Throughout this paper, we assume that V and W are real vector spaces and k is a real number satisfying k ∉ {−1, 0, 1} unless there are specifications for them. For any given mapping f : V → W, we will set

for all x, y, z ∈ V.

Every solution of functional equation

is called an additive mapping, and each solution of functional equation

is called a quadratic mapping, while every solution of functional equation

is called a cubic mapping.

If a mapping can be expressed by the sum of an additive mapping, a quadratic mapping, and a cubic mapping, then we call the mapping a cubic-quadratic-additive mapping. If a mapping can be expressed by the sum of a constant, an additive mapping, a quadratic mapping, and a cubic mapping, then we call the mapping a general cubic mapping. A functional equation is called a cubic-quadratic-additive type functional equation if each solution of that equation is a cubic-quadratic-additive mapping. A functional equation is called a general cubic type functional equation when each of its solutions is a general cubic mapping.

In 1940, Ulam [13] raised an important problem concerning the stability of group homomorphisms: Under what conditions is the approximate solution of an equation necessarily close to the exact solution of the equation? Just the following year, Hyers [8] solved the problem of Ulam only in the case of the Cauchy additive functional equation Af(x, y) = f(x + y) − f(x) − f(y) = 0. Indeed, Hyers proved the following statement for any previously given constant ɛ > 0: every solution of inequality ||Af(x, y)|| ≤ ɛ (for all x and y) can be approximated by an exact solution (an additive function). In this case, the Cauchy additive functional equation is said to satisfy the Hyers-Ulam stability.

About three decades later, Rassias [12] generalized Hyers’ result and then Găvruta [7] extended Rassias’ result by allowing unbounded control functions. The concept of stability introduced by Rassias and Găvruta is known today as the generalized Hyers-Ulam stability of functional equations.

Jun et al. [9] and Lee [10] investigated the stability of the general cubic functional equation D₃f(x, y) = 0 for k = 2. Independently from them, Gordji [3] investigated the stability of the cubic-quadratic-additive functional equation D₅f(x, y) = 0. Moreover, Gordji et al. [2, 3, 4, 5, 6] investigated the stability of the general cubic functional equation D₃f(x, y) = 0 for any integer k ∉ {−1, 0, 1}. However, they could not prove the uniqueness of the exact solution because they divided the related function into even and odd parts and proved their stability separately, while we prove in this paper the stability in an integrated way. At the same time, we prove the uniqueness of the exact solution. This is an advantage of this paper in comparison with the papers [2, 3, 4, 5, 6] of other mathematicians.

In this paper, we will prove the generalized Hyers-Ulam stability of the functional equations D_mf(x, y) = 0 for m ∈ {1, 2, 3, 4, 5} and D_mf(x, y, z) = 0 for m ∈ {6, 7}.

The following theorem is a special version of Baker’s theorem when δ = 0 (refer to [1]).

Theorem 2.1. ([1, Theorem 1])

Given an m ∈ ℕ, assume that V and W are vector spaces over ℚ, ℝ or ℂ and that α₀, β₀, . . . , α_m, β_m are scalars satisfying α_jβ_ℓ − α_ℓβ_j ≠ 0 whenever 0 ≤ j < ℓ ≤ m. If the functions f_ℓ : V → W, ℓ ∈ {0, 1, . . . , m}, satisfy the equation

for all x, y ∈ V, then each f_ℓ is a generalized polynomial mapping of degree at most m − 1.

Baker [1] also states that if f : V → W is a generalized polynomial mapping of degree at most m − 1, then f can be expressed as f(x)=x0+∑ℓ=1m-1aℓ*(x) for x ∈ V, where aℓ* is a monomial mapping of degree ℓ and f has a property f(rx)=x0+∑ℓ=1m-1rℓaℓ*(x) for x ∈ V and r ∈ ℚ. The monomial mapping of degree 1, 2 and 3 are called an additive mapping, a quadric mapping, and a cubic mapping, respectively. The generalized polynomial mapping of degree 1, 2 and 3, on the other hand, are called a Jensen mapping, a general quadric mapping, and a general cubic mapping, respectively.

In summary, the following corollary can be obtained from Baker’s theorem.

Corollary 2.2

Let V and W be vector spaces over ℚ, ℝ or ℂ, and let r be a rational number satisfying r ∉ {−1, 0, 1}. Given an m ∈ ℕ, assume that n₁, . . . , n_m are positive integers and that c_ℓ,i, d_ℓ,i, α₀, β₀, . . . , α_m, β_m (ℓ ∈ {1, . . . , m} and i ∈ {1, . . . , n_ℓ}) are scalars satisfying α_jβ_ℓ − α_ℓβ_j ≠ 0 whenever 0 ≤ j < ℓ ≤ m. If a mapping f : V → W satisfies the equation f(rx) = r^kf(x) for all x ∈ V and the equation

for all x, y ∈ V, then f is a monomial mapping of degree k.

Lemma 2.3

Let V be a real vector space and let (Y, || · ||) be a real Banach space. Given an m ∈ {1, 2, 3, 4, 5}, assume that a mapping f : V → Y satisfies f(0) = 0 and inequality

for all x, y ∈ V. Then the following inequalities

hold for all x ∈ V, where μ_m, ν_m : V → ℝ are defined by

Lemma 2.4

Given m ∈ {1, 2, 3, 4, 5}, assume that a mapping f : V → Y satisfies f(0) = 0 and D_mf(x, y) = 0 for all x, y ∈ V. Then the equalities

are true for all x ∈ V, where

Theorem 2.5

Let m ∈ {1, 2, 3, 4, 5} be fixed and let ϕ : V × V → [0,∞) be a function satisfying the condition

for all x, y ∈ V. If a mapping f : V → Y satisfies f(0) = 0 and inequality(2.1)for all x, y ∈ V, then there exists a unique mapping F : V → Y satisfying

for all x, y ∈ V and

for all x ∈ V.

Theorem 2.6

Let m ∈ {1, 2, 3, 4, 5} be fixed and let ϕ : V × V → [0,∞) be a function satisfying the condition

for all x, y ∈ V. If a mapping f : V → Y satisfies f(0) = 0 and inequality(2.1)for all x, y ∈ V, then there exists a unique mapping F : V → Y satisfying equality(2.6)for all x, y ∈ V and

for all x ∈ V.

Theorem 2.7

Let m ∈ {1, 2, 3, 4, 5} be fixed and let ϕ : V × V → [0,∞) be a function satisfying the conditions

for all x, y ∈ V. If a mapping f : V → Y satisfies f(0) = 0 and inequality(2.1)for all x, y ∈ V, then there exists a unique mapping F : V → Y satisfying equality(2.6)for all x, y ∈ V and

for all x ∈ V.

Theorem 2.8

Let m ∈ {1, 2, 3, 4, 5} be fixed and let ϕ : V × V → [0,∞) be a function satisfying the conditions

for all x, y ∈ V. If a mapping f : V → Y satisfies f(0) = 0 and inequality(2.1)for all x, y ∈ V, then there exists a unique mapping F : V → Y satisfying equality(2.6)for all x, y ∈ V as well as inequality

for all x ∈ V.

Lemma 2.9

Given an m ∈ {6, 7} and a function ϕ : V³ → [0,∞), assume that a mapping f : V → Y satisfies f(0) = 0 and

for all x, y, z ∈ V. Then inequalities of(2.2)are true for all x ∈ V, where μ_m, ν_m : V → ℝ are defined by

for all x ∈ V.

Lemma 2.10

Given an m ∈ {6, 7} and a function ϕ : V³ → [0,∞), assume that a mapping f : V → Y satisfies f(0) = 0 and D_mf(x, y, z) = 0 for all x, y, z ∈ V. Then equalities in(2.4)are true for all x ∈ V.

Theorem 2.11

Let m ∈ {6, 7} be fixed and let ϕ : V³ → [0,∞) be a function satisfying the condition

for all x, y, z ∈ V. If a mapping f : V → Y satisfies f(0) = 0 and inequality(2.14)for all x, y, z ∈ V, then there exists a unique mapping F : V → Y satisfying

for all x, y, z ∈ V as well as inequality(2.7)for all x ∈ V.

Again, Y is a real Banach space.

Theorem 2.12

Let m ∈ {6, 7} be fixed and let ϕ : V³ → [0,∞) be a function satisfying the condition

for all x, y, z ∈ V. If a mapping f : V → Y satisfies f(0) = 0 and inequality(2.14)for all x, y, z ∈ V, then there exists a unique mapping F : V → Y satisfying equality(2.16)for all x, y, z ∈ V and inequality(2.9)for all x ∈ V.

Theorem 2.13

Let m ∈ {6, 7} be fixed and let ϕ : V³ → [0,∞) be a function satisfying the conditions

for all x, y, z ∈ V. If a mapping f : V → Y satisfies f(0) = 0 and inequality(2.14)for all x, y, z ∈ V, then there exists a unique mapping F : V → Y satisfying equality(2.16)for all x, y, z ∈ V and inequality(2.11)for all x ∈ V.

Theorem 2.14

Let m ∈ {6, 7} be fixed and let ϕ : V³ → [0,∞) be a function satisfying the conditions

for all x, y, z ∈ V. If a mapping f : V → Y satisfies f(0) = 0 and inequality(2.14)for all x, y, z ∈ V, then there exists a unique mapping F : V → Y satisfying equality(2.16)for all x, y ∈ V and inequality(2.13)for all x ∈ V.

Subjects similar to those covered in this paper have been studied previously (see [3, 5, 6, 4, 2]). However, in these works, the proof of the uniqueness of the exact solution was not possible, because the stability of the related functions was proved separately for the even and odd parts. We feel that the uniqueness of the solution is important, and that the division into even and odd parts is somewhat unnatural.

In this paper, we were able to prove the stability of the related function in an integrated way, without dividing the related function into even and odd parts. This allows us to prove the uniqueness of the exact solution. We see this a significant improvement over the results of [3, 5, 6, 4, 2].

Because of space constraints, let us look at only one example that uses the results of this paper. If we put m = 7, n = 2, c₁ = 1, c₂ = −1, c₃ = −2, c₄ = 2, c₅ = 6, c₆ = 2, c₇ = −2, a₁₁ = 1, a₁₂ = 2, a₂₁ = 1, a₂₂ = −2, a₃₁ = 1, a₃₂ = 1, a₄₁ = 1, a₄₂ = −1, a₅₁ = 0, a₅₂ = 1, a₆₁ = 0, a₆₂ = −1, a₇₁ = 0, a₇₂ = 2 in (1.1) in [11], then the expression (1.1) in [11] becomes D₂f(x, y) in this paper. We have proved the (generalized) Hyers-Ulam stability of the functional equation D₂f(x, y) = 0 in Theorem 2.5 of this paper. If we set ϕ(x, y) = ɛ > 0, then it follows from Theorem 2.5 that there exists a unique function F : V → Y satisfying D₂f(x, y) = 0 for all x, y ∈ V and

for all x ∈ V. In the text of Lemma 2.3, we can see the definitions of μ₂ and ν₂.