ALMOST ADDITIVE-QUADRATIC-CUBIC MAPPINGS IN MODULAR SPACES

We introduce and obtain the general solution of a class of generalized mixed additive, quadratic and cubic functional equations. We investigate the stability of such modified functional equations in the modular space Xρ by applying the ∆2-condition and the Fatou property (in some results) on the modular function ρ. Furthermore, a counterexample for the even case (quadratic mapping) is presented.


Introduction
In 1940, Ulam raised the first stability problem before a Mathematical Colloquium at the University of Wisconsin, where he presented a list of unsolved problems that were published years later in the book [38]. He posed the question of whether there exists an exact homomorphism near an approximate homomorphism. An answer to the problem was partially given by Hyers [12] for linear functional equations in the setting of Banach spaces. Later on, various generalizations and extension of Hyers' result were ascertained by Aoki [1] for additive mappings, Th. M. Rassias [34] for linear mappings by considering an unbounded Cauchy difference, and also J. M. Rassias [32], Gajda [11] in different versions. Over the last seven decades, the Ulam problem was tackled by numerous authors, and its solutions via miscellaneous forms of functional equations like additive [37], quadratic [36,8], cubic [18,26], quartic [3,20], and mixed type functional equations [4,6,9,28,41] which involve only these types of functional equations were discussed.
The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is related to symmetric bi-additive functions. Since f (x) = cx 2 is its solution, it is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation is said to be a quadratic mapping.
In 2001, J. M. Rassias [33] introduced the cubic functional equation f (x + 2y) − 3f (x + y) + 3f (x) − f (x − y) = 6f (y) (1.1) and established the solution of the Hyers-Ulam stability problem for these cubic mappings. It is easy to see that the function f (x) = ax 3 satisfies (1.1). The alternative cubic functional equation has been introduced by Jun and Kim in [13]. They found out the general solution and established the Hyers-Ulam stability for the functional equation (1.2). Moreover, in [14] they introduced the cubic functional equation and investigated the Hyers-Ulam stability problem for it. In [7], Bodaghi et al. introduced a generalization of the cubic functional equation as follows: for an integer n ≥ 1. They determined the general solution and proved the Hyers-Ulam stability problem for the functional equation (1.3) related to cubic Jordan * -derivations. The stability of the equation (1.3) in various spaces is studied in [5]. The cubic functional equation which is the generalization form of the mentioned versions was given in [2] as follows: where m, n are integer numbers with m ≥ 2. Nakano [27] initiated the study of modulars on linear spaces and the related theory of modular linear spaces as generalizations of metric spaces. After that, Luxemburg [21], Mazur, Musielak and Orlicz [23,24,25] thoroughly developed it extensively. Since then, the theory of modulars and modular spaces has been widely applied in the study of interpolation theory [22,19] and various Orlicz spaces [29]. A modular yields less properties than a norm does, but it makes more sense in many special situations. When we work in a modular space, it is frequently assumed that the modular satisfies extra additional properties like some relaxed continuity or some ∆ 2 -condition. As for the mentioned condition, Khamsi [15] studied the stability of quasicontraction mappings in modular spaces without ∆ 2 -condition by using the fixed point theorem. The stability results of additive functional equations in modular spaces equipped with the Fatou property and ∆ 2condition were investigated by Sadeghi [35], who used Khamsi's fixed point theorem. In addition, the stability of quadratic functional equations in modular spaces satisfying the Fatou property without using the ∆ 2 -condition was investigated in [40]. Recently, Park, Bodaghi, and Kim [30] investigated the stability of additive and Jensen-additive functional equations without using the ∆ 2 -condition by a fixed point method. An alternative generalized Hyers-Ulam stability theorem of a modified quadratic functional equation in a modular space using the ∆ 3 -condition without the Fatou property on a modular function can be found in [16]. In addition, a refined stability result and alternative stability results for additive and quadratic functional equations using the direct way (Hyers' method) in modular spaces are given in [17].
In [39], Wang et al. obtained the general solution and investigated the Ulam stability problem for the following mixed additive, quadratic and cubic functional equation in Banach spaces. A different form of a mixed additive, quadratic and cubic functional equation which is introduced in [10] is as follows: where m is a fixed integer with m = 0, ±1. In this article, for two vector spaces X and Y , we consider the mapping f : X → Y which is a common solution of the functional equations for the fixed integer n and any integer m such that m, n = 0, ±1 and m + n = 0. We note that each element of this class of functional equations can be considered as a combination of (1.4) and (1.5) in a general form. It is easily verified that the function f (x) = αx 3 + βx 2 + γx is a solution of the functional equation (1.6) for all m, n with m, n = 0, ±1 and m + n = 0. We obtain the general solution and study the Hyers-Ulam-Rassias stability of the equation (1.6) in the setting of modular spaces.

Preliminary notations
In this section, we recall some basic facts concerning modular spaces and some preliminary results. Definition 2.1. Let X be a linear space over a field K (R or C). A generalized function ρ : X → [0, ∞] is called a modular if it satisfies the following three conditions for elements α, β ∈ K, x, y ∈ X: If the condition (iii) is replaced by ρ(αx+βy) ≤ α t ρ(x)+β t ρ(y) when α t +β t = 1 and α, β ≥ 0 with a t ∈ (0, 1], then ρ is called a t-convex modular. 1-convex modulars are called convex modulars. For a modular ρ, there corresponds a linear subspace X ρ of X, given by X ρ := {x ∈ X : ρ(λx) → 0 as λ → 0}. In this case X ρ is called a modular space.
Then, ρ ψ is a modular and the corresponding modular space is called an Orlicz space and denoted by One can check that L ψ is ρ ψ -complete.
A modular function ρ is said to satisfy the ∆ s -condition if there exists κ > 0 such that ρ(sx) ≤ κρ(x) for all x ∈ X ρ . Throughout this paper, we say that the constant κ is a ∆ s -constant related to the ∆ s -condition. Suppose that ρ is convex and satisfies the ∆ s -condition with ∆ s -constant κ. If κ < s, then ρ(x) ≤ κρ x s ≤ κ s ρ(x), which implies ρ = 0. Hence, we must have the ∆ s -constant κ ≥ s if ρ is convex modular. It is said that the modular ρ has the Fatou property if and only if ρ(x) ≤ lim inf n→∞ ρ(x n ) whenever the sequence {x n } is ρ-convergent to x in the modular space X ρ .

Solution of equation (1.6)
In this section, we find out the general solution of (1.6). Here, we consider the following mixed additive, quadratic and cubic functional equation for the fixed integer n such that n = 0, ±1. It is easy to check that the equation (3.1) is a generalized form of (1.4) and that f (x) = ax 3 + bx 2 + cx is a solution of it.
for fixed integer m with m = 0, ±1. It follows from (3.2) and our assumption that for all x, y ∈ X. Adding both sides of (3.3) and (3.4), we obtain (1.6).
In the sequel, by "the mapping f : X → Y satisfies the functional equation (1.6)", we mean that f satisfies (1.6) for the fixed integer n and any integer m such that m, n = 0, ±1 and m + n = 0. Proof. (i) By our assumption, the equation (1.6) can be rewritten as follows: for all x ∈ X. Putting m = n in (3.5) and using (3.6), we have for all x, y ∈ X. Since n = 1, it follows from (3.7) that f ( This means that f is a quadratic mapping. (ii) The hypothesis shows that the equation (1.6) is equivalent to Letting y = x and m = 2n in (3.8) and using the oddness of f , we get for all x ∈ X (and for the rest of this proof, all the equations are valid for all x ∈ X). Putting y = x and m = 3n in (3.8), we obtain It follows from (3.11) and (3.13) that (3.14) Since n = 0, the equalities (3.9) and (3.14) imply that Adding (3.12) to (3.16), we have From the relations (3.10) and (3.17), we arrive at From the relation (3.19), the desired results can be obtained.
Here and subsequently, given f : X → Y , for simplicity, we define the difference operator Λ m,n f : for all x, y ∈ X, the fixed integer n and any integer m such that m = 0, ±1, m + n = 0.

Stability of equation (1.6) in modular spaces
In this section, we prove the generalized Hyers-Ulam-Rassias stability of the mixed type additive-quadratic-cubic functional equation (1.6). From now on, we assume that V is a real vector space and X ρ is a complete modular space that satisfies the ∆ 2 -condition and has the Fatou property unless otherwise stated explicitly. In the next theorem we prove the stability of the functional equation (1.6) as a quadratic functional equation (the even case of (1.6)) in the modular spaces.
for all x, y ∈ V . Suppose that f : V → X ρ is an even mapping satisfying f (0) = 0 (when s = 1) and the inequality for all x, y ∈ V . Then, there exists a unique quadratic mapping Q : V → X ρ such that Proof. We first consider s = 1. Note that in this case f (0) = 0 is assumed.
Replacing (x, y, m) by (0, x, n) in (4.2), we get for all x ∈ V . Substituting (x, y, m) by (x, x, n) in (4.2), we obtain for all x ∈ V . The relation (4.6) can be modified as follows: for all x ∈ V . The relations (4.5) and (4.7) imply that for all x ∈ V . Also, it follows from (4.5) that for all x ∈ V . Plugging (4.8) into (4.9), we find for all x ∈ V , where Φ(x) is defined in (4.4). The inequality (4.10) implies that for all x ∈ V . Once more, by induction on k, one can prove the following functional inequality: for all x ∈ V . Now, changing x by 2 l x in (4.11), we have for all x ∈ V . Since the right-hand side of the above inequality tends to zero as l goes to infinity, the sequence f (2 k x) 4 k is a ρ-Cauchy sequence in X ρ and so the mentioned sequence is ρ-convergent on X ρ . Thus, we may define the map- for all x ∈ V . In other words, Replacing (x, y) by (2 k x, 2 k y) in (4.2), and dividing the resulting inequality by 4 k , we get for all x, y ∈ V . For the fixed and arbitrary integer m = 0, ±1, put t = 9 + 6|mn(m + n)| + 2(m 2 + n 2 ). By the property ρ(αt) ≤ αρ(t) when 0 < α ≤ 1, we

ALMOST ADDITIVE-QUADRATIC-CUBIC MAPPINGS IN MODULAR SPACES 367
have for all x, y ∈ V and all positive integers k. Taking the limit as k → ∞, we see that ρ 1 t Λ m,n Q(x, y) = 0, and hence Λ m,n Q(x, y) = 0 for all x, y ∈ V . This means that Q is a quadratic mapping. Now, it follows from the Fatou property of modular ρ that for all x ∈ V , which shows that the relation (4.3) holds. For the uniqueness of Q, we assume that there exists another quadratic mapping Q 0 : V → X ρ such that for all x ∈ V such that Q 0 (x * ) = Q(x * ) for some x * ∈ V . In other words, there is a positive constant δ > 0 such that ρ(Q 0 (x * ) − Q(x * )) > δ. On the other hand, there is a positive integer p 0 ∈ N such that 1 4 ∞ j=p0 Φ(2 j x) 4 j < δ. Since Q and Q 0 are quadratic mappings, we have Q 0 (2 p0 x) = 4 p0 Q 0 (x) and Q(2 p0 x) = 4 p0 Q(x).

So,
which is a contradiction. Now, assume that s = −1. Since κ ≥ 2, (4.1) implies that f (0) = 0. It follows from (4.10) that for all x ∈ V . By the convexity of the modular ρ, ∆ 2 -condition and (4.12), we have for all x ∈ V . Replacing x by x 2 l in (4.13), we get for all x ∈ V . It follows from (4.1) and (4.14) that the sequence This means that the mentioned sequence is ρconvergent to Q(x). Using the ∆ 2 -condition without applying the Fatou property, we obtain for all x ∈ V . Letting k → ∞, we see that (4.3) holds. The rest of the proof is similar to the case s = 1. This completes the proof.
The following corollaries are the direct consequences of Theorem 4.1 concerning the stability of (1.6). 2 . If f : V → X ρ is an even mapping satisfying for all x, y ∈ V , then there exists a unique quadratic mapping Q : V → X ρ such that Proof. We first note that f (0) = 0. Putting φ(x, y) = θ( x r + y r ) in Theorem 4.1, one can obtain the first and second inequalities for s = 1 and s = −1, respectively. 2 . If f : V → X ρ is an even mapping satisfying ρ(Λ m,n f (x, y)) ≤ θ x p y q for all x, y ∈ V , then there exists a unique quadratic mapping Q : V → X ρ such that Proof. Refer to the proof of Corollary 4.2. The idea of the following example is taken from [11]. We show the hypothesis r = 2 can not be removed in Corollary 4.2.
Suppose that the function f : R → R is defined through We have f (0) = 0 and f is an even function. Furthermore, ψ is continuous and bounded by µ. Since f is a uniformly convergent series of continuous functions, it is continuous and bounded. Indeed, for each x ∈ R, we have |f (x)| ≤ 4 3 µ. We wish to show that for all x, y ∈ R. Obviously, (4.15) holds for x = y = 0. Assume that x, y ∈ R with x 2 + y 2 < 1 16t 2 . Thus, there exists a positive integer N such that (4.16) So, |x| < 1 4t and |y| < 1 4t , and hence |x| + |y| < 1 2t . Therefore, Since ψ is quadratic on (−1, 1), by Proposition 3.2, |Λ m,n ψ(2 k x, 2 k y)| = 0 for all k ∈ {0, 1, 2, . . . , N − 1}. The last equality and the relation (4.16) imply that Therefore, f satisfies (4.15) for all x, y ∈ R. Now suppose, contrary to our claim, that there exists a number b ∈ [0, ∞) and a quadratic function Q : R → R such that |f (x) − Q(x)| < bx 2 for all x ∈ R. Hence, there is a constant c ∈ R such that Q(x) = cx 2 for all x ∈ R. So for all x ∈ R. On the other hand, consider p ∈ N such that (p + 1)µ > |c| + b. If x is a real number in 0, 1 2 N −1 , then 2 k x ∈ (0, 1) for all k = 0, 1, . . . , N − 1. So, for such x, we get The above relation contradicts (4.17).
We have the next result which is analogous to Theorem 4.1 for the functional equation (1.6) in the odd case.
Proof. Replacing (x, y, m) by (2x, x, n) in (4.18), we have for all x ∈ V . Changing (x, y, m) into (x, 2x, n) in (4.18) and using the oddness of f , we obtain for all x ∈ V . It follows from (4.20) and (4.21) that for all x ∈ V . Letting y = x and m = 2n in (4.18) and applying the oddness of f , we get for all x ∈ V . From the relations (4.22) and (4.23) we arrive at for all x ∈ V . Substituting (x, y, m) by (3x, x, n) in (4.18), we deduce that for all x ∈ V . Replacing (x, y, m) by (x, 3x, n) in (4.18), we find for all x ∈ V . Plugging (4.25) into (4.26), we have for all x ∈ V . Putting y = x and m = 3n in (4.18), we obtain for all x ∈ V . The relations (4.27) and (4.28) imply that for all x ∈ V . One can conclude from (4.24) and (4.29) that for all x ∈ V . From the relation (4.30), we have for all x ∈ V , where Γ(x) is defined in (4.19). We can write the relation (4.31) as . For the rest of the proof, one can repeat the process in the proof of Theorem 4.1 after relation (4.10) to obtain the desired result.
The following results are some consequences of Theorem 4.6 concerning the stability of (1.6) when f is an odd mapping. Since the proofs are similar to the proofs of Corollaries 4.2 and 4.3, we omit them. for all x, y ∈ V , then there exists a unique cubic mapping C : V → X ρ such that for all x ∈ V , where M is defined in (4.32).
Corollary 4.11. Let V be a normed space and X ρ a ρ-complete convex modular space. Assume θ, p, q > 0 and r = p + q with r = 3, log 2 . If f : V → X ρ is an odd mapping satisfying ρ(Λ m,n f (x, y)) ≤ θ x p y q for all x, y ∈ V , then there exists a unique cubic mapping C : V → X ρ such that for all x ∈ V , where N is defined in (4.33).
Here, by using Theorems 4.1, 4.6 and 4.9, we prove the generalized Hyers-Ulam-Rassias stability of the mixed type additive, quadratic and cubic functional equation (1.6) when f is an arbitrary mapping. for all x, y ∈ V . Suppose that f : V → X ρ is a mapping satisfying f (0) = 0 (when s = 1) and the inequality for all x, y ∈ V . Then, there exists a unique additive mapping A : V → X ρ , a unique quadratic mapping Q : V → X ρ , and a unique cubic mapping C : V → X ρ such that for all x ∈ V . The relation (4.41) implies that for all x ∈ V . Now, by (4.38) and (4.42) we can obtain the inequality (4.34), where A(x) = − 1 6 A 0 (x) and C(x) = 1 6 C 0 (x).
The following corollaries are the direct consequences of Theorem 4.1 concerning the stability of (1.6). for all x, y ∈ V , then there exist a unique additive mapping A : V → X ρ , a unique quadratic mapping Q : V → X ρ , and a unique cubic mapping C : V → X ρ such that 4n 2 |n−1|(4−2 r ) + 1 12|n| 3 (8−2 r ) θMκ x r , 0 < r < 1,   Corollary 4.14. Let V be a normed space and X ρ a ρ-complete convex modular space. Assume θ, p, q > 0 and r = p + q with r = t, log κ 2t 2 t 2 when t ∈ {1, 2, 3}. If f : V → X ρ is a mapping satisfying ρ(Λ m,n f (x, y)) ≤ θ x p y q for all x, y ∈ V , then there exist a unique additive mapping A : V → X ρ , a unique quadratic mapping Q : V → X ρ , and a unique cubic mapping C :  for all x ∈ V , where N is defined in (4.33).