (S) from the following functional equations:

on the abelian group. As corollaries, we obtain the superstability of (S) from the following functional equations:

Furthermore, the obtained results can be extended to the Banach space.

In this paper, let be a uniquely 2-divisible Abelian group, the field of complex numbers, and the field of real numbers. Whenever we deal with (C), only needs Abelian which is not 2-divisibility.

We may assume that and are nonzero functions, is a nonnegative real constant, and is a mapping. For simplicity, we will form the notations of the equation as follows:

2. Superstability of the Functional Equations

In this section, we will investigate the superstability of the (hyperbolic) sine functional equation (S) from the functional equations (C_fgfg), (C_fggf), (T_fgfg), and (T_fggf) under the conditions from which the differences of each equation are bounded by and

Theorem 2.1. Suppose that satisfy the inequality
Then, either with is bounded or satisfies (S). Particularly, if satisfies (C^λ), then and are the solutions of the Wilson type equation (C_fg^λ).

Proof. Let be the unbounded solution of the inequality (2.1). Then, there exists a sequence in such that as .
Taking in the inequality (2.1), dividing both sides by and passing to the limit as we obtain the following:
Using (2.1), we have and thus, for all .
We conclude that, for every , there exists a limit function where the function satisfies the equation
Applying the case in (2.6), which implies that is odd and keeping this in mind, by means of (2.6), we infer the equality
Putting in (2.6), we obtain the equation
This, in return, leads to the equation being valid for all which, in the light of the unique 2-divisibility of , states nothing else but (S).
Particularly, if satisfies (C^λ), the limit states nothing else but ; thus, (2.6) validates the required equation (C_fg^λ).

Corollary 2.2. Suppose that satisfy the inequality
Then, either with is bounded or satisfies (S).

Proof. Replacing by in (2.1) of Theorem 2.1, an obvious slight change in the proof steps applied in Theorem 2.1 allows us to show that satisfies (S).
Namely, for be unbounded, there exists a sequence in such that as . Taking in the inequality (2.1), dividing both sides by and passing to the limit as we obtain A similar procedure to that applied after formula (2.2) yields the required result by using of (2.6).

Theorem 2.3. Suppose that satisfy the inequality
Then, either with is bounded or satisfies (S). Particularly, if satisfies (C^λ), then and are the solutions of equation (C_gf^λ).

Proof. For the unbounded , we can choose a sequence in such that as .
The same reasoning to the proof applied in Theorem 2.1 for (2.12) with gives
Substituting and for in (2.12), and dividing by then it gives us the existence of a limit function where the function satisfies the equation
Applying the case in (2.15), it implies that is odd.
A similar procedure to that applied after formula (2.6) allows us to show that satisfies (S).
Particularly, if satisfies (C^λ), the limit states nothing else but thus, the required equation (C_gf^λ) holds from (2.15).

Corollary 2.4. Suppose that satisfy the inequality
Then, either with is bounded or satisfies (S).

Proof. Substituting for in (2.12) of Theorem 2.3, the next of the proof runs along that of the above theorem.

Theorem 2.5. Suppose that satisfy the inequality
Then, either with is bounded or satisfies (S). Particularly, if satisfies (C^λ), then and are solutions of the Wilson type equation (C_gf^λ).

Proof. For the unbounded of the inequality (2.17), we can choose a sequence in such that as .
Taking in the inequality (2.17), dividing both sides by , and passing to the limit as we obtain
In (2.17), replacing by and , replacing by , and dividing by , it then gives us the existence of a limit function where the function satisfies the equation
Applying the case in (2.20), it implies that is odd. Since (2.20) equals to (2.6), an obvious slight change in the proof steps applied after formula (2.15) allows us to see that satisfies (S). Particularly, if satisfies (C^λ), then the limit states nothing else but , thus, the required equation (C_fg^λ) holds from (2.20).

Corollary 2.6. Suppose that satisfy the inequality
Then, either with is bounded or satisfies (S).

Proof. Substituting for in (2.17) of Theorem 2.5, as Corollary 2.4, we then obtain the required result from the above theorem.

Theorem 2.7. Suppose that satisfy the inequality
Then, either with is bounded or satisfies (S). Particularly, if satisfies (C^λ), then and are the solutions of equation (C_fg^λ).

Proof. For the unbounded , we can choose a sequence in such that as .
For (2.22) with , the same reasoning as the proof applied in Theorem 2.1 gives us
In (2.22), replacing by and replacing by , and dividing by , it then gives us the existence of a limit function where the function satisfies the equation
Since (2.25) is the same as (2.6), the next proof runs along that of Theorem 2.1.

Corollary 2.8. Suppose that satisfy the inequality
Then, either with is bounded or satisfies (S).

Proof. Substituting for in (2.22) of Theorem 2.7, the next proof runs along that of the above theorem.

The cases , in the following result follow the procedure applied in Theorems 2.1 and 2.5, respectively. In the obtained result, applying the cases and , then they are founded in [2, 4–6].

Corollary 2.9. Suppose that satisfy the inequality
Then, either is bounded or satisfies (C^λ).

Proof. For being unbounded, we can choose two sequences and in such that and as .(i) case
Taking in inequality (2.27), dividing it by , and passing to the limit as we obtain
In (2.27), replacing by and and dividing by , it then gives, with the application of (2.28), that satisfies (C^λ).(ii) case
For the chosen sequence , the procedure as (i) implies
In (2.27), replacing by and and replacing by , the other procedure is the same as (i).

Since the proofs of the following results (Theorems 2.10–2.16 and Corollaries 2.11–2.17) for the functional equations (T_fgfg), (T_fggf), (T_fgff), and (T_fggg) are, respectively, the same processes those of Theorems 2.1–2.7 and Corollaries 2.2–2.8, as we will skip their proofs.

Theorem 2.10. Suppose that satisfy the inequality
Then, either with is bounded or satisfies (S). Particularly, if satisfies (C^λ), then and are solutions of the Wilson type equation (C_fg^λ).

Corollary 2.11. Suppose that satisfy the inequality
Then, either with is bounded or satisfies (S).

Theorem 2.12. Suppose that satisfy the inequality
Then, either with is bounded or satisfies (S). Particularly, if satisfies (C^λ), then and are solutions of the Wilson type equation (C_gf^λ).

Corollary 2.13. Suppose that satisfy the inequality
Then, either with is bounded or satisfies (S).

Theorem 2.14. Suppose that satisfy the inequality
Then, either with is bounded or satisfies (S). Particularly, if satisfies (C^λ), then and are solutions of the Wilson type equation (C_gf^λ).

Corollary 2.15. Suppose that satisfy the inequality
Then, either with is bounded or satisfies (S).

Theorem 2.16. Suppose that satisfy the inequality
Then, either with is bounded or satisfies (S). Particularly, if satisfies (C^λ), then and are the solutions of equation (C_fg^λ).

Corollary 2.17. Suppose that satisfy the inequality
Then, either with is bounded or satisfies (S).

The cases and in the following result follow the procedure applied in Theorems 2.10 and 2.14, respectively. In the obtained result, applying the cases and , then they are founded in [10–12].

Corollary 2.18. Suppose that satisfy the inequality
Then is bounded.

Proof. For being unbounded, we can choose two sequences and in such that and as .(i) case
First, going though the same process of the case (i) of Corollary 2.9, then we obtain that satisfies (C^λ).
Secondly, from the chosen sequence , we obtain
Replacing by and in (2.38), then we obtain, from their difference, the inequality which gives, with the application of (2.39), the equation
Thus, since the function satisfies two equations (C^λ) and (), the function states nothing else but zero. It is a contradiction assuming that is nonzero. Thus is bounded.(ii) case
For the chosen sequence , the same procedure as in the above case (i) gives us the required result.

Remark 2.19. (a) Substituting for of the second term of the stability inequalities in all the results of Section 2, then we obtain the same number of corollaries, which are the stability of the (hyperbolic) cosine type functional equations (C_fg^λ, C_gf^λ), and the (hyperbolic) trigonometric type functional equations .
(b) Applying the case in all the results of Section 2 and (a)'s application, then we obtain the same number of corollaries. Some of their stabilities were founded in papers [6, 7, 9–11].
(c) Applying in all the results of Section 2 and (a)'s application, then we obtain the same number of corollaries. Some of their stabilities were founded in papers [6, 7, 9–12].
(d) Applying and in all the results of Section 2, (a)'s, (b)'s, and (c)'s applications, then we obtain the same number of corollaries. Some of their stabilities were founded in papers [5–7, 9–12].

3. Extension to the Banach Space

In all the results presented in Section 2, the range of functions on the abelian group can be extended to the Banach space. For simplicity, we will only prove the plus case of (3.1) of Theorem 3.1. The other cases are similar to this, thus their proofs will be omitted.

Theorem 3.1. Let be a semisimple commutative Banach space. Assume that satisfy one of each inequalities for all For an arbitrary linear multiplicative functional
then,
(i) case (3.1), either with is bounded or satisfies (S). Particularly, if satisfies (C^λ), then and are the solutions of the Wilson type equation (C_fg^λ). (ii) case (3.2), either with is bounded or satisfies (S). Particularly, if satisfies (C^λ), then and are the solutions of the Wilson type equation (C_gf^λ).

Proof. (i) As and have the same procedure, we will only show the plus case in (3.1).
Assume that (3.1) holds and arbitrarily fixes a linear multiplicative functional . As is well known, we have hence, for every , we have which states that the superpositions and yield a solution of inequality (2.1). Since, by assumption, the superposition with is unbounded, an appeal to Theorem 2.1 shows that the two results hold.
First, the function solves (S). In other words, bearing the linear multiplicativity of in mind, for all , the difference defined by 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 above formula coincides with the singleton that is, as claimed.
Second, in particular, if