Definition 1 (see [29].)Let V be a vector space over $$(ℂ or ℝ). A generalized functional ρ : V⟶[0, ∞] is called a modular if for arbitrary u, v ∈ V, ρ satisfies

• (a)

  ρ(u) = 0 if and only if u = 0

• (b)

  ρ(βu) = ρ(u) for every scalar β with |β| = 1

• (c)

  ρ(βu + γv) ≤ ρ(u) + ρ(v), whenever β, γ ≥ 0 and β + γ = 1

If we replace (c) by.

(c’) ρ(βu + γv) ≤ βρ(u) + γρ(v), whenever β, γ ≥ 0 and β + γ = 1, then, the modular ρ is called convex. A modular ρ defines a corresponding modular space, i.e., the vector space V_ρ given by:

Definition 2 (see [29].)If V_ρ is a modular space and the sequence {v_n} in V_ρ, then

• (i)

  $$ if ρ(v_n − v)⟶0 as n⟶∞

• (ii)

  {v_n} is known as ρ-Cauchy if ρ(v_l − v_n)⟶0 as l, n⟶∞

• (iii)

  A subset A⊆V_ρ is known as ρ-complete iff every ρ-Cauchy sequence is ρ-convergent in A

Definition 3 (see [29].)Let V_ρ be a modular space and a non-empty subset A⊆V_ρ. The mapping J : A⟶A is referred to as a quasicontraction, if there is k < 1 satisfies

for any l, m ∈ A.

Definition 4 (see [29].)Let V_ρ be a modular space, a nonempty subset A⊆V_ρ, and a function J : A⟶A, the J orbit around a point v is

the quantity is then related to J and is referred to as the orbital diameter of J at v. If Y_ρ(J) < ∞, in particular, one says that J has an orbit of v that is limited to v.

Theorem 5. If an even mapping ϕ : V⟶W satisfies the functional equation (5) for all v₁, v₂, ⋯, v_n ∈ V, then, the function ϕ is quartic.

Proof. In the view of evenness, we obtain ϕ(−v) = ϕ(v) for all v in V. Now, setting v₁ = v₂ = ⋯ = v_n = 0 in equation (5), we have ϕ(0) = 0. Replacing (v₁, v₂, ⋯, v_n) by (v, 0, ⋯, 0) in equation (5), we get

for all v ∈ V. Replacing v by 3v in equation (6), we have for all v ∈ V. Again, replacing v by 3v in equation (7), we obtain for all v ∈ V. For any nonnegative integer n ≥ 0, we can generalize the result that for all v ∈ V. Similarly, we have for all v ∈ V. Next, replacing (v₁, v₂, ⋯, v_n) by (v₁, v₁, v₂, 0, ⋯, 0), we obtain for all v₁, v₂ ∈ V. Hence, the function ϕ is quartic.

Theorem 6. Let ψ : Vⁿ⟶[0, +∞) be a function such that

for all v₁, v₂, ⋯, v_n ∈ V, with L < 1. If an even mapping ϕ : V⟶W_ρ with ϕ(0) = 0 and such that for all v₁, v₂, ⋯, v_n ∈ V, then, there exists a unique quartic mapping Q₄ : V⟶W_ρ satisfying for all v ∈ V.

Proof. We consider the set

and define the function $$ on Λ as follows:

Now, we show that $$ is a convex modular on Λ. It is easy to verify that $$ satisfies the axioms (a) and (b) of a modular. Next, we will show that $$ is convex, and hence, (c^′) is satisfied. Let ε > 0 be given, then, there exist real constants θ₁ > 0 and θ₂ > 0 such that

Also

for all v ∈ V. If β + γ = 1 and β, γ ≥ 0, then, we get so we get

This concludes that $$ is convex modular on Λ. Now, we show that $$ is $$-complete.

Let {p_n} is a $$-Cauchy sequence in $$ and let ε > 0. Then, there exists a positive integer n₀ ∈ ℕ such that

for all n, m ≥ n₀. Then for all v ∈ V and all n, m ≥ n₀. Therefore, {p_n(v)} is a ρ-Cauchy sequence in W_ρ. Since W_ρ is ρ-complete, so {p_n(v)} is convergent in W_ρ, for each v ∈ V. Hence, we can define a function p : V⟶W_ρ by for all v ∈ V. Since ρ satisfies the Fatou property, it follows from (25) that so for all n ≥ n₀. Thus, {p_n} is $$-converges. Hence, $$ is $$-complete.

Next, we show that $$ satisfies the Fatou property. Suppose that {p_n} is a sequence in $$ which is $$-convergent to an element $$.

Let ε > 0 be given. For each n ∈ ℕ, let θ_n be a real constant such that

So

for all v ∈ V. Since ρ satisfies the Fatou property, we get

Thus, we obtain

Hence, $$ satisfies the Fatou property. Consider the function $$ by

for all v ∈ V and all $$. Let $$ and let θ ∈ [0, 1] be an arbitrary constant with $$. From the definition of $$, we obtain for all v ∈ V. By inequality (15) and the above inequality, we get for all v ∈ V. Hence, i.e., Ψ is a $$-contraction. Next, we show that Ψ has a bounded orbit at ϕ. Replacing (v₁, v₂, ⋯, v_n) by (v, 0, ⋯, 0) in (16), we get for all v ∈ V. Replacing v with 3v in (37), we get for all v ∈ V. By using (37) and (38), we get for all v ∈ V. By induction, we can easily see that for all v ∈ V. It follows from inequality (40) that for all v ∈ V and all n, m ∈ ℕ. By the definition of $$, we conclude that which implies the boundedness of an orbit of Ψ at ϕ. It follows from Theorem 1.5 [29] that the sequence {Ψⁿϕ}$$-converges to $$. Now, by the $$-contractivity of Ψ, we have

Passing to the limit n⟶∞ and applying the Fatou property of $$, we obtain that

Therefore, Q₄ is a fixed point of Ψ. Replacing (v₁, v₂, ⋯, v_n) by (3^lv₁, 3^lv₂, ⋯, 3^lv_n) in (16), we get

for all v₁, v₂, ⋯, v_n ∈ V. Therefore

Employing the limit l⟶∞, we get

for all v₁, v₂, ⋯, v_n ∈ V. It follows from Theorem 2 that Q₄ is quartic. By using (40), we get (17).

To prove the uniqueness of Q₄, let $$ be another quartic mapping satisfying (17). Then, Q4^′ is a fixed point of Ψ.

which implies that $$. This proves that $$. Therefore, the function Q₄ is unique. This completes the proof.

Corollary 7. Let a mapping ψ : Vⁿ⟶[0, +∞) such that

for all v₁, v₂, ⋯, v_n ∈ V with L < 1. Suppose that a mapping ϕ : V⟶W with ϕ(0) = 0 and such that for all v₁, v₂, ⋯, v_n ∈ V, then, there exists a unique quartic mapping Q₄ : V⟶W satisfying for all v ∈ V.

Proof. It is known that every normed space is modular space with the modular ρ(v) = ∥v∥ and satisfies the Δ₃-condition with k = 3.

Remark 8. If we replace ψ(v₁, v₂, ⋯, v_n) by $$ and letting L = 3^p−4 in Corollary 7, we obtain the stability results for the sum of norms that

for all v ∈ V, where α and p are constants with p < 4.

Remark 9. If we replace ψ(v₁, v₂, ⋯, v_n) by $$ and letting L = 3^np−4 in Corollary 7, we obtain the stability results for the sum of product of norms that

for all v ∈ V, where α and p are constants with np < 4.

Theorem 10. Let ψ : Vⁿ⟶[0, +∞) be a function such that

for all v₁, v₂, ⋯, v_n ∈ V with L < 1. Suppose that ϕ : V⟶W_ρ with ϕ(0) = 0 and satisfies (16), then, there exists a unique quartic mapping Q₄ : V⟶W_ρ satisfying for all v ∈ V.

Proof. We consider the set

and define the function $$ on Λ as follows:

Similar to the proof of Theorem 6, we have

• (1)

  $$ is a convex modular on Λ

• (2)

  $$ is $$-complete

• (3)

  $$ satisfies the Fatou property

Now, we consider the function $$ defined by

for all v ∈ V and all $$. Let $$ and let θ ∈ [0, 1] be an arbitrary constant with $$. From the definition of $$, we have for all v ∈ V. By the assumption and the last inequality, we get

Hence,

i.e., Ψ is a $$-contraction.

Next, we prove then that Ψ has a bounded orbit at ϕ. Replacing (v₁, v₂, ⋯, v_n) by (v, 0, ⋯, 0) in (16), we get

for all v ∈ V. Replacing v with v/3 in (64), we get for all v ∈ V. Replacing v with v/3 in (65), we get for all v ∈ V. By using (64), (65), and (66), we get for all v ∈ V. By induction, we can easily see that for all v ∈ V. It follows from inequality (68) that for all v ∈ V and all n, m ∈ ℕ. By the definition of $$, we conclude that which implies the boundedness of an orbit of Ψ at ϕ. It follows from Theorem 1.5 [29] that the sequence {Ψⁿϕ}$$-converges to $$.

Now, by the $$-contractivity of Ψ, we have

Employing the limit n⟶∞ and applying the Fatou property of $$, we obtain that

Therefore, Q₄ is a fixed point of Ψ. Replacing (v₁, v₂, ⋯, v_n) by (v₁/3^l, v₂/3^l, ⋯, v_n/3^l) in (16), we get

for all v₁, v₂, ⋯, v_n ∈ V. Therefore

Passing to the limit l⟶∞, we get

for all v₁, v₂, ⋯, v_n ∈ V. It follows from Theorem 2 that Q₄ is quartic. By using (68), we get (57).

In order to prove the uniqueness of Q₄, consider another quartic solution $$ that satisfy the inequality (17). Then, $$ is a fixed point of Ψ.

which implies that $$ or $$. Hence, the proof is now completed.

Corollary 11. Let a mapping ψ : Vⁿ⟶[0, +∞) such that

for all v₁, v₂, ⋯, v_n ∈ V, with L < 1. Suppose that ϕ : V⟶W with ϕ(0) = 0 and satisfies (51), then there exists a unique quartic mapping Q₄ : V⟶W satisfying for all v ∈ V.

Proof. It is known that every normed space is modular space with the modular ρ(v) = ∥v∥ and satisfies the Δ₃-condition with k = 3.

Remark 12. If we replace ψ(v₁, v₂, ⋯, v_n) by $$ and letting L = 3^4−p in Corollary 11, we obtain the stability results for the sum of norms that

for all v ∈ V, where α and p are constants with p > 4.

Remark 13. If we replace ψ(v₁, v₂, ⋯, v_n) by $$ and letting L = 3^4−np in Corollary 11, we obtain the stability results for the sum of product of norms that

for all v ∈ V, where α and p are constants with np > 4.

Remark 14. If a function ϕ : ℝ⟶V satisfies the functional equation (5), then, the following assertions hold:

• (1)

  ϕ(q^k/4v) = q^kϕ(v), q ∈ ℚ, k ∈ ℤ and v ∈ ℝ

• (2)

  ϕ(v) = v⁴ϕ(1), v ∈ ℝ if the function ϕ is continuous

Example 15. Let a mapping ϕ : ℝ⟶ℝ be defined as follows:

where then, the mapping ϕ : ℝ⟶ℝ satisfies for all v₁, v₂, ⋯, v_n ∈ ℝ, but a quartic mapping Q₄ : ℝ⟶ℝ does not exist satisfies for all v ∈ ℝ, where θ and ε are a constant.

Proof. It is easy to show that ϕ is bounded by 81/80θ on ℝ. If $$ or 0, then

Thus, (84) is valid. Next, suppose that

then, there exists an integer m > 0 satisfies

So that 3^4m|v₁| < 1/3⁴, 3^4m|v₂| < 1/3⁴, ⋯, 3^4m|v_n| < 1/3⁴ and

Also, for a = 0, 1, ⋯, m − 1,

Next, by inequality (88), we obtain that

It follows from (88) that

Thus, the function ϕ satisfies the inequality (84). Assume on a contrary that there exist a quartic solution Q₄ : ℝ⟶ℝ satisfying (85). For every v in ℝ, since ϕ is continuous and bounded, Q₄ is limited to an open interval of origin and continuous origin.

In the view of Remark 14, Q₄ must be Q₄(v) = cv⁴, v ∈ ℝ. So we obtain

Suppose, we can choose m > 0 with mθ > ε + |c|. If v ∈ (0, 1/3^m−1), then, 3^av ∈ (0, 1) for all a = 0, 1, ⋯, m − 1, we obtain

which contradicts.