Approximate Cubic Lie Derivations on ρ-Complete Convex Modular Algebras

In 1940, S. M. Ulam [1] raised the question concerning the stability of group homomorphisms. Let G be a group and let G󸀠 be a metric group with the metric d(⋅, ⋅). Given ε > 0, does there exist δ > 0 such that if a mapping f : G 󳨀→ G󸀠 satisfies the inequality d (f (xy) , f (x) f (y)) < δ (1) for all x, y ∈ G, then there exists a homomorphism F : G 󳨀→ G󸀠 with d(f(x), F(x)) < ε for all x ∈ G? D. H. Hyers [2] has solved the problem of Ulam for the case of additive mappings in 1941. The result was generalized by T. Aoki [3] in 1950, by Th.M. Rassias [4] in 1978, by J. M. Rassias [5] in 1992, and by P. Gǎvruta [6] in 1994. Over the past few decades, many mathematicians have investigated the generalized Hyers–Ulam stability theorems of various functional equations [7–12]. Now, we recall some basic definitions and remarks of modular spaces with modular functions, which are primitive notions corresponding to norms or metrics, as in the following [13–15]. Definition 1. Let χ be a linear space. (a) A function ρ : χ 󳨀→ [0,∞] is called a modular if, for arbitrary x, y ∈ χ, (1) ρ(x) = 0 if and only if x = 0, (2) ρ(αx) = ρ(x) for every scalar α with |α| = 1, (3) ρ(αx + βy) ≤ ρ(x) + ρ(y) for any scalars α, β, where α + β = 1 and α, β ≥ 0;


Introduction
In 1940, S. M. Ulam [1] raised the question concerning the stability of group homomorphisms. Let be a group and let be a metric group with the metric (⋅, ⋅). Given > 0, does there exist > 0 such that if a mapping : → satisfies the inequality ( ( ) , ( ) ( )) < (1) for all , ∈ , then there exists a homomorphism : → with ( ( ), ( )) < for all ∈ ? D. H. Hyers [2] has solved the problem of Ulam for the case of additive mappings in 1941. The result was generalized by T. Aoki [3] in 1950, by Th.M. Rassias [4] in 1978, by J. M. Rassias [5] in 1992, and by P. Gǎvruta [6] in 1994. Over the past few decades, many mathematicians have investigated the generalized Hyers-Ulam stability theorems of various functional equations [7][8][9][10][11][12]. Now, we recall some basic definitions and remarks of modular spaces with modular functions, which are primitive notions corresponding to norms or metrics, as in the following [13][14][15].  then we say that is a convex modular. It is well known that a modular defines a corresponding modular space, i.e., the linear space given by Let be a convex modular. Then, we remark the modular space can be a Banach space equipped with a norm called the Luxemburg norm, defined by If is a modular on , we note that ( ) is an increasing function in ≥ 0 for each fixed ∈ ; that is, ( ) ≤ ( ), whenever 0 ≤ < . In addition, if is a convex modular on , then ( ) ≤ ( ) for all ∈ and 0 ≤ ≤ 1. Moreover, we see that ( ) ≤ | | ( ) for all ∈ and | | ≤ 1.

Definition 2.
Let be a modular space and let { } be a sequence in . Then, (3) a subset of is called -complete if and only if any -Cauchy sequence in is -convergent to an element in .
They say that the modular has the Fatou property if and only if ( ) ≤ liminf →∞ ( ) whenever the sequence { } is -convergent to . A modular function is said to satisfy the Δ 2 -condition if there exists > 0 such that (2 ) ≤ ( ) for all ∈ .
In 2014, G. Sadeghi [16] has demonstrated generalized Hyers-Ulam stability via the fixed point method of a generalized Jensen functional equation ( + ) = ( ) + ℎ( ) in convex modular spaces with the Fatou property satisfying the Δ 2 -condition with 0 < ≤ 2. In [15], the authors have proved the generalized Hyers-Ulam stability of quadratic functional equations via the extensive studies of fixed point theory in the framework of modular spaces whose modulars are convex and lower semicontinuous but do not satisfy any relatives of Δ 2 -conditions (see also [17,18]). Recently, the authors [14,19,20] have investigated stability theorems of functional equations in modular spaces without using the Fatou property and Δ 2 -condition. In 2001, J. M. Rassias [21] has introduced to study Hyers-Ulam stability of the following cubic functional equation: which is equivalent to whose general solution is characterized as ( ) = ( , , ) where is symmetric and additive for each fixed one variable [22]. For this reason, every solution of the cubic functional equation is said to be a cubic mapping. Now, we say that is called a (convex) modular algebra if the fundamental space is an algebra over K = R or C with (convex) modular subject to ( ) ≤ ( ) ( ) for all , ∈ . A subset of a convex modular algebra is called -complete if and only if any -Cauchy sequence in is -convergent to an element in . Throughout the paper, will be a -complete convex modular algebra and the symbol [ , ] will denote the commutator − . We say that a mapping is cubic homogeneous if ( ) = 3 ( ) for all vectors and all scalars , and a cubic homogeneous [23,24].
In this article, we first investigate generalized Hyers-Ulam stability of the equation in -complete convex modular algebras without using the Fatou property and Δ 2 -condition and then present alternatively generalized Hyers-Ulam stability of (6) using necessarily Δ 2 -condition without the Fatou property in -complete convex modular algebras.

Generalized Hyers-Ulam Stability of (6)
First of all, we remark that (6) is equivalent to the original cubic functional equation, and so every solution of (6) is a cubic mapping.
For notational convenience, we let the difference operators of cubic equation (6) and of cubic derivation be as follows: for all , in a linear space and ∈ Λ fl { ∈ C : | | = 1}. In the following, we present a generalized Hyers-Ulam stability via direct method of the system = 0 and = 0 in -complete convex modular algebras without using both the Fatou property and Δ 2 -condition.

Theorem 3. Suppose that a mapping :
→ satisfies for all , , ∈ and ∈ Λ. If for each ∈ the mapping → ( ) from R to is continuous, then there exists a unique cubic Lie derivation 1 : → which satisfies equation (6) and for all ∈ .
Proof. Putting = and = 1 in (8), we obtain which yields for all ∈ . Since ∑ −1 =0 (1/8 +1 ) ≤ 1, we prove the following functional inequality: for all ∈ by using the property of convex modular . Now, replacing by 2 in (13), we have which converges to zero as → ∞ by assumption (9). Thus the above inequality implies that the sequence { (2 )/2 3 } is -Cauchy for all ∈ and so it is convergent in since the space is -complete. Thus, we may define a mapping 1 : → as for all ∈ .
In fact, if we put ( , , ) fl (2 , 2 , 0) in (8) and then divide the resulting inequality by 2 3 , one obtains for all , ∈ , where ≥ 16| | + 3 is a fixed positive real. Thus we figure out by use of the first remark for all , ∈ , ∈ Λ and all positive integers . Taking the limit as → ∞, one obtains ((1/ ) 1 ( , )) = 0, and so 1 ( , ) = 0 for all , ∈ . Hence, taking = 1 in 1 ( , ) = 0, it follows that 1 satisfies (6) and so it is 4 Journal of Function Spaces cubic. On the other hand, since ∑ =0 (1/2 3( +1) + 1/2 3 ) ≤ 1 for all ∈ N, it follows from (12) and the first remark that without applying the Fatou property of the modular for all ∈ and all ∈ N, from which we obtain the approximation of by the cubic mapping 1 as follows: for all ∈ by taking → ∞ in the last inequality.
As a corollary of Theorem 3, we obtain the following stability result of cubic equation (6) associated with cubic Lie derivation on the Banach algebra , which may be considered as endowed with modular =‖ ⋅ ‖.
We observe that if the modular satisfies the Δ 2condition, then ≥ 1 for nontrivial modular , and ≥ 2 for nontrivial convex modular . See [13][14][15][16]. In the following theorem, we prove generalized Hyers-Ulam stability of the system = 0 and = 0 using necessarily Δ 2condition, which permits the existence of -Cauchy sequence in .
Theorem 5. Let be a -complete convex modular space with Δ 2 -condition. Suppose there exist two functions 1 , 2 : 2 → [0, ∞) for which a mapping : → satisfies for all , ∈ and ∈ Λ. If for each ∈ the mapping → ( ) from R to is continuous, then there exists a unique cubic Lie derivation 2 : → satisfying (6) and for all ∈ .
Journal of Function Spaces 7 for all , ∈ , from which 2 ( , ) = 0 by taking → ∞ and so 2 is a cubic Lie derivation. Claim 3. 2 is unique. To show the uniqueness of 2 , let us assume that there exists a cubic Lie derivation 2 : → which satisfies the approximation (28). Since 2 and 2 are cubic mappings, we see from the equalities 2 3 2 (2 − ) = 2 ( ) and 2 3 2 (2 − ) = 2 ( ) that which tends to zero as → ∞ for all ∈ . Hence the mapping 2 is a unique cubic Lie derivation satisfying (28).