On the Hyers-Ulam Stability of a General Mixed Additive and Cubic Functional Equation in n-Banach Spaces

Abstract

The objective of the present paper is to determine the generalized Hyers-Ulam stability of the mixed additive-cubic functional equation in n-Banach spaces by the direct method. In addition, we show under some suitable conditions that an approximately mixed additive-cubic function can be approximated by a mixed additive and cubic mapping.

1. Introduction and Preliminaries

A basic question in the theory of functional equations is as follows: when is it true that a function, which approximately satisfies a functional equation, must be close to an exact solution of the equation?

If the problem accepts a unique solution, we say the equation is stable (see [1]). The study of stability problems for functional equations is related to a question of Ulam [2] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [3]. The result of Hyers was generalized by Aoki [4] for approximate additive mappings and by Rassias [5] for approximate linear mappings by allowing the Cauchy difference operator CD𝑓(𝑥,𝑦)=𝑓(𝑥+𝑦)−[𝑓(𝑥)+𝑓(𝑦)] to be controlled by 𝜖(‖𝑥‖^𝑝+‖𝑦‖^𝑝). In 1994, a generalization of Rassias’ theorem was obtained by Găvruţa [6], who replaced 𝜖(‖𝑥‖^𝑝+‖𝑦‖^𝑝) by a general control function 𝜑(𝑥,𝑦). On the other hand, several further interesting discussions, modifications, extensions, and generalizations of the original problem of Ulam have been proposed (see, e.g. [7–12] and the references therein).

Recently, Park [9] investigated the approximate additive mappings, approximate Jensen mappings, and approximate quadratic mappings in 2-Banach spaces and proved the generalized Hyers-Ulam stability of the Cauchy functional equation, the Jensen functional equation, and the quadratic functional equation in 2-Banach spaces. This is the first result for the stability problem of functional equations in 2-Banach spaces.

In [11, 12], we introduced the following mixed additive-cubic functional equation for fixed integers 𝑘 with 𝑘≠0, ±1:𝑓𝑘𝑥+𝑦+𝑓𝑘𝑥−𝑦=𝑘𝑓𝑥+𝑦+𝑘𝑓𝑥−𝑦+2𝑓(𝑘𝑥)−2𝑘𝑓(𝑥), (1.1) with 𝑓(0)=0, and investigated the generalized Hyers-Ulam stability of (1.1) in quasi-Banach spaces and non-Archimedean fuzzy normed spaces, respectively.

In this paper, we investigate, approximate mixed additive-cubic mappings in 𝑛-Banach spaces. That is, we prove the generalized Hyers-Ulam stability of a general mixed additive-cubic equation (1.1) in 𝑛-Banach spaces by the direct method.

The concept of 2-normed spaces was initially developed by Gähler [13, 14] in the middle of 1960s, while that of 𝑛-normed spaces can be found in [15, 16]. Since then, many others have studied this concept and obtained various results; see for instance [15, 17–19].

We recall some basic facts concerning 𝑛-normed spaces and some preliminary results.

Definition 1.1. Let 𝑛∈ℕ, and let 𝑋 be a real linear space with dim 𝑋≥𝑛 and ‖⋅,…,⋅‖∶𝑋^𝑛→ℝ a function satisfying the following properties:(N1)‖𝑥₁,𝑥₂,…,𝑥_𝑛‖=0 if and only if 𝑥₁,𝑥₂,…,𝑥_𝑛 are linearly dependent,(N2)‖𝑥₁,𝑥₂,…,𝑥_𝑛‖ is invariant under permutation,(N3)‖𝛼𝑥₁,𝑥₂,…,𝑥_𝑛‖=|𝛼|‖𝑥₁,𝑥₂,…,𝑥_𝑛‖,(N4)‖𝑥+𝑦,𝑥₂,…,𝑥_𝑛‖≤‖𝑥,𝑥₂,…,𝑥_𝑛‖+‖𝑦,𝑥₂,…,𝑥_𝑛‖ for all 𝛼∈ℝ and 𝑥,𝑦,𝑥₁,𝑥₂,…,𝑥_𝑛∈X. Then the function ‖⋅,…,⋅‖ is called an 𝑛-norm on 𝑋 and the pair (𝑋,‖⋅,…,⋅‖) is called an 𝑛-normed space.

Example 1.2. For 𝑥₁,𝑥₂,…,𝑥_𝑛∈ℝ^𝑛, the Euclidean 𝑛-norm ‖𝑥₁,𝑥₂,…,𝑥_𝑛‖_𝐸 is defined by 𝑥₁,𝑥₂,…,𝑥_𝑛𝐸=det𝑥_𝑖𝑗𝑥=abs¹¹⋯𝑥_1𝑛𝑥⋮⋱⋮_𝑛1⋯𝑥_𝑛𝑛, (1.2) where 𝑥_𝑖=(𝑥_𝑖1,…,𝑥_𝑖𝑛)∈ℝ^𝑛 for each 𝑖=1,2,…,𝑛.

Example 1.3. The standard 𝑛-norm on 𝑋, a real inner product space of dimension dim 𝑋≥𝑛, is as follows: 𝑥₁,𝑥₂,…,𝑥_𝑛𝑆=𝑥₁,𝑥₁⋯𝑥₁,𝑥_𝑛𝑥⋮⋱⋮_𝑛,𝑥₁⋯𝑥_𝑛,𝑥_𝑛^1/2, (1.3) where ⟨⋅,⋅⟩ denotes the inner product on 𝑋. If 𝑋=ℝ^𝑛, then this 𝑛-norm is exactly the same as the Euclidean 𝑛-norm ‖𝑥₁,𝑥₂,…,𝑥_𝑛‖_𝐸 mentioned earlier. For 𝑛=1, this 𝑛-norm is the usual norm ‖𝑥₁‖=⟨𝑥₁,𝑥₁⟩^1/2.

Definition 1.4. A sequence {𝑥_𝑘} in an 𝑛-normed space 𝑋 is said to converge to some 𝑥∈𝑋 in the 𝑛-norm if lim_𝑘→∞𝑥_𝑘−𝑥,𝑦₂,…,𝑦_𝑛=0, (1.4) for every 𝑦₂,…,𝑦_𝑛∈𝑋.

Definition 1.5. A sequence {𝑥_𝑘} in an 𝑛-normed space 𝑋 is said to be a Cauchy sequence with respect to the 𝑛-norm if lim_{𝑘,𝑙→∞}𝑥_𝑘−𝑥_𝑙,𝑦₂,…,𝑦_𝑛=0, (1.5) for every 𝑦₂,…,𝑦_𝑛∈𝑋. If every Cauchy sequence in 𝑋 converges to some 𝑥∈𝑋, then 𝑋 is said to be complete with respect to the 𝑛-norm. Any complete 𝑛-normed space is said to be an 𝑛-Banach space.

Now we state the following results as lemma (see [9] for the details).

Lemma 1.6. Let 𝑋 be an 𝑛-normed space. Then,(1)For 𝑥_𝑖∈𝑋(𝑖=1,…,𝑛) and 𝛾, a real number, 𝑥₁,…,𝑥_𝑖,…,𝑥_𝑗,…,𝑥_𝑛=𝑥₁,…,𝑥_𝑖,…,𝑥_𝑗+𝛾𝑥_𝑖,…,𝑥_𝑛 (1.6) for all 1≤𝑖≠𝑗≤𝑛,(2)|‖𝑥,𝑦₂,…,𝑦_𝑛‖−‖𝑦,𝑦₂,…,𝑦_𝑛‖|≤‖𝑥−𝑦,𝑦₂,…,𝑦_𝑛‖ for all 𝑥,𝑦,𝑦₂,…,𝑦_𝑛∈𝑋,(3)if ‖𝑥,𝑦₂,…,𝑦_𝑛‖=0 for all 𝑦₂,…,𝑦_𝑛∈𝑋, then 𝑥=0,(4)for a convergent sequence {𝑥_𝑗} in 𝑋, lim_𝑗→∞𝑥_𝑗,𝑦₂,…,𝑦_𝑛=lim_𝑗→∞𝑥_𝑗,𝑦₂,…,𝑦_𝑛 (1.7) for all 𝑦₂,…,𝑦_𝑛∈𝑋.

2. Approximate Mixed Additive-Cubic Mappings

In this section, we investigate the generalized Hyers-Ulam stability of the generalized mixed additive-cubic functional equation in 𝑛-Banach spaces. Let 𝑋 be a linear space and 𝑌 an 𝑛-Banach space. For convenience, we use the following abbreviation for a given mapping 𝑓∶𝑋→𝑌:𝐷𝑓𝑥,𝑦∶=𝑓𝑘𝑥+𝑦+𝑓𝑘𝑥−𝑦−𝑘𝑓𝑥+𝑦−𝑘𝑓𝑥−𝑦−2𝑓(𝑘𝑥)+2𝑘𝑓(𝑥) (2.1) for all 𝑥,𝑦∈𝑋.

Theorem 2.1. Let 𝑋 be a linear space and 𝑌 an 𝑛-Banach space. Let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 for which there is a function 𝜑∶𝑋^𝑛+1→[0,∞) such that ^∞_𝑗=012^𝑗𝜑2^𝑗𝑥,2^𝑗𝑦,𝑢₂,…,𝑢_𝑛<∞,𝐷𝑓(𝑥,𝑦),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,𝑦,𝑢₂,…,𝑢_𝑛 (2.2)(2.3) for all 𝑥,𝑦,𝑢₂,…,𝑢_𝑛∈𝑋. Then, there is a unique additive mapping 𝐴∶𝑋→𝑌 such that 𝑓(2𝑥)−8𝑓(𝑥)−𝐴(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤^∞_𝑗=012^𝑗+1𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛 (2.4) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋, where 𝜑𝑥,𝑢₂,…,𝑢_𝑛1∶=𝑘³𝑘|𝜑()−𝑘(|+1)𝑥,2𝑘+1𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,(2𝑘−1)𝑥,𝑢₂,…,𝑢_𝑛+𝜑3𝑥,𝑥,𝑢₂,…,𝑢_𝑛+8𝑘²+1𝜑𝑥,𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,3𝑘𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,𝑘𝑥,𝑢₂,…,𝑢_𝑛+𝑘²𝜑2𝑥,2𝑥,𝑢₂,…,𝑢_𝑛+𝜑2𝑥,2𝑘𝑥,𝑢₂,…,𝑢_𝑛+2𝜑𝑥,(𝑘+1)𝑥,𝑢₂,…,𝑢_𝑛+2𝜑𝑥,(𝑘−1)𝑥,𝑢₂,…,𝑢_𝑛+2𝜑2𝑥,𝑥,𝑢₂,…,𝑢_𝑛+2𝜑2𝑥,𝑘𝑥,𝑢₂,…,𝑢_𝑛𝑥+8𝜑2,𝑘𝑥2,𝑢₂,…,𝑢_𝑛|𝑘|𝜑𝑥+82,(2𝑘−1)𝑥2,𝑢₂,…,𝑢_𝑛|𝑘|𝜑𝑥+82,(2𝑘+1)𝑥2,𝑢₂,…,𝑢_𝑛𝑥+8𝜑2,3𝑘𝑥2,𝑢₂,…,𝑢_𝑛+|𝑘|+1||𝜑()𝑘−10,𝑘+1𝑥,𝑢₂,…,𝑢_𝑛+8𝑘²+1||𝜑()𝑘−10,𝑘−1𝑥,𝑢₂,…,𝑢_𝑛+2||𝜑𝑘−10,𝑥,𝑢₂,…,𝑢_𝑛+|𝑘|||𝜑()𝑘−10,3𝑘−1𝑥,𝑢₂,…,𝑢_𝑛+𝑘²||𝜑()𝑘−10,2𝑘−1𝑥,𝑢₂,…,𝑢_𝑛+𝑘²+|𝑘|−1||𝜑𝑘−10,2𝑘𝑥,𝑢₂,…,𝑢_𝑛+8|𝑘|||𝜑()𝑥𝑘−10,3𝑘−12,𝑢₂,…,𝑢_𝑛+8|𝑘|||𝜑()𝑥𝑘−10,𝑘+12,𝑢₂,…,𝑢_𝑛+8𝑘²+2|𝑘|−8||𝜑𝑘−10,𝑘𝑥,𝑢₂,…,𝑢_𝑛. (2.5)

Proof. Letting 𝑥=0 in (2.3), we get 𝑓(𝑦)+𝑓(−𝑦),𝑢₂,…,𝑢_𝑛𝑌≤1||𝜑𝑘−10,𝑦,𝑢₂,…,𝑢_𝑛 (2.6) for all 𝑦,𝑢₂,…,𝑢_𝑛∈𝑋. Putting 𝑦=𝑥 in (2.3), we have 𝑓((𝑘+1)𝑥)+𝑓((𝑘−1)𝑥)−𝑘𝑓(2𝑥)−2𝑓(𝑘𝑥)+2𝑘𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,𝑥,𝑢₂,…,𝑢_𝑛 (2.7) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Thus 𝑓(2(𝑘+1)𝑥)+𝑓(2(𝑘−1)𝑥)−𝑘𝑓(4𝑥)−2𝑓(2𝑘𝑥)+2𝑘𝑓(2𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑2𝑥,2𝑥,𝑢₂,…,𝑢_𝑛 (2.8) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Letting 𝑦=𝑘𝑥 in (2.3), we get 𝑓(2𝑘𝑥)−𝑘𝑓((𝑘+1)𝑥)−𝑘𝑓(−(𝑘−1)𝑥)−2𝑓(𝑘𝑥)+2𝑘𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,𝑘𝑥,𝑢₂,…,𝑢_𝑛 (2.9) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Letting 𝑦=(𝑘+1)𝑥 in (2.3), we have 𝑓((2𝑘+1)𝑥)+𝑓(−𝑥)−𝑘𝑓((𝑘+2)𝑥)−𝑘𝑓(−𝑘𝑥)−2𝑓(𝑘𝑥)+2𝑘𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,(𝑘+1)𝑥,𝑢₂,…,𝑢_𝑛 (2.10) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Letting 𝑦=(𝑘−1)𝑥 in (2.3), we have 𝑓((2𝑘−1)𝑥)−(𝑘+2)𝑓(𝑘𝑥)−𝑘𝑓(−(𝑘−2)𝑥)+(2𝑘+1)𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,(𝑘−1)𝑥,𝑢₂,…,𝑢_𝑛 (2.11) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Replacing 𝑥 and 𝑦 by 2𝑥 and 𝑥 in (2.3), respectively, we get 𝑓((2𝑘+1)𝑥)+𝑓((2𝑘−1)𝑥)−2𝑓(2𝑘𝑥)−𝑘𝑓(3𝑥)+2𝑘𝑓(2𝑥)−𝑘𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑2𝑥,𝑥,𝑢₂,…,𝑢_𝑛 (2.12) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Replacing 𝑥 and 𝑦 by 3𝑥 and 𝑥 in (2.3), respectively, we get 𝑓((3𝑘+1)𝑥)+𝑓((3𝑘−1)𝑥)−2𝑓(3𝑘𝑥)−𝑘𝑓(4𝑥)−𝑘𝑓(2𝑥)+2𝑘𝑓(3𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑3𝑥,𝑥,𝑢₂,…,𝑢_𝑛 (2.13) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Replacing 𝑥 and 𝑦 by 2𝑥 and 𝑘𝑥 in (2.3), respectively, we have 𝑓(3𝑘𝑥)+𝑓(𝑘𝑥)−𝑘𝑓((𝑘+2)𝑥)−𝑘𝑓(−(𝑘−2)𝑥)−2𝑓(2𝑘𝑥)+2𝑘𝑓(2𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑2𝑥,𝑘𝑥,𝑢₂,…,𝑢_𝑛 (2.14) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Setting 𝑦=(2𝑘+1)𝑥 in (2.3), we have 𝑓((3𝑘+1)𝑥)+𝑓(−(𝑘+1)𝑥)−𝑘𝑓(2(𝑘+1)𝑥)−𝑘𝑓(−2𝑘𝑥)−2𝑓(𝑘𝑥)+2𝑘𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,(2𝑘+1)𝑥,𝑢₂,…,𝑢_𝑛 (2.15) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Letting 𝑦=(2𝑘−1)𝑥 in (2.3), we have 𝑓((3𝑘−1)𝑥)+𝑓(−(𝑘−1)𝑥)−𝑘𝑓(−2(𝑘−1)𝑥)−𝑘𝑓(2𝑘𝑥)−2𝑓(𝑘𝑥)+2𝑘𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,(2𝑘−1)𝑥,𝑢₂,…,𝑢_𝑛 (2.16) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Letting 𝑦=3𝑘𝑥 in (2.3), we have 𝑓(4k𝑥)+𝑓(−2𝑘𝑥)−𝑘𝑓((3𝑘+1)𝑥)−𝑘𝑓(−(3𝑘−1)𝑥)−2𝑓(𝑘𝑥)+2𝑘𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,3𝑘𝑥,𝑢₂,…,𝑢_𝑛 (2.17) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. By (2.6), (2.7), (2.13), (2.15), and (2.16), we get 𝑘𝑓(2(𝑘+1)𝑥)+𝑘𝑓(−2(𝑘−1)𝑥)+6𝑓(𝑘𝑥)−2𝑓(3𝑘𝑥)−𝑘𝑓(4𝑥)+2𝑘𝑓(3𝑥)−6𝑘𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,(2𝑘+1)𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,(2𝑘−1)𝑥,𝑢₂,…,𝑢_𝑛+𝜑3𝑥,𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,𝑥,𝑢₂,…,𝑢_𝑛+1||𝜑()𝑘−10,𝑘+1𝑥,𝑢₂,…,𝑢_𝑛+1||𝜑()𝑘−10,𝑘−1𝑥,𝑢₂,…,𝑢_𝑛+|𝑘|||𝜑𝑘−10,2𝑘𝑥,𝑢₂,…,𝑢_𝑛 (2.18) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. By (2.6), (2.10), and (2.11), we have 𝑓((2𝑘+1)𝑥)+𝑓((2𝑘−1)𝑥)−𝑘𝑓((𝑘+2)𝑥)−𝑘𝑓(−(𝑘−2)𝑥)−4𝑓(𝑘𝑥)+4𝑘𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,(𝑘+1)𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,(𝑘−1)𝑥,𝑢₂,…,𝑢_𝑛+1||𝜑𝑘−10,𝑥,𝑢₂,…,𝑢_𝑛+𝑘𝜑𝑘−10,𝑘𝑥,𝑢₂,…,𝑢_𝑛 (2.19) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. It follows from (2.12) and (2.19) that 𝑘𝑓((𝑘+2)𝑥)+𝑘𝑓(−(𝑘−2)𝑥)−2𝑓(2𝑘𝑥)+4𝑓(𝑘𝑥)−𝑘𝑓(3𝑥)+2𝑘𝑓(2𝑥)−5𝑘𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,(𝑘+1)𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,(𝑘−1)𝑥,𝑢₂,…,𝑢_𝑛+𝜑2𝑥,𝑥,𝑢₂,…,𝑢_𝑛+1||𝜑𝑘−10,𝑥,𝑢₂,…,𝑢_𝑛+𝑘𝜑𝑘−10,𝑘𝑥,𝑢₂,…,𝑢_𝑛 (2.20) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. By (2.14) and (2.20), we have 𝑓(3𝑘𝑥)−4𝑓(2𝑘𝑥)+5𝑓(𝑘𝑥)−𝑘𝑓(3𝑥)+4𝑘𝑓(2𝑥)−5𝑘𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,(𝑘+1)𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,(𝑘−1)𝑥,𝑢₂,…,𝑢_𝑛+𝜑2𝑥,𝑥,𝑢₂,…,𝑢_𝑛+𝜑2𝑥,𝑘𝑥,𝑢₂,…,𝑢_𝑛+1||𝜑𝑘−10,𝑥,𝑢₂,…,𝑢_𝑛+𝑘𝜑𝑘−10,𝑘𝑥,𝑢₂,…,𝑢_𝑛 (2.21) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. By (2.6), (2.15), (2.16), and (2.17), we have 𝑘𝑓(−(𝑘+1)𝑥)−𝑘𝑓(−(𝑘−1)𝑥)−𝑘²𝑓(2(𝑘+1)𝑥)+𝑘²𝑓(−2(𝑘−1)𝑥)+𝑘²𝑓(2𝑘𝑥)−𝑘²−1𝑓(−2𝑘𝑥)+𝑓(4𝑘𝑥)−2𝑓(𝑘𝑥)+2𝑘𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤|𝑘|𝜑𝑥,(2𝑘+1)𝑥,𝑢₂,…,𝑢_𝑛+|𝑘|𝜑𝑥,(2𝑘−1)𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,3𝑘𝑥,𝑢₂,…,𝑢_𝑛+𝑘𝜑()𝑘−10,3𝑘−1𝑥,𝑢₂,…,𝑢_𝑛 (2.22) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. It follows from (2.6), (2.8), (2.9), and (2.22) that 𝑓(4𝑘𝑥)−2𝑓(2𝑘𝑥)−𝑘³𝑓(4𝑥)+2𝑘³𝑓(2𝑥),𝑢₂,…,𝑢_𝑛𝑌≤|𝑘|𝜑𝑥,(2𝑘+1)𝑥,𝑢₂,…,𝑢_𝑛+|𝑘|𝜑𝑥,(2𝑘−1)𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,3𝑘𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,𝑘𝑥,𝑢₂,…,𝑢_𝑛+𝑘²𝜑2𝑥,2𝑥,𝑢₂,…,𝑢_𝑛+𝑘𝜑()𝑘−10,3𝑘−1𝑥,𝑢₂,…,𝑢_𝑛+𝑘𝜑()𝑘−10,𝑘+1𝑥,𝑢₂,…,𝑢_𝑛+𝑘²||𝜑()𝑘−10,2𝑘−1𝑥,𝑢₂,…,𝑢_𝑛+𝑘²−1||𝜑𝑘−10,2𝑘𝑥,𝑢₂,…,𝑢_𝑛 (2.23) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Hence, 𝑓(2𝑘𝑥)−2𝑓(𝑘𝑥)−𝑘³𝑓(2𝑥)+2𝑘³𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤|𝑘|𝜑𝑥2,(2𝑘+1)𝑥2,𝑢₂,…,𝑢_𝑛+|𝑘|𝜑𝑥2,(2𝑘−1)𝑥2,𝑢₂,…,𝑢_𝑛𝑥+𝜑2,3𝑘𝑥2,𝑢₂,…,𝑢_𝑛𝑥+𝜑2,𝑘𝑥2,𝑢₂,…,𝑢_𝑛+𝑘²𝜑𝑥,𝑥,𝑢₂,…,𝑢_𝑛+𝑘𝜑()𝑥𝑘−10,3𝑘−12,𝑢₂,…,𝑢_𝑛+𝑘𝜑()𝑥𝑘−10,𝑘+12,𝑢₂,…,𝑢_𝑛+𝑘²||𝜑()𝑘−10,𝑘−1𝑥,𝑢₂,…,𝑢_𝑛+𝑘²−1||𝜑𝑘−10,𝑘𝑥,𝑢₂,…,𝑢_𝑛 (2.24) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. By (2.9), we have 𝑓(4𝑘𝑥)−𝑘𝑓(2(𝑘+1)𝑥)−𝑘𝑓(−2(𝑘−1)𝑥)−2𝑓(2𝑘𝑥)+2𝑘𝑓(2𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑2𝑥,2𝑘𝑥,𝑢₂,…,𝑢_𝑛 (2.25) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. From (2.23) and (2.25), we have 𝑘𝑓(2(𝑘+1)𝑥)+𝑘𝑓(−2(𝑘−1)𝑥)−𝑘³𝑓(4𝑥)+2𝑘³−2𝑘𝑓(2𝑥)_𝑌≤|𝑘|𝜑𝑥,(2𝑘+1)𝑥,𝑢₂,…,𝑢_𝑛+|𝑘|𝜑𝑥,(2𝑘−1)𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,3𝑘𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,𝑘𝑥,𝑢₂,…,𝑢_𝑛+𝑘²𝜑2𝑥,2𝑥,𝑢₂,…,𝑢_𝑛+𝜑2𝑥,2𝑘𝑥,𝑢₂,…,𝑢_𝑛+𝑘𝜑()𝑘−10,3𝑘−1𝑥,𝑢₂,…,𝑢_𝑛+𝑘𝜑()𝑘−10,𝑘+1𝑥,𝑢₂,…,𝑢_𝑛+𝑘²||𝜑()𝑘−10,2𝑘−1𝑥,𝑢₂,…,𝑢_𝑛+𝑘²−1||𝜑𝑘−10,2𝑘𝑥,𝑢₂,…,𝑢_𝑛 (2.26) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Also, from (2.18) and (2.26), we get 2𝑓(3𝑘𝑥)−6𝑓(𝑘𝑥)+𝑘−𝑘³𝑓(4𝑥)−2𝑘𝑓(3𝑥)+2𝑘³−2𝑘𝑓(2𝑥)+6𝑘𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤(|𝑘|+1)𝜑𝑥,(2𝑘+1)𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,(2𝑘−1)𝑥,𝑢₂,…,𝑢_𝑛+𝜑3𝑥,𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,3𝑘𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,𝑘𝑥,𝑢₂,…,𝑢_𝑛+𝑘²𝜑2𝑥,2𝑥,𝑢₂,…,𝑢_𝑛+𝜑2𝑥,2𝑘𝑥,𝑢₂,…,𝑢_𝑛+|𝑘|+1||𝜑()𝑘−10,𝑘+1𝑥,𝑢₂,…,𝑢_𝑛+1||𝜑()𝑘−10,𝑘−1𝑥,𝑢₂,…,𝑢_𝑛+𝑘²+|𝑘|−1||𝜑𝑘−10,2𝑘𝑥,𝑢₂,…,𝑢_𝑛+𝑘𝜑()𝑘−10,3𝑘−1𝑥,𝑢₂,…,𝑢_𝑛+𝑘²||𝜑()𝑘−10,2𝑘−1𝑥,𝑢₂,…,𝑢_𝑛 (2.27) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋.
On the other hand, it follows from (2.21) and (2.27) that8𝑓(2𝑘𝑥)−16𝑓(𝑘𝑥)+𝑘−𝑘³𝑓(4𝑥)+2𝑘³−10𝑘𝑓(2𝑥)+16𝑘𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤(|𝑘|+1)𝜑𝑥,(2𝑘+1)𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,(2𝑘−1)𝑥,𝑢₂,…,𝑢_𝑛+𝜑3𝑥,𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,3𝑘𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,𝑘𝑥,𝑢₂,…,𝑢_𝑛+𝑘²𝜑2𝑥,2𝑥,𝑢₂,…,𝑢_𝑛+𝜑2𝑥,2𝑘𝑥,𝑢₂,…,𝑢_𝑛+2𝜑𝑥,(𝑘+1)𝑥,𝑢₂,…,𝑢_𝑛+2𝜑𝑥,(𝑘−1)𝑥,𝑢₂,…,𝑢_𝑛+2𝜑2𝑥,𝑥,𝑢₂,…,𝑢_𝑛+2𝜑2𝑥,𝑘𝑥,𝑢₂,…,𝑢_𝑛+2||𝜑𝑘−10,𝑥,𝑢₂,…,𝑢_𝑛+2|𝑘|||𝜑𝑘−10,𝑘𝑥,𝑢₂,…,𝑢_𝑛+|𝑘|+1||𝜑()𝑘−10,𝑘+1𝑥,𝑢₂,…,𝑢_𝑛+1||𝜑()𝑘−10,𝑘−1𝑥,𝑢₂,…,𝑢_𝑛+𝑘²+|𝑘|−1||𝜑𝑘−10,2𝑘𝑥,𝑢₂,…,𝑢_𝑛+𝑘𝜑()𝑘−10,3𝑘−1𝑥,𝑢₂,…,𝑢_𝑛+𝑘²||𝜑()𝑘−10,2𝑘−1𝑥,𝑢₂,…,𝑢_𝑛 (2.28) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Therefore by (2.24) and (2.28), we get 𝑓(4𝑥)−10𝑓(2𝑥)+16𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤1𝑘³×𝑘|𝜑()−𝑘(|+1)𝑥,2𝑘+1𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,(2𝑘−1)𝑥,𝑢₂,…,𝑢_𝑛+𝜑3𝑥,𝑥,𝑢₂,…,𝑢_𝑛+8𝑘²+1𝜑𝑥,𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,3𝑘𝑥,𝑢₂,…,𝑢_𝑛+𝜑𝑥,𝑘𝑥,𝑢₂,…,𝑢_𝑛+𝑘²𝜑2𝑥,2𝑥,𝑢₂,…,𝑢_𝑛+𝜑2𝑥,2𝑘𝑥,𝑢₂,…,𝑢_𝑛+2𝜑𝑥,(𝑘+1)𝑥,𝑢₂,…,𝑢_𝑛+2𝜑𝑥,(𝑘−1)𝑥,𝑢₂,…,𝑢_𝑛+2𝜑2𝑥,𝑥,𝑢₂,…,𝑢_𝑛+2𝜑2𝑥,𝑘𝑥,𝑢₂,…,𝑢_𝑛𝑥+8𝜑2,𝑘𝑥2,𝑢₂,…,𝑢_𝑛|𝑘|𝜑𝑥+82,(2𝑘−1)𝑥2,𝑢₂,…,𝑢_𝑛|𝑘|𝜑𝑥+82,(2𝑘+1)𝑥2,𝑢₂,…,𝑢_𝑛𝑥+8𝜑2,3𝑘𝑥2,𝑢₂,…,𝑢_𝑛+|𝑘|+1||𝜑()𝑘−10,𝑘+1𝑥,𝑢₂,…,𝑢_𝑛+8𝑘²+1||𝜑()𝑘−10,𝑘−1𝑥,𝑢₂,…,𝑢_𝑛+2||𝜑𝑘−10,𝑥,𝑢₂,…,𝑢_𝑛+𝑘𝜑()𝑘−10,3𝑘−1𝑥,𝑢₂,…,𝑢_𝑛+𝑘²||𝜑()𝑘−10,2𝑘−1𝑥,𝑢₂,…,𝑢_𝑛+𝑘²+|𝑘|−1||𝜑𝑘−10,2𝑘𝑥,𝑢₂,…,𝑢_𝑛+8|𝑘|||𝜑()𝑥𝑘−10,3𝑘−12,𝑢₂,…,𝑢_𝑛+8|𝑘|||𝜑()𝑥𝑘−10,𝑘+12,𝑢₂,…,𝑢_𝑛+8𝑘²+2|𝑘|−8||𝜑𝑘−10,𝑘𝑥,𝑢₂,…,𝑢_𝑛∶=𝜑𝑥,𝑢₂,…,𝑢_𝑛 (2.29) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋.
Now, let 𝑔∶𝑋→𝑌 be the mapping defined by 𝑔(𝑥)∶=𝑓(2𝑥)−8𝑓(𝑥) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Then, (2.29) means that𝑓(4𝑥)−10𝑓(2𝑥)+16𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,𝑢₂,…,𝑢_𝑛 (2.30) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Also, we get 𝑔(2𝑥)−2𝑔(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,𝑢₂,…,𝑢_𝑛 (2.31) for all 𝑥∈𝑋. Replacing 𝑥 by 2^𝑗𝑥 in (2.31) and dividing both sides of (2.31) by 2^𝑗+1, we get 12^𝑗𝑔(2^𝑗1𝑥)−2^𝑗+1𝑔(2^𝑗+1𝑥),𝑢₂,…,𝑢_𝑛𝑌≤12^𝑗+1𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛 (2.32) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋 and all integers 𝑗≥0. For all integers 𝑙,𝑚 with 0≤𝑙<𝑚, we have 12^𝑙𝑔2^𝑙𝑥−12^𝑚𝑔2^𝑚𝑥,𝑢₂,…,𝑢_𝑛𝑌≤^𝑚−1_𝑗=𝑙12^𝑗𝑔2^𝑗𝑥−12^𝑗+1𝑔2^𝑗+1𝑥,𝑢₂,…,𝑢_𝑛𝑌≤^𝑚−1_𝑗=𝑙12^𝑗+1𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛 (2.33) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. So, we get lim_{𝑙,𝑚→∞}12^𝑙𝑔(2^𝑙1𝑥)−2^𝑚𝑔(2^𝑚𝑥),𝑢₂,…,𝑢_𝑛𝑌=0 (2.34) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. This shows that the sequence {(1/2^𝑗)𝑔(2^𝑗𝑥)} is a Cauchy sequence in 𝑌. Since 𝑌 is an 𝑛-Banach space, the sequence {(1/2^𝑗)𝑔(2^𝑗𝑥)} converges. So, we can define a mapping 𝐴∶𝑋→𝑌 by 𝐴(𝑥)∶=lim_𝑗→∞12^𝑗𝑔2^𝑗𝑥 (2.35) for all 𝑥∈𝑋. Putting 𝑙=0, then passing the limit 𝑚→∞ in (2.33), and using Lemma 1.6(4), we get 𝑔(𝑥)−𝐴(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤^∞_𝑗=012^𝑗+1𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛 (2.36) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋.
Now we show that 𝐴 is additive. By Lemma 1.6, (2.2), (2.32), and (2.35), we have𝐴(2𝑥)−2𝐴(𝑥),𝑢₂,…,𝑢_𝑛𝑌=lim_𝑗→∞12^𝑗𝑔2^𝑗+1𝑥−12^𝑗−1𝑔2^𝑗𝑥,𝑢₂,…,𝑢_𝑛𝑌=2lim_𝑗→∞12^𝑗+1𝑔2^𝑗+1𝑥−12^𝑗𝑔2^𝑗𝑥,𝑢₂,…,𝑢_𝑛𝑌≤lim_𝑗→∞12^𝑗𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛=0 (2.37) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. By Lemma 1.6(3), 𝐴(2𝑥)=2𝐴(𝑥) for all 𝑥∈𝑋. Also, by Lemma 1.6(4), (2.2), (2.3), and (2.35), we get 𝐷𝐴(𝑥,𝑦),𝑢₂,…,𝑢_𝑛𝑌=lim_𝑗→∞12^𝑗𝐷𝑔2^𝑗𝑥,2^𝑗𝑦,𝑢₂,…,𝑢_𝑛𝑌=lim_𝑗→∞12^𝑗𝐷𝑓2^𝑗+1𝑥,2^𝑗+1𝑦−8𝐷𝑓2^𝑗𝑥,2^𝑗𝑦,𝑢₂,…,𝑢_𝑛𝑌≤lim_𝑗→∞12^𝑗𝐷𝑓2^𝑗+1𝑥,2^𝑗+1𝑦,𝑢₂,…,𝑢_𝑛𝑌+8𝐷𝑓2^𝑗𝑥,2^𝑗𝑦,𝑢₂,…,𝑢_𝑛𝑌≤lim_𝑗→∞12^𝑗𝜑2^𝑗+1𝑥,2^𝑗+1𝑦,𝑢₂,…,𝑢_𝑛+8𝜑2^𝑗𝑥,2^𝑗𝑦,𝑢₂,…,𝑢_𝑛=0 (2.38) for all 𝑥,𝑦,𝑢₂,…,𝑢_𝑛∈𝑋. By Lemma 1.6(3), 𝐷𝐴(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝑋. Hence, the mapping 𝐴 satisfies (1.1). By [11, Lemma 2.3], the mapping 𝑥→𝐴(2𝑥)−8𝐴(𝑥) is additive. Therefore, 𝐴(2𝑥)=2𝐴(𝑥) implies that the mapping 𝐴 is additive.
To prove the uniqueness of 𝐴, let 𝐵∶𝑋→𝑌 be another additive mapping satisfying (2.4). Fix 𝑥∈𝑋. Clearly, 𝐴(2^𝑙𝑥)=2^𝑙𝐴(𝑥) and 𝐵(2^𝑙𝑥)=2^𝑙𝐵(𝑥) for all 𝑙∈ℕ. It follows from (2.4) that𝐴(𝑥)−𝐵(𝑥),𝑢₂,…,𝑢_𝑛𝑌=𝐴2^𝑙𝑥2^𝑙−𝐵2^𝑙𝑥2^𝑙,𝑢₂,…,𝑢_𝑛𝑌≤12^𝑙𝑓2^𝑙+1𝑥−8𝑓2^𝑙𝑥−𝐴2^𝑙𝑥,𝑢₂,…,𝑢_𝑛𝑌+𝐵2^𝑙𝑥−𝑓2^𝑙+1𝑥+8𝑓2^𝑙𝑥,𝑢₂,…,𝑢_𝑛𝑌≤12^𝑙∞_𝑗=012^𝑗𝜑2^𝑗+𝑙𝑥,𝑢₂,…,𝑢_𝑛≤^∞_𝑗=012^𝑗+𝑙𝜑2^𝑗+𝑙𝑥,𝑢₂,…,𝑢_𝑛=^∞_𝑗=𝑙12^𝑗𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛 (2.39) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋, and 𝑙∈ℕ. By (2.2), we see that the right-hand side of the above inequality tends to 0 as 𝑙→∞. Therefore, ‖𝐴(𝑥)−𝐵(𝑥),𝑢₂,…,𝑢_𝑛‖_𝑌=0 for all 𝑢₂,…,𝑢_𝑛∈𝑋. By Lemma 1.6, we can conclude that 𝐴(𝑥)=𝐵(𝑥) for all 𝑥∈𝑋. So, 𝐴=𝐵. This proves the uniqueness of 𝐴.

Theorem 2.2. Let 𝑋 be a linear space and 𝑌 an 𝑛-Banach space. Let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 for which there is a function 𝜑∶𝑋^𝑛+1→[0,∞) such that ^∞_𝑗=12^𝑗𝜑𝑥2^𝑗,𝑦2^𝑗,𝑢₂,…,𝑢_𝑛<∞,𝐷𝑓(𝑥,𝑦),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,𝑦,𝑢₂,…,𝑢_𝑛 (2.40) for all 𝑥,𝑦,𝑢₂,…,𝑢_𝑛∈𝑋. Then, there is a unique additive mapping 𝐴∶𝑋→𝑌 such that 𝑓(2𝑥)−8𝑓(𝑥)−𝐴(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤^∞_𝑗=12^𝑗−1𝜑𝑥2^𝑗,𝑢₂,…,𝑢_𝑛 (2.41) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋, where 𝜑(𝑥,𝑢₂,…,𝑢_𝑛) is defined as in Theorem 2.1.

Proof. The proof is similar to the proof of Theorem 2.1.

Corollary 2.3. Let 𝑋 be a normed space and 𝑌 an 𝑛-Banach space. Let 𝜃∈[0,∞),𝑝,𝑟₂,…,𝑟_𝑛∈(0,∞) such that 𝑝≠1, and let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 such that 𝐷𝑓(𝑥,𝑦),𝑢₂,…,𝑢_𝑛𝑌≤𝜃‖𝑥‖^𝑝_𝑋+𝑦^𝑝_𝑋𝑢₂^𝑟2_𝑋⋯𝑢_𝑛^𝑟𝑛_𝑋 (2.42) for all 𝑥,𝑦,𝑢₂,…,𝑢_𝑛∈𝑋. Then, there exists a unique additive mapping 𝐴∶𝑋→𝑌 such that 𝑓(2𝑥)−8𝑓(𝑥)−𝐴(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜃𝜖‖𝑥‖^𝑝_𝑋𝑢₂^𝑟2_𝑋⋯𝑢_𝑛^𝑟𝑛_𝑋(2−2^𝑝)𝑘³−𝑘 (2.43) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋, where 𝜖=1+|𝑘|+2^3−𝑝|𝑘|(2𝑘+1)^𝑝+(2𝑘−1)^𝑝+2|𝑘|+13+3^𝑝+3|𝑘|^𝑝+16𝑘²+3^𝑝|𝑘|^𝑝+2^𝑝+1𝑘²+2^𝑝5+|𝑘|^𝑝+2|𝑘+1|^𝑝+2|𝑘−1|^𝑝+2^3−𝑝2+|𝑘|+|𝑘|^𝑝+3^𝑝|𝑘|^𝑝+(|𝑘|+1)|𝑘+1|^𝑝||+2𝑘−1_3−𝑝|𝑘|||||𝑘−1𝑘+1^𝑝+1+8𝑘²+2^𝑝𝑘²|𝑘−1|^𝑝−1+2^𝑝|𝑘|^𝑝𝑘²+|𝑘|−1||+2𝑘−1||+|𝑘2𝑘−1|^3−𝑝+1||||𝑘−13𝑘−1^𝑝+8𝑘²+2|𝑘|−8|||𝑘|𝑘−1^𝑝. (2.44)

Proof. Define 𝜑(𝑥,𝑦)=𝜃(‖𝑥‖^𝑝_𝑋+‖𝑦‖^𝑝_𝑋)‖𝑢₂‖^𝑟2_𝑋⋯‖𝑢_𝑛‖^𝑟𝑛_𝑋 for all 𝑥,𝑦,𝑢₂,…,𝑢_𝑛∈𝑋, and apply Theorems 2.1 and 2.2.

The following example shows that the assumption 𝑝≠1 cannot be omitted in Corollary 2.3.

Example 2.4. Let 𝑋=ℂ be a linear space over ℝ. Define ‖⋅,⋅‖∶𝑋×𝑋→ℝ by ‖𝑥₁,𝑥₂‖=|𝑎₁𝑏₂−𝑎₂𝑏₁|, where 𝑥_𝑗=𝑎_𝑗+𝑏_𝑗𝑖∈ℂ, 𝑎_𝑗,𝑏_𝑗∈ℝ, 𝑗=1,2 (𝑖=√−1 is the imaginary unit). Then, (𝑋,‖⋅,⋅‖) is a 2-normed linear space.
Let 𝜙∶ℂ→ℂ defined by𝜙(𝑥)=|𝑥||𝑥|𝑥,for<1,1,for≥1. (2.45)
Consider the function 𝑓∶ℂ→ℂ defined by𝑓(𝑥)=^∞_𝑚=0𝛼^−𝑚𝜙𝛼^𝑚𝑥 (2.46) for all 𝑥∈ℂ, where 𝛼>|𝑘|. Then, 𝑓 satisfies the functional inequality 𝐷𝑓𝑥,𝑦,𝑢≤4𝛼²(|𝑘|+1)|𝑥|+𝑦|𝑢|𝛼−1 (2.47) for all 𝑥,𝑦,𝑢∈ℂ, but there do not exist an additive mapping 𝐴∶ℂ→ℂ and a constant 𝑑>0 such that ‖𝑓(𝑥)−𝐴(𝑥),𝑢‖≤𝑑|𝑥||𝑢| for all 𝑥,𝑢∈ℂ.
It is clear that |𝑓(𝑥)|≤𝛼/(𝛼−1) for all 𝑥∈ℂ. If |𝑥|+|𝑦|=0 or |𝑥|+|𝑦|≥1/𝛼 for all 𝑥,𝑦∈ℂ, then the inequality (2.47) holds. Now suppose that 0<|𝑥|+|𝑦|<1/𝛼. Then, there exists an integer 𝑛≥1 such that1𝛼^𝑛+1≤|𝑥|+𝑦<1𝛼^𝑛. (2.48) Hence, 𝛼^𝑚|𝑘𝑥±𝑦|<1,𝛼^𝑚|𝑥±𝑦|<1,𝛼^𝑚|𝑥|<1 for all 𝑚=0,1,…,𝑛−1. From the definition of 𝑓 and (2.48), we obtain that =𝐷𝑓𝑥,𝑦,𝑢_∞𝑚=𝑛𝛼^−𝑚𝜙𝛼^𝑚𝑘𝑥+𝑦+^∞_𝑚=𝑛𝛼^−𝑚𝜙𝛼^𝑚𝑘𝑥−𝑦−𝑘^∞_𝑚=𝑛𝛼^−𝑚𝜙𝛼^𝑚𝑥+𝑦−𝑘^∞_𝑚=𝑛𝛼^−𝑚𝜙𝛼^𝑚𝑥−𝑦−2^∞_𝑚=𝑛𝛼^−𝑚𝜙𝛼^𝑚𝑘𝑥+2𝑘^∞_𝑚=𝑛𝛼^−𝑚𝜙𝛼^𝑚𝑥≤,𝑢4𝛼²(|𝑘|+1)|𝑥|+𝑦|𝑢|.𝛼−1 (2.49) Therefore, 𝑓 satisfies (2.47). Now, we claim that the functional equation (1.1) is not stable for 𝑝=1 in Corollary 2.3. Suppose on the contrary that there exist an additive mapping 𝐴∶ℂ→ℂ and a constant 𝑑>0 such that ‖𝑓(𝑥)−𝐴(𝑥),𝑢‖≤𝑑|𝑥||𝑢| for all 𝑥,𝑢∈ℂ. Then, there exists a constant 𝑐∈ℂ such that 𝐴(𝑥)=𝑐𝑥 for all rational numbers 𝑥. So, we obtain that 𝑓(𝑥),𝑢≤(𝑑+|𝑐|)|𝑥||𝑢| (2.50) for all rational numbers 𝑥 and all 𝑢∈ℂ. Let 𝑠∈ℕ with 𝑠+1>𝑑+|𝑐|. If 𝑥 is a rational number in (0,𝛼^−𝑠) and 𝑢=𝑏𝑖 (𝑏∈ℝ), then 𝛼^𝑚𝑥∈(0,1) for all 𝑚=0,1,…,𝑠, and we get 𝑓(𝑥),𝑢=^∞_𝑚=0𝜙𝛼^𝑚𝑥𝛼^𝑚,𝑢≥^𝑠_𝑚=0𝜙𝛼^𝑚𝑥𝛼^𝑚|𝑏|=(𝑠+1)𝑥|𝑏|>(𝑑+|𝑐|)𝑥|𝑏|=(𝑑+|𝑐|)|𝑥||𝑢|, (2.51) which contradicts (2.50).

Theorem 2.5. Let 𝑋 be a linear space and 𝑌 an 𝑛-Banach space. Let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 for which there is a function 𝜑∶𝑋^𝑛+1→[0,∞) such that ^∞_𝑗=018^𝑗𝜑2^𝑗𝑥,2^𝑗𝑦,𝑢₂,…,𝑢_𝑛<∞,𝐷𝑓(𝑥,𝑦),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,𝑦,𝑢₂,…,𝑢_𝑛 (2.52)(2.53) for all 𝑥,𝑦,𝑢₂,…,𝑢_𝑛∈𝑋. Then, there is a unique cubic mapping 𝐶∶𝑋→𝑌 such that 𝑓(2𝑥)−2𝑓(𝑥)−𝐶(x),𝑢₂,…,𝑢_𝑛𝑌≤^∞_𝑗=018^𝑗+1𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛 (2.54) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋, where 𝜑(𝑥,𝑢₂,…,𝑢_𝑛) is defined as in Theorem 2.1.

Proof. As in the proof of Theorem 2.1, we have 𝑓(4𝑥)−10𝑓(2𝑥)+16𝑓(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,𝑢₂,…,𝑢_𝑛 (2.55) for all 𝑥∈𝑋, where 𝜑(𝑥,𝑢₂,…,𝑢_𝑛) is defined as in Theorem 2.1.
Now, let ℎ∶𝑋→𝑌 be the mapping defined by ℎ(𝑥)∶=𝑓(2𝑥)−2𝑓(𝑥). By (2.55), we haveℎ(2𝑥)−8ℎ(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,𝑢₂,…,𝑢_𝑛 (2.56) for all 𝑥∈𝑋. Replacing 𝑥 by 2^𝑗𝑥 in (2.56) and dividing both sides of (2.56) by 8^𝑗+1, we get 18^𝑗ℎ(2^𝑗1𝑥)−8^𝑗+1ℎ(2^𝑗+1𝑥),𝑢₂,…,𝑢_𝑛𝑌≤18^𝑗+1𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛 (2.57) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋 and all integers 𝑗≥0. For all integers 𝑙,𝑚 with 0≤𝑙<𝑚, we have 18^𝑙ℎ(2^𝑙1𝑥)−8^𝑚ℎ(2^𝑚𝑥),𝑢₂,…,𝑢_𝑛𝑌≤^𝑚−1_𝑗=𝑙18^𝑗ℎ(2^𝑗1𝑥)−8^𝑗+1ℎ(2^𝑗+1𝑥),𝑢₂,…,𝑢_𝑛𝑌≤^𝑚−1_𝑗=𝑙18^𝑗+1𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛 (2.58) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. So, we get lim_{𝑙,𝑚→∞}18^𝑙ℎ(2^𝑙1𝑥)−8^𝑚ℎ(2^𝑚𝑥),𝑢₂,…,𝑢_𝑛𝑌=0 (2.59) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. This shows that the sequence {(1/8^𝑗)ℎ(2^𝑗𝑥)} is a Cauchy sequence in 𝑌. Since 𝑌 is an 𝑛-Banach space, the sequence {(1/8^𝑗)ℎ(2^𝑗𝑥)} converges. So, we can define a mapping 𝐶∶𝑋→𝑌 by 𝐶(𝑥)∶=lim_𝑗→∞18^𝑗ℎ2^𝑗𝑥 (2.60) for all 𝑥∈𝑋. Putting 𝑙=0, then passing the limit 𝑚→∞ in (2.58), and using Lemma 1.6(4), we get ℎ(𝑥)−𝐶(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤^∞_𝑗=018^𝑗+1𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛 (2.61) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋.
Now we show that C is cubic. By Lemma 1.6, (2.52), (2.58), and (2.60), we have𝐶(2𝑥)−8𝐶(𝑥),𝑢₂,…,𝑢_𝑛𝑌=lim_𝑗→∞18^𝑗ℎ(2^𝑗+11𝑥)−8^𝑗−1ℎ(2^𝑗𝑥),𝑢₂,…,𝑢_𝑛𝑌=8lim_𝑗→∞18^𝑗+1ℎ(2^𝑗+11𝑥)−8^𝑗ℎ(2^𝑗𝑥),𝑢₂,…,𝑢_𝑛𝑌≤lim_𝑗→∞18^𝑗𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛=0 (2.62) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. By Lemma 1.6(3), 𝐶(2𝑥)=8𝐶(𝑥) for all 𝑥∈𝑋. Also, by Lemma 1.6(4), (2.52), (2.53), and (2.60), we get 𝐷𝐶(𝑥,𝑦),𝑢₂,…,𝑢_𝑛𝑌=lim_𝑗→∞18^𝑗𝐷ℎ(2^𝑗𝑥,2^𝑗𝑦),𝑢₂,…,𝑢_𝑛𝑌=lim_𝑗→∞18^𝑗𝐷𝑓(2^𝑗+1𝑥,2^𝑗+1𝑦)−2𝐷𝑓(2^𝑗𝑥,2^𝑗𝑦),𝑢₂,…,𝑢_𝑛𝑌≤lim_𝑗→∞18^𝑗𝐷𝑓(2^𝑗+1𝑥,2^𝑗+1𝑦),𝑢₂,…,𝑢_𝑛𝑌+2𝐷𝑓(2^𝑗𝑥,2^𝑗𝑦),𝑢₂,…,𝑢_𝑛𝑌≤lim_𝑗→∞18^𝑗𝜑2^𝑗+1𝑥,2^𝑗+1𝑦,𝑢₂,…,𝑢_𝑛+2𝜑2^𝑗𝑥,2^𝑗𝑦,𝑢₂,…,𝑢_𝑛=0 (2.63) for all 𝑥,𝑦,𝑢₂,…,𝑢_𝑛∈𝑋. By Lemma 1.6(3), 𝐷𝐶(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝑋. Hence the mapping 𝐶 satisfies (1.1). By [11, Lemma 2.3], the mapping 𝑥→𝐶(2𝑥)−2𝐶(𝑥) is cubic. Therefore, 𝐶(2𝑥)=8𝐶(𝑥) implies that the mapping 𝐶 is cubic.
To prove the uniqueness of 𝐶, let 𝑆∶𝑋→𝑌 be another cubic mapping satisfying (2.54). Fix 𝑥∈𝑋. Clearly, 𝐶(2^𝑙𝑥)=8^𝑙𝐴(𝑥) and 𝑆(2^𝑙𝑥)=8^𝑙𝑆(𝑥) for all 𝑙∈ℕ. It follows from (2.54) that𝐶(𝑥)−𝑆(𝑥),𝑢₂,…,𝑢_𝑛𝑌=𝐶2^𝑙𝑥8^𝑙−𝑆2^𝑙𝑥8^𝑙,𝑢₂,…,𝑢_𝑛𝑌≤18^𝑙𝑓(2^𝑙+1𝑥)−2𝑓(2^𝑙𝑥)−𝐶(2^𝑙𝑥),𝑢₂,…,𝑢_𝑛𝑌+𝑆(2^𝑙𝑥)−𝑓(2^𝑙+1𝑥)+2𝑓(2^𝑙𝑥),𝑢₂,…,𝑢_𝑛𝑌≤18^𝑙∞_𝑗=018^𝑗𝜑2^𝑗+𝑙𝑥,𝑢₂,…,𝑢_𝑛≤^∞_𝑗=018^𝑗+𝑙𝜑2^𝑗+𝑙𝑥,𝑢₂,…,𝑢_𝑛=^∞_𝑗=𝑙18^𝑗𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛 (2.64) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋, and 𝑙∈ℕ. By (2.52), we see that the right-hand side of the above inequality tends to 0 as 𝑙→∞. Therefore, ‖𝐶(𝑥)−𝑆(𝑥),𝑢₂,…,𝑢_𝑛‖_𝑌=0 for all 𝑢₂,…,𝑢_𝑛∈𝑋. By Lemma 1.6, we can conclude that 𝐶(𝑥)=𝑆(𝑥) for all 𝑥∈𝑋. So 𝐶=𝑆. This proves the uniqueness of 𝐶.

Theorem 2.6. Let 𝑋 be a linear space and 𝑌 an 𝑛-Banach space. Let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 for which there is a function 𝜑∶𝑋^𝑛+1→[0,∞) such that ^∞_𝑗=18^𝑗𝜑𝑥2^𝑗,𝑦2^𝑗,𝑢₂,…,𝑢_𝑛<∞,𝐷𝑓(𝑥,𝑦),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,𝑦,𝑢₂,…,𝑢_𝑛 (2.65) for all 𝑥,𝑦,𝑢₂,…,𝑢_𝑛∈𝑋. Then, there is a unique cubic mapping 𝐶∶𝑋→𝑌 such that 𝑓(2𝑥)−2𝑓(𝑥)−𝐶(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤^∞_𝑗=18^𝑗−1𝜑𝑥2^𝑗,𝑢₂,…,𝑢_𝑛 (2.66) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋, where 𝜑(𝑥,𝑢₂,…,𝑢_𝑛) is defined as in Theorem 2.1.

Proof. The proof is similar to the proof of Theorem 2.5.

Corollary 2.7. Let 𝑋 be a normed space and 𝑌 an 𝑛-Banach space. Let 𝜃∈[0,∞),𝑝,𝑟₂,…,𝑟_𝑛∈(0,∞) such that 𝑝≠3, and let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 such that 𝐷𝑓(𝑥,𝑦),𝑢₂,…,𝑢_𝑛𝑌≤𝜃‖𝑥‖^𝑝_𝑋+𝑦^𝑝_𝑋𝑢₂^𝑟2_𝑋⋯𝑢_𝑛^𝑟𝑛_𝑋 (2.67) for all 𝑥,𝑦,𝑢₂,…,𝑢_𝑛∈𝑋. Then, there exists a unique cubic mapping 𝐶∶𝑋→𝑌 such that 𝑓(2𝑥)−2𝑓(𝑥)−𝐶(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝜃𝜖‖𝑥‖^𝑝_𝑋𝑢₂^𝑟2_𝑋⋯𝑢_𝑛^𝑟𝑛_𝑋(8−2^𝑝)𝑘³−𝑘 (2.68) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋, where 𝜖 is defined as in Corollary 2.3.

Proof. Define 𝜑(𝑥,𝑦)=𝜃(‖𝑥‖^𝑝_𝑋+‖𝑦‖^𝑝_𝑋)‖𝑢₂‖^𝑟2_𝑋⋯‖𝑢_𝑛‖^𝑟𝑛_𝑋 for all 𝑥,𝑦,𝑢₂,…,𝑢_𝑛∈𝑋, and apply Theorems 2.5 and 2.6.

The following example shows that the the generalized Hyers-Ulam stability problem for the case of 𝑝=3 was excluded in Corollary 2.7.

Example 2.8. Let 𝑋=ℂ be a linear space over ℝ, and let ‖⋅,⋅‖∶𝑋×𝑋→ℝ be defined as in Example 2.4. Then, (𝑋,‖⋅,⋅‖) is a 2-normed linear space.
Let 𝜙∶ℂ→ℂ be defined by𝜙(𝑥)=𝑥³|𝑥||𝑥|,for<1,1,for≥1. (2.69)
Consider the function 𝑓∶ℂ→ℂ defined by𝑓(𝑥)=^∞_𝑚=0𝛼^−3𝑚𝜙𝛼^𝑚𝑥 (2.70) for all 𝑥∈ℂ, where 𝛼>|𝑘|. Then, 𝑓 satisfies the functional inequality 𝐷𝑓𝑥,𝑦,𝑢≤4𝛼⁶(|𝑘|+1)𝛼³|𝑥|−1³+𝑦³|𝑢| (2.71) for all 𝑥,𝑦,𝑢∈ℂ, but there do not exist a cubic mapping 𝐶∶ℂ→ℂ and a constant 𝑑>0 such that ‖𝑓(𝑥)−𝐶(𝑥),𝑢‖≤𝑑|𝑥|³|𝑢| for all 𝑥,𝑢∈ℂ.
It is clear that |𝑓(𝑥)|≤𝛼³/(𝛼³−1) for all 𝑥∈ℂ. If |𝑥|³+|𝑦|³=0 or |𝑥|³+|𝑦|³≥1/𝛼³ for all 𝑥,𝑦∈ℂ, then inequality (2.71) holds. Now suppose that 0<|𝑥|³+|𝑦|³<1/𝛼³. Then, there exists an integer 𝑛≥1 such that1𝛼^3(𝑛+1)≤|𝑥|³+𝑦³<1𝛼^3𝑛. (2.72) Hence, 𝛼^𝑚|𝑘𝑥±𝑦|<1,𝛼^𝑚|𝑥±𝑦|<1,𝛼^𝑚|𝑥|<1 for all 𝑚=0,1,…,𝑛−1. From the definition of 𝑓 and (2.72), we obtain that 𝐷𝑓𝑥,𝑦,𝑢=^∞_𝑚=𝑛𝛼^−3𝑚𝜙𝛼^𝑚𝑘𝑥+𝑦+^∞_𝑚=𝑛𝛼^−3𝑚𝜙𝛼^𝑚𝑘𝑥−𝑦−𝑘^∞_𝑚=𝑛𝛼^−3𝑚𝜙𝛼^𝑚𝑥+𝑦−𝑘^∞_𝑚=𝑛𝛼^−3𝑚𝜙𝛼^𝑚𝑥−𝑦−2^∞_𝑚=𝑛𝛼^−3𝑚𝜙𝛼^𝑚𝑘𝑥+2𝑘^∞_𝑚=𝑛𝛼^−3𝑚𝜙𝛼^𝑚𝑥≤,𝑢4𝛼⁶(|𝑘|+1)𝛼³|𝑥|−1³+𝑦³|𝑢|. (2.73) Therefore, 𝑓 satisfies (2.71). Now, we claim that the functional equation (1.1) is not stable for 𝑝=3 in Corollary 2.7. Suppose on the contrary that there exist a cubic mapping 𝐶∶ℂ→ℂ and a constant 𝑑>0 such that ‖𝑓(𝑥)−𝐶(𝑥),𝑢‖≤𝑑|𝑥|³|𝑢| for all 𝑥,𝑢∈ℂ. Then, there exists a constant 𝛽∈ℂ such that 𝐶(𝑥)=𝛽𝑥³ for all rational numbers 𝑥. So, we obtain that 𝑓(𝑥),𝑢≤𝑑+𝛽|𝑥|³|𝑢| (2.74) for all rational numbers 𝑥 and all 𝑢∈ℂ. Let 𝑠∈ℕ with 𝑠+1>𝑑+|𝛽|. If 𝑥 is a rational number in (0,𝛼^−𝑠) and 𝑢=𝑏𝑖 (𝑏∈ℝ), then 𝛼^𝑚𝑥∈(0,1) for all 𝑚=0,1,…,𝑠, and we get 𝑓(𝑥),𝑢=^∞_𝑚=0𝜙𝛼^𝑚𝑥𝛼^3𝑚,𝑢≥^𝑠_𝑚=0𝜙𝛼^𝑚𝑥𝛼^3𝑚|𝑏|=(𝑠+1)𝑥³|𝑏|>𝑑+𝛽𝑥³|𝑏|=𝑑+𝛽|𝑥|³|𝑢|, (2.75) which contradicts (2.74).

Theorem 2.9. Let 𝑋 be a linear space and 𝑌 an 𝑛-Banach space. Let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 for which there is a function 𝜑∶𝑋^𝑛+1→[0,∞) such that ^∞_𝑗=012^𝑗𝜑2^𝑗𝑥,2^𝑗𝑦,𝑢₂,…,𝑢_𝑛<∞,𝐷𝑓𝑥,𝑦,𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,𝑦,𝑢₂,…,𝑢_𝑛 (2.76)(2.77) for all 𝑥,𝑦,𝑢₂,…,𝑢_𝑛∈𝑋. Then, there exist a unique additive mapping 𝐴∶𝑋→𝑌 and a unique cubic mapping 𝐶∶𝑋→𝑌 such that 𝑓(𝑥)−𝐴(𝑥)−𝐶(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤16^∞_𝑗=012^𝑗+1+18^𝑗+1𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛 (2.78) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋, where 𝜑(𝑥,𝑢₂,…,𝑢_𝑛) is defined as in Theorem 2.1.

Proof. By Theorems 2.1 and 2.5, there exist an additive mapping 𝐴^′∶𝑋→𝑌 and a cubic mapping 𝐶′∶𝑋→𝑌 such that 𝑓(2𝑥)−8𝑓(𝑥)−𝐴^′(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤^∞_𝑗=012^𝑗+1𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛,𝑓(2𝑥)−2𝑓(𝑥)−𝐶^′(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤^∞_𝑗=018^𝑗+1𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛 (2.79) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Hence, 𝑓(𝑥)+16𝐴^′(𝑥)−16𝐶^′(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤16^∞_𝑗=012^𝑗+1+18^𝑗+1𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛 (2.80) for all 𝑥∈𝑋. So, we obtain (2.78) by letting 𝐴(𝑥)=−(1/6)𝐴^′(𝑥) and 𝐶(𝑥)=(1/6)𝐶^′(𝑥) for all 𝑥∈𝑋.
To prove the uniqueness of 𝐴 and 𝐶, let 𝐴^′′,𝐶^′′∶𝑋→𝑌 be another additive and cubic mapping satisfying (2.78). Fix 𝑥∈𝑋. Let 𝐴₁=𝐴−𝐴^′′ and 𝐶₁=𝐶−𝐶^′′. So,𝐴₁(𝑥)+𝐶₁(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤𝑓(𝑥)−𝐴(𝑥)−𝐶(𝑥),𝑢₂,…,𝑢_𝑛𝑌+𝑓(𝑥)−𝐴^′′(𝑥)−𝐶^′′(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤13^∞_𝑗=012^𝑗+1+18^𝑗+1𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛 (2.81) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Then (2.76) implies that lim_𝑛→∞18^𝑛𝐴₁2^𝑛𝑥+𝐶₁2^𝑛𝑥,𝑢₂,…,𝑢_𝑛𝑌=0 (2.82) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋. Thus, 𝐶₁=0. So, it follows from (2.81) that 𝐴₁(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤13^∞_𝑗=012^𝑗+1+18^𝑗+1𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛 (2.83) for all 𝑢₂,…,𝑢_𝑛∈𝑋. Therefore, 𝐴₁=0.

Similarly to Theorem 2.9, one can prove the following result.

Theorem 2.10. Let 𝑋 be a linear space and 𝑌 an 𝑛-Banach space. Let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 for which there is a function 𝜑∶𝑋^𝑛+1→[0,∞) such that ^∞_𝑗=08^𝑗𝜑𝑥2^𝑗,𝑦2^𝑗,𝑢₂,…,𝑢_𝑛<∞,𝐷𝑓(𝑥,𝑦),𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,𝑦,𝑢₂,…,𝑢_𝑛 (2.84) for all 𝑥,𝑦,𝑢₂,…,𝑢_𝑛∈𝑋. Then, there exist a unique additive mapping 𝐴∶𝑋→𝑌 and a unique cubic mapping 𝐶∶𝑋→𝑌 such that 𝑓(𝑥)−𝐴(𝑥)−𝐶(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤16^∞_𝑗=12^𝑗−1+8^𝑗−1𝜑𝑥2^𝑗,𝑢₂,…,𝑢_𝑛 (2.85) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋, where 𝜑(𝑥,𝑢₂,…,𝑢_𝑛) is defined as in Theorem 2.1.

Proof. The proof is similar to the proof of Theorem 2.9 and the result follows from Theorems 2.2 and 2.6.

Theorem 2.11. Let 𝑋 be a linear space and 𝑌 an 𝑛-Banach space. Let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 for which there is a function 𝜑∶𝑋^𝑛+1→[0,∞) such that ^∞_𝑗=12^𝑗𝜑𝑥2^𝑗,𝑦2^𝑗,𝑢₂,…,𝑢_𝑛<∞,^∞_𝑗=018^𝑗𝜑2^𝑗𝑥,2^𝑗𝑦,𝑢₂,…,𝑢_𝑛<∞,𝐷𝑓𝑥,𝑦,𝑢₂,…,𝑢_𝑛𝑌≤𝜑𝑥,𝑦,𝑢₂,…,𝑢_𝑛 (2.86) for all 𝑥,𝑦,𝑢₂,…,𝑢_𝑛∈𝑋. Then, there exist a unique additive mapping 𝐴∶𝑋→𝑌 and a unique cubic mapping 𝐶∶𝑋→𝑌 such that 𝑓(𝑥)−𝐴(𝑥)−𝐶(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤16^∞_𝑗=12^𝑗−1𝜑𝑥2^𝑗,𝑢₂,…,𝑢_𝑛+^∞_𝑗=018^𝑗+1𝜑2^𝑗𝑥,𝑢₂,…,𝑢_𝑛 (2.87) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋, where 𝜑(𝑥,𝑢₂,…,𝑢_𝑛) is defined as in Theorem 2.1.

Proof. The proof is similar to the proof of Theorem 2.9 and the result follows from Theorems 2.2 and 2.5.

Corollary 2.12. Let 𝑋 be a normed space and 𝑌 an 𝑛-Banach space. Let 𝜃∈[0,∞),𝑟₂,…,𝑟_𝑛∈(0,∞), 𝑝∈(0,1)∪(1,3)∪(3,∞), and let 𝑓∶𝑋→𝑌 be a mapping with 𝑓(0)=0 such that 𝐷𝑓𝑥,𝑦,𝑢₂,…,𝑢_𝑛𝑌≤𝜃‖𝑥‖^𝑝_𝑋+𝑦^𝑝_𝑋𝑢₂^𝑟2_𝑋⋯𝑢_𝑛^𝑟𝑛_𝑋 (2.88) for all 𝑥,𝑦,𝑢₂,…,𝑢_𝑛∈𝑋. Then, there exist a unique additive mapping 𝐴∶𝑋→𝑌 and a unique cubic mapping C∶𝑋→𝑌 such that 𝑓(𝑥)−𝐴(𝑥)−𝐶(𝑥),𝑢₂,…,𝑢_𝑛𝑌≤16𝑘³1−𝑘|2−2^𝑝|+1|8−2^𝑝|𝜃𝜖‖𝑥‖^𝑝_𝑋𝑢₂^𝑟2_𝑋⋯𝑢_𝑛^𝑟𝑛_𝑋 (2.89) for all 𝑥,𝑢₂,…,𝑢_𝑛∈𝑋, where 𝜖 is defined as in Corollary 2.3.

Proof. Define 𝜑(𝑥,𝑦)=𝜃(‖𝑥‖^𝑝_𝑋+‖𝑦‖^𝑝_𝑋)‖𝑢₂‖^𝑟2_𝑋⋯‖𝑢_𝑛‖^𝑟𝑛_𝑋 for all 𝑥,𝑦,𝑢₂,…,𝑢_𝑛∈𝑋, and apply Theorems 2.9–2.11.

Remark 2.13. The generalized Hyers-Ulam stability problem for the cases of 𝑝=1 and 𝑝=3 was excluded in Corollary 2.12 (see Examples 2.4 and 2.8).

Acknowledgments

The authors would like to thank the Editor Professor Krzysztof Ciepliński and anonymous referees for their valuable comments and suggestions. The first author was supported by the National Natural Science Foundation of China (NNSFC) (grant No. 11171022).