Approximation of Mixed-Type Functional Equations in Menger PN-Spaces

Abstract

Let 𝑋 and 𝑌 be vector spaces. We show that a function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies Δ𝑓(𝑥₁,…,𝑥_𝑛)=0 for all 𝑥₁,…,𝑥_𝑛∈𝑋, if and only if there exist functions 𝐶∶𝑋×𝑋×𝑋→𝑌, 𝐵∶𝑋×𝑋→𝑌 and 𝐴∶𝑋→𝑌 such that 𝑓(𝑥)=𝐶(𝑥,𝑥,𝑥)+𝐵(𝑥,𝑥)+𝐴(𝑥) for all 𝑥∈𝑋, where the function 𝐶 is symmetric for each fixed one variable and is additive for fixed two variables, 𝐵 is symmetric bi-additive, 𝐴 is additive and Δ𝑓(𝑥₁,…,𝑥_𝑛)=∑^𝑛_𝑘=2(∑^𝑘_𝑖1=2∑_𝑖^𝑘+1_2=𝑖1+1⋯∑^𝑛_{𝑖𝑛−𝑘+1=𝑖𝑛−𝑘+1})𝑓(∑^𝑛_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑛−𝑘+1}𝑥_𝑖−∑^{𝑛−𝑘+1}_𝑟=1𝑥_𝑖𝑟)+𝑓(∑^𝑛_𝑖=1𝑥_𝑖)−2^𝑛−2∑^𝑛_𝑖=2(𝑓(𝑥₁+𝑥_𝑖)+𝑓(𝑥₁−𝑥_𝑖))+2^𝑛−1(𝑛−2)𝑓(𝑥₁) (𝑛∈ℕ, 𝑛≥3) for all 𝑥₁,…,𝑥_𝑛∈𝑋. Furthermore, we solve the stability problem for a given function 𝑓 satisfying Δ𝑓(𝑥₁,…,𝑥_𝑛)=0, in the Menger probabilistic normed spaces.

1. Introduction and Preliminaries

Menger [1] introduced the notion of a probabilistic metric space in 1942 and since then the theory of probabilistic metric spaces has developed in many directions [2]. The idea of Menger was to use distribution functions instead of nonnegative real numbers as values of the metric. The notion of a probabilistic metric space corresponds to situations when we do not know exactly the distance between two points, but we know probabilities of possible values of this distance. A probabilistic generalization of metric spaces appears to be interest in the investigation of physical quantities and physiological thresholds. It is also of fundamental importance in probabilistic functional analysis. Probabilistic normed spaces were introduced by Sherstnev in 1962 [3] by means of a definition that was closely modelled on the theory of (classical) normed spaces and used to study the problem of best approximation in statistics. In the sequel, we will adopt the usual terminology, notation and conventions of the theory of probabilistic normed spaces as in [2, 4–6, 6, 7, 7–18].

Throughout this paper, let Δ⁺ be the space of distribution functions, that is, Δ⁺{{}⟶[](0)(),∶=𝐹∶ℝ∪−∞,∞0,1∶𝐹isleft-continuous,nondecreasingonℝ,𝐹=0,𝐹+∞=1 (1.1) and the subset 𝐷⁺⊆Δ⁺ is the set: 𝐷⁺=𝐹∈Δ⁺∶𝑙⁻𝐹(+∞)=1, (1.2) where, 𝑙⁻𝑓(𝑥) denotes the left limit of the function 𝑓 at the point 𝑥. The space Δ⁺ is partially ordered by the usual point-wise ordering of functions, that is, 𝐹≤𝐺 if and only if 𝐹(𝑡)≤𝐺(𝑡) for all 𝑡∈ℝ. The maximal element for Δ⁺ in this order is the distribution function given by 𝜀₀(𝑡)=0,if𝑡≤0,1,if𝑡>0. (1.3)

Definition 1.1 (see [2]). A mapping  𝑇∶[0,1]×[0,1]→[0,1]   is a continuous triangular norm (briefly, a continuous  𝑡-norm) if 𝑇 satisfies the following conditions:(1)𝑇 is commutative and associative;(2)𝑇 is continuous;(3)𝑇(𝑎,1)=𝑎 for all 𝑎∈[0,1];(4)𝑇(𝑎,𝑏)≤𝑇(𝑐,𝑑) whenever 𝑎≤𝑐 and 𝑏≤𝑑 for all 𝑎,𝑏,𝑐,𝑑∈[0,1].

Two typical examples of continuous 𝑡-norms are 𝑇_𝑃(𝑎,𝑏)=𝑎𝑏, 𝑇_𝑀(𝑎,𝑏)=min(𝑎,𝑏). Recall (see [19, 20]) that if 𝑇 is a 𝑡-norm and {𝑥_𝑛} is a given sequence of numbers in [0,1], 𝑇^𝑛_𝑖=1𝑥_𝑖 is defined recurrently by 𝑇^𝑛_𝑖=1𝑥_𝑖=𝑥₁𝑇𝑇,if𝑛=1,_{𝑛−1𝑖=1}𝑥_𝑖,𝑥_𝑛,if𝑛≥2. (1.4)  𝑇^∞_𝑖=𝑛+1𝑥_𝑖 is defined as 𝑇^∞_𝑖=1𝑥_𝑛+𝑖.

Definition 1.2. A Menger probabilistic normed space (briefly, Menger PN-space) is a triple  (𝑋,Λ,𝑇) where,  𝑋  is a vector space,  𝑇  is a continuous  𝑡-norm, and  Λ  is a mapping from  𝑋  into  𝐷⁺  such that the following conditions hold: (PN₁)Λ_𝑥(0)=0 for all 𝑥∈𝑋;(PN₂)Λ_𝑥(𝑡)=𝜀₀(𝑡) for all 𝑡>0 if and only if 𝑥=0;(PN₃)Λ_𝛼𝑥(𝑡)=Λ_𝑥(𝑡/|𝛼|) for all 𝑥∈𝑋, 𝛼≠0 and all 𝑡>0;(PN₄)Λ_𝑥+𝑦(𝑡+𝑠)≥𝑇(Λ_𝑥(𝑡),Λ_𝑦(𝑠)) for all 𝑥,𝑦∈𝑋 and all 𝑡,𝑠≥0.

Clearly, every Menger PN-space is probabilistic metric space having a metrizable uniformity on 𝑋 if sup_𝑎<1𝑇(𝑎,𝑎)=1.

Definition 1.3. Let (𝑋,Λ,𝑇) be a Menger PN-space. (i)A sequence {𝑥_𝑛} in 𝑋 is said to be convergent to 𝑥 in 𝑋 if, for every 𝜖>0 and 𝜆>0, there exists positive integer 𝑁 such that Λ_{𝑥𝑛−𝑥}(𝜖)>1−𝜆 whenever 𝑛≥𝑁.(ii)A sequence {𝑥_𝑛} in 𝑋 is called Cauchy sequence if, for every 𝜖>0 and 𝜆>0, there exists positive integer 𝑁 such that Λ_{𝑥𝑛−𝑥𝑚}(𝜖)>1−𝜆 whenever 𝑛≥𝑚≥𝑁.(iii)A Menger PN-space (𝑋,Λ,𝑇) is said to be complete if and only if every Cauchy sequence in 𝑋 is convergent to a point in 𝑋.

Theorem 1.4. If (𝑋,Λ,𝑇) is a Menger PN-space and {𝑥_𝑛} is a sequence such that 𝑥_𝑛→𝑥, then lim_𝑛→∞Λ_𝑥𝑛(𝑡)=Λ_𝑥(𝑡).

The concept of stability of a functional equation arises when one replaces a functional equation by an inequality which acts as a perturbation of the equation. The first stability problem concerning group homomorphisms was raised by Ulam [21] in 1940 and affirmatively solved by Hyers [22]. The result of Hyers was generalized by Aoki [23] for approximate additive function and by Rassias [24] for approximate linear functions by allowing the difference Cauchy equation ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖ to be controlled by 𝜀(‖𝑥‖^𝑝+‖𝑦‖^𝑝). Taking into consideration a lot of influence of Ulam, Hyers, and Rassias on the development of stability problems of functional equations, the stability phenomenon that was proved by Rassias is called the Hyers-Ulam-Rassias stability. In 1994, a generalization of Rassias’ theorem was obtained by Găvruţa [25], who replaced 𝜀(‖𝑥‖^𝑝+‖𝑦‖^𝑝) by a general control function 𝜑(𝑥,𝑦). The functional equation, 𝑓𝑥₁+𝑥₂+𝑓𝑥₁−𝑥₂=2𝑓𝑥₁+2𝑓𝑥₂, (1.5) is related to symmetric biadditive function and is called a quadratic functional equation and every solution of the quadratic equation (1.5) is said to be a quadratic function. For more details about the results concerning such problems, the reader is referred to [26–28].

The functional equation, 𝑓2𝑥₁+𝑥₂+𝑓2𝑥₁−𝑥₂=2𝑓𝑥₁+𝑥₂+2𝑓𝑥₁−𝑥₂+12𝑓𝑥₁, (1.6) is called the cubic functional equation, since the function 𝑓(𝑥)=𝑐𝑥³ is its solution. Every solution of the cubic functional equation is said to be a cubic mapping. The stability results for the cubic functional equation were proved by Jun and Kim [29].

Eshaghi Gordji and Khodaei [30] have established the general solution and investigated the Hyers-Ulam-Rassias stability for a mixed type of cubic, quadratic, and additive functional equations, with 𝑓(0)=0, 𝑓𝑥₁+𝑘𝑥₂+𝑓𝑥₁−𝑘𝑥₂=𝑘²𝑓𝑥₁+𝑥₂+𝑘²𝑓𝑥₁−𝑥₂+21−𝑘²𝑓𝑥₁ (1.7) in quasi-Banach spaces, where 𝑘 is nonzero integer numbers with 𝑘≠±1. It is easy to see that the function 𝑓(𝑥)=𝑎𝑥+𝑏𝑥²+𝑐𝑥³ is a solution of the functional equation (1.7), see [31, 32]. The stability of different functional equations in probabilistic normed spaces, RN-spaces, and fuzzy normed spaces has been studied in [6, 7, 33–37].

Now, we introduce the new mixed type of cubic, quadratic, and additive functional equation in 𝑛-variables as follows: ^𝑛_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑛_{𝑖𝑛−𝑘+1=𝑖𝑛−𝑘+1}𝑓^𝑛_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑛−𝑘+1}𝑥_𝑖−^{𝑛−𝑘+1}_𝑟=1𝑥_𝑖𝑟+𝑓^𝑛_𝑖=1𝑥_𝑖=2^𝑛𝑛−2_𝑖=2𝑓𝑥₁+𝑥_𝑖+𝑓𝑥₁−𝑥_𝑖−2^𝑛−1(𝑛−2)𝑓𝑥₁, (1.8) where 𝑛≥3 and 𝑓(0)=0. As a special case, if 𝑛=3 in (1.8), then (1.8) reduces to ²_𝑖1³_{=2𝑖2=𝑖1+1}𝑓³_{𝑖=1,𝑖≠𝑖1,𝑖2}𝑥_𝑖−²_𝑟=1𝑥_𝑖𝑟+³_𝑖1=2𝑓³_{𝑖=1,𝑖≠𝑖1}𝑥_𝑖−𝑥_𝑖1+𝑓³_𝑖=1𝑥_𝑖=2³_𝑖=2𝑓𝑥₁+𝑥_𝑖+𝑓𝑥₁−𝑥_𝑖−2²𝑓𝑥₁, (1.9) that is, 𝑓𝑥₁−𝑥₂−𝑥₃+𝑓𝑥₁−𝑥₂+𝑥₃+𝑓𝑥₁+𝑥₂−𝑥₃+𝑓𝑥₁+𝑥₂+𝑥₃=2𝑓𝑥₁+𝑥₂+𝑓𝑥₁−𝑥₂+𝑓𝑥₁+𝑥₃+𝑓𝑥₁−𝑥₃−4𝑓𝑥₁. (1.10) The main purpose of this paper is to prove the stability for (1.8), in Menger probabilistic normed spaces.

2. Results in Menger Probabilistic Normed Spaces

We start our work with a general solution for the mixed functional equation (1.8) and then investigate the stability of this equation in Menger PN-space.

Theorem 2.1. Let 𝑋 and 𝑌 be vector spaces. A function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies (1.8) for all 𝑥₁,…,𝑥_𝑛∈𝑋 if and only if there exist functions 𝐶∶𝑋×𝑋×𝑋→𝑌, 𝐵∶𝑋×𝑋→𝑌 and 𝐴∶𝑋→𝑌 such that 𝑓(𝑥)=𝐶(𝑥,𝑥,𝑥)+𝐵(𝑥,𝑥)+𝐴(𝑥) for all 𝑥∈𝑋, where the function 𝐶 is symmetric for each fixed one variable and is additive for fixed two variables, 𝐵 is symmetric biadditive and 𝐴 is additive.

Proof. If there exists a function 𝐶 that is symmetric for each fixed one variable and is additive for fixed two variables, 𝐵 is biadditive, and 𝐴 is additive, then by a simple computation one can show that the functions 𝑥↦𝐶(𝑥,𝑥,𝑥), 𝑥↦𝐵(𝑥,𝑥), and 𝑥↦𝐴(𝑥) satisfy the functional equation (1.8). Therefore, the function 𝑓 satisfies (1.8).
Conversely, let 𝑓 with 𝑓(0)=0 satisfies (1.8). Hence, according to (1.8), we get ²_𝑖1³_{=2𝑖2=𝑖1+1}⋯^𝑛_{𝑖𝑛−1=𝑖𝑛−2+1}𝑓^𝑛_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑛−1}𝑥_𝑖−^𝑛−1_𝑟=1𝑥_𝑖𝑟+³_𝑖1⁴_{=2𝑖2=𝑖1+1}⋯^𝑛_{𝑖𝑛−2=𝑖𝑛−3+1}𝑓^𝑛_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑛−2}𝑥_𝑖−^𝑛−2_𝑟=1𝑥_𝑖𝑟+⋯+^𝑛_𝑖1=2𝑓^𝑛_{𝑖=1,𝑖≠𝑖1}𝑥_𝑖−𝑥_𝑖1+𝑓^𝑛_𝑖=1𝑥_𝑖=2^𝑛𝑛−2_𝑖=2𝑓𝑥₁+𝑥_𝑖+𝑓𝑥₁−𝑥_𝑖−2^𝑛−1(𝑛−2)𝑓𝑥₁ (2.1) for all 𝑥₁,…,𝑥_𝑛∈𝑋. Setting 𝑥_𝑖=0(𝑖=4,…,𝑛) in (2.1), we have 𝑓𝑥₁−𝑥₂−𝑥₃+(𝑛−3)𝑓𝑥₁−𝑥₂−𝑥₃+𝑓𝑥₁−𝑥₂+𝑥₃+𝑓𝑥₁+𝑥₂−𝑥₃+2𝑓𝑥𝑛−3₁−𝑥₂−𝑥₃+𝑓𝑥𝑛−3𝑛−4¹−𝑥₂+𝑥₃+𝑓𝑥𝑛−3𝑛−4¹+𝑥₂−𝑥₃+𝑓𝑥𝑛−3𝑛−3¹+𝑥₂+𝑥₃+3𝑓𝑥𝑛−3₁−𝑥₂−𝑥₃+𝑓𝑥𝑛−3𝑛−5¹−𝑥₂+𝑥₃+𝑓𝑥𝑛−3𝑛−5¹+𝑥₂−𝑥₃+𝑓𝑥𝑛−3𝑛−4¹+𝑥₂+𝑥₃+𝑓𝑥+⋯𝑛−3𝑛−3¹−𝑥₂−𝑥₃+1𝑓𝑥𝑛−3₁−𝑥₂+𝑥₃+1𝑓𝑥𝑛−3₁+𝑥₂−𝑥₃+2𝑓𝑥𝑛−3₁+𝑥₂+𝑥₃+𝑓𝑥₁−𝑥₂+𝑥₃+𝑓𝑥₁+𝑥₂−𝑥₃+(𝑛−3)𝑓𝑥₁+𝑥₂+𝑥₃+𝑓𝑥₁+𝑥₂+𝑥₃=2^3𝑛−2_𝑖=2𝑓𝑥₁+𝑥_𝑖+𝑓𝑥₁−𝑥_𝑖−2^𝑛−1𝑓𝑥₁, (2.2) that is, 1+^𝑛−3_ℓ=1ℓ𝑓𝑥𝑛−3₁−𝑥₂−𝑥₃+𝑓𝑥₁−𝑥₂+𝑥₃+𝑓𝑥₁+𝑥₂−𝑥₃+𝑓𝑥₁+𝑥₂+𝑥₃=2^𝑛−2𝑓𝑥₁+𝑥₂+𝑓𝑥₁−𝑥₂+𝑓𝑥₁+𝑥₃+𝑓𝑥₁−𝑥₃−2^𝑛−1𝑓𝑥₁ (2.3) for all 𝑥₁,𝑥₂,𝑥₃∈𝑋. On the other hand, we have the relation: 1+^𝑛−3_ℓ=1ℓ=𝑛−3^𝑛−3_𝑖=0ℓ𝑛−3=2^𝑛−3 (2.4) for all 𝑛≥3. Hence, we obtain from (2.3) and (2.4) that 𝑓𝑥₁−𝑥₂−𝑥₃+𝑓𝑥₁−𝑥₂+𝑥₃+𝑓𝑥₁+𝑥₂−𝑥₃+𝑓𝑥₁+𝑥₂+𝑥₃=2𝑓𝑥₁+𝑥₂+𝑓𝑥₁−𝑥₂+𝑓𝑥₁+𝑥₃+𝑓𝑥₁−𝑥₃−4𝑓𝑥₁ (2.5) for all 𝑥₁,𝑥₂,𝑥₃∈𝑋. Replacing 𝑥₃ by 𝑥₂ in (2.5), one gets 𝑓𝑥₁+2𝑥₂+𝑓𝑥₁−2𝑥₂=4𝑓𝑥₁+𝑥₂+4𝑓𝑥₁−𝑥₂−6𝑓𝑥₁ (2.6) for all 𝑥₁,𝑥₂∈𝑋. Therefore, 𝑓 satisfies (1.7) for 𝑘=2. By Theorem  2.3 of [30], there exist an additive function 𝐴∶𝑋→𝑌, symmetric biadditive function 𝐵∶𝑋×𝑋→𝑌, and 𝐶∶𝑋×𝑋×𝑋→𝑌 such that 𝑓(𝑥)=𝐶(𝑥,𝑥,𝑥)+𝐵(𝑥,𝑥)+𝐴(𝑥) for all 𝑥∈𝑋, where the function 𝐶 is symmetric for each fixed one variable and is additive for fixed two variables.

From now on, let 𝑋 be a real linear space and let (𝑌,Λ,𝑇) be a complete Menger PN-space. For convenience, we use the following abbreviation for a given function 𝑓∶𝑋→𝑌:Δ𝑓𝑥₁,…,𝑥_𝑛=^𝑛_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑛_{𝑖𝑛−𝑘+1=𝑖𝑛−𝑘+1}𝑓^𝑛_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑛−𝑘+1}𝑥_𝑖−^{𝑛−𝑘+1}_𝑟=1𝑥_𝑖𝑟+𝑓^𝑛_𝑖=1𝑥_𝑖−2^𝑛𝑛−2_𝑖=2𝑓𝑥₁+𝑥_𝑖+𝑓𝑥₁−𝑥_𝑖+2^𝑛−1(𝑛−2)𝑓𝑥₁ (2.7) for all 𝑥₁,…,𝑥_𝑛∈𝑋.

Theorem 2.2. Let 𝜉∶𝑋^𝑛→𝐷⁺(𝑛∈ℕ, 𝑛≥3 and 𝜉(𝑥₁,…,𝑥_𝑛) is denoted by 𝜉_{𝑥1,…,𝑥𝑛}) be a function such that lim_𝑚→∞𝜉_{2𝑚𝑥1,…,2𝑚𝑥𝑛}2^2𝑚𝑡=1 (2.8) for all 𝑥₁,…,𝑥_𝑛∈𝑋, 𝑡>0 and lim_𝑚→∞𝑇^∞_ℓ=1𝜉_{0,…,0,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥}2^{2𝑚+ℓ+𝑛−2}𝑡=1 (2.9) for all 𝑥∈𝑋 and 𝑡>0. Suppose that an even function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality: Λ_{Δ𝑓(𝑥1,…,𝑥𝑛)}(𝑡)≥𝜉_{𝑥1,…,𝑥𝑛}(𝑡) (2.10) for all 𝑥₁,…,𝑥_𝑛∈𝑋 and 𝑡>0. Then, there exists a unique quadratic function 𝑄∶𝑋→𝑌 such that Λ_{𝑓(𝑥)−𝑄(𝑥)}(𝑡)≥𝑇^∞_ℓ=1𝜉_{0,…,0,2ℓ−1𝑥,2ℓ−1𝑥}2^ℓ+𝑛−2𝑡 (2.11) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Setting 𝑥_𝑖=0(𝑖=1,…,𝑛−2) and 𝑥_𝑛−1=𝑥_𝑛=𝑥 in (2.10) and then use 𝑓(0)=0, we obtain that Λ_{1+∑𝑛−3ℓ=1ℓ𝑛−3(𝑓(2𝑥)+𝑓(−2𝑥))−2𝑛−1(𝑓(𝑥)+𝑓(−𝑥))}(𝑡)≥𝜉_{0,…,0,𝑥,𝑥}(𝑡) (2.12) for all 𝑥∈𝑋 and 𝑡>0. By using evenness of 𝑓 and the relation 1+∑^𝑛−3_ℓ=1ℓ^𝑛−3=∑^𝑛−3_𝑖=0ℓ^𝑛−3=2^𝑛−3, we get Λ_{2𝑛−2𝑓(2𝑥)−2𝑛𝑓(𝑥)}(𝑡)≥𝜉_{0,…,0,𝑥,𝑥}(𝑡) (2.13) for all 𝑥∈𝑋 and 𝑡>0. So, Λ_{𝑓(2𝑥)/22−𝑓(𝑥)}(𝑡)≥𝜉_{0,…,0,𝑥,𝑥}2^𝑛𝑡≥𝜉_{0,…,0,𝑥,𝑥}2^𝑛−1𝑡 (2.14) for all 𝑥∈𝑋 and 𝑡>0, which implies that Λ_{𝑓(2ℓ+1𝑥)/22(ℓ+1)−𝑓(2ℓ𝑥)/22ℓ}(𝑡)≥𝜉_{0,…,0,2ℓ𝑥,2ℓ𝑥}2^𝑛+2ℓ𝑡 (2.15) for all 𝑥∈𝑋, 𝑡>0 and ℓ∈ℕ. It follows from (2.15) and (PN₄) that Λ_{𝑓(22𝑥)/24−𝑓(𝑥)}(𝑡)≥𝑇Λ_{𝑓(22𝑥)/24−𝑓(2𝑥)/22}𝑡2,Λ_{𝑓(2𝑥)/22−𝑓(𝑥)}𝑡2≥𝑇𝜉_{0,…,0,2𝑥,2𝑥}2^𝑛+1𝑡,𝜉_{0,…,0,𝑥,𝑥}2^𝑛−1𝑡≥𝑇𝜉_{0,…,0,2𝑥,2𝑥}2^𝑛𝑡,𝜉_{0,…,0,𝑥,𝑥}2^𝑛−1𝑡,Λ_{𝑓(23𝑥)/26−𝑓(𝑥)}(𝑡)≥𝑇Λ_{𝑓(23𝑥)/26−𝑓(2𝑥)/22}𝑡2,Λ_{𝑓(2𝑥)/22−𝑓(𝑥)}𝑡2≥𝑇𝑇Λ_{𝑓(23𝑥)/26−𝑓(22𝑥)/24}𝑡4,Λ_{𝑓(22𝑥)/24−𝑓(2𝑥)/22}𝑡4,Λ_{𝑓(2𝑥)/22−𝑓(𝑥)}𝑡2≥𝑇𝑇𝜉_{0,…,0,22𝑥,22𝑥}2^𝑛+2𝑡,𝜉_{0,…,0,2𝑥,2𝑥}2^𝑛𝑡,𝜉_{0,…,0,𝑥,𝑥}2^𝑛−1𝑡≥𝑇𝑇𝜉_{0,…,0,22𝑥,22𝑥}2^𝑛+1𝑡,𝜉_{0,…,0,2𝑥,2𝑥}2^𝑛𝑡,𝜉_{0,…,0,𝑥,𝑥}2^𝑛−1𝑡=𝑇𝜉_{0,…,0,𝑥,𝑥}2^𝑛−1𝑡,𝑇𝜉_{0,…,0,2𝑥,2𝑥}2^𝑛𝑡,𝜉_{0,…,0,22𝑥,22𝑥}2^𝑛+1𝑡=𝑇𝑇𝜉_{0,…,0,𝑥,𝑥}2^𝑛−1𝑡,𝜉_{0,…,0,2𝑥,2𝑥}2^𝑛𝑡,𝜉_{0,…,0,22𝑥,22𝑥}2^𝑛+1𝑡 (2.16) for all 𝑥∈𝑋 and 𝑡>0. Thus, Λ_{𝑓(2𝑚𝑥)/22𝑚−𝑓(𝑥)}(𝑡)≥𝑇^𝑚_ℓ=1𝜉_{0,…,0,2ℓ−1𝑥,2ℓ−1𝑥}2^ℓ+𝑛−2𝑡 (2.17) for all 𝑥∈𝑋 and 𝑡>0. In order to prove the convergence of the sequence {𝑓(2^𝑚𝑥)/2^2𝑚}, we replace 𝑥 with 2^𝑚′𝑥 in (2.17) to find that Λ_{𝑓(2′𝑚+𝑚𝑥)/2′)2(𝑚+𝑚−𝑓(2𝑚′𝑥)/2′2𝑚}(𝑡)≥𝑇^𝑚_ℓ=1𝜉_{0,…,0,2𝑚′+ℓ−1𝑥,2𝑚′+ℓ−1𝑥}2^{2𝑚′+ℓ+𝑛−2}𝑡 (2.18) for all 𝑥∈𝑥 and 𝑡>0. Since the right-hand side of the inequality tends to 1 as 𝑚′ and 𝑚 tend to infinity, the sequence {𝑓(2^𝑚𝑥)/2^2𝑚} is a Cauchy sequence. Therefore, one can define the function 𝑄∶𝑋→𝑌 by 𝑄(𝑥)∶=lim_𝑚→∞(1/2^2𝑚)𝑓(2^𝑚𝑥) for all 𝑥∈𝑋. Now, if we replace 𝑥₁,…,𝑥_𝑛 with 2^𝑚𝑥₁,…,2^𝑚𝑥_𝑛 in (2.10), respectively, it follows that Λ_{Δ𝑓(2𝑚𝑥1,…,2𝑚𝑥𝑛)/22𝑚}(𝑡)≥𝜉_{2𝑚𝑥1,…,2𝑚𝑥𝑛}2^2𝑚𝑡 (2.19) for all 𝑥₁,…,𝑥_𝑛∈𝑥 and 𝑡>0. By letting 𝑚→∞ in (2.19), we find that Λ_{Δ𝑄(𝑥1,…,𝑥𝑛)}(𝑡)=1 for all 𝑡>0, which implies that Δ𝑄(𝑥₁,…,𝑥_𝑛)=0 thus 𝑄 satisfies (1.8). Hence, by Theorem 2.1 (see [30, Lemma  2.1]), the function 𝑄∶𝑋→𝑌 is quadratic.
To prove (2.11), take the limit as 𝑚→∞ in (2.17).
Finally, to prove the uniqueness of the quadratic function 𝑄 subject to (2.11), let us assume that there exists a quadratic function 𝑄^′ which satisfies (2.11). Since 𝑄(2^𝑚𝑥)=2^2𝑚𝑄(𝑥) and 𝑄^′(2^𝑚𝑥)=2^2𝑚𝑄^′(𝑥) for all 𝑥∈𝑋 and 𝑚∈ℕ, from (2.11), it follows that Λ_{𝑄(𝑥)−𝑄′(𝑥)}(𝑡)=Λ_{𝑄(2𝑚𝑥)−𝑄′(2𝑚𝑥)}2^2𝑚𝑡≥𝑇Λ_{𝑄(2𝑚𝑥)−𝑓(2𝑚𝑥)}2^2𝑚−1𝑡,Λ_{𝑓(2𝑚𝑥)−𝑄′(2𝑚𝑥)}2^2𝑚−1𝑡≥𝑇𝑇^∞_ℓ=1𝜉_{0,…,0,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥}2^{2𝑚+ℓ+𝑛−2}𝑡,𝑇^∞_ℓ=1𝜉_{0,…,0,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥}2^{2𝑚+ℓ+𝑛−2}𝑡 (2.20) for all 𝑥∈𝑋 and 𝑡>0. By letting 𝑚→∞ in (2.20), we find that 𝑄=𝑄^′.

Theorem 2.3. Let 𝜉∶𝑋^𝑛→𝐷⁺ be a function such that lim_𝑚→∞𝑇𝜉_{2𝑚𝑥1,…,2𝑚𝑥𝑛}2^𝑚𝑡,𝜉_{2𝑚𝑥1,…,2𝑚𝑥𝑛}2^𝑚−4𝑡=1 (2.21) for all 𝑥₁,…,𝑥_𝑛∈𝑋, 𝑡>0 and lim_𝑚→∞𝑇^∞_ℓ=1𝑇𝜉_{2𝑚+ℓ𝑥,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,0,…,0,}2^{𝑚+𝑛−5}𝑡,𝜉_{2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,0,…,0}2^{𝑚+𝑛−7}𝑡=1 (2.22) for all 𝑥∈𝑋 and 𝑡>0. Suppose that an odd function 𝑓∶𝑋→𝑌 satisfies (2.10) for all 𝑥₁,…,𝑥_𝑛∈𝑋 and 𝑡>0. Then, there exists a unique additive function 𝐴∶𝑋→𝑌 such that Λ_{𝑓(2𝑥)−8𝑓(𝑥)−𝐴(𝑥)}(𝑡)≥𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}2^𝑛−4𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}2^𝑛−6𝑡 (2.23) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Setting 𝑥₁=𝑥₂=𝑥₃=𝑥 and 𝑥_𝑖=0(𝑖=4,…,𝑛) in (2.10), we obtain Λ_{∑𝑛−3ℓ=0ℓ𝑛−3(𝑓(3𝑥)+2𝑓(𝑥)+𝑓(−𝑥))−2𝑛−1𝑓(2𝑥)+2𝑛−1𝑓(𝑥)}(𝑡)≥𝜉_{𝑥,𝑥,𝑥,0,…,0}(𝑡) (2.24) for all 𝑥∈𝑋 and 𝑡>0. By using oddness of 𝑓 and the relation ∑^𝑛−3_𝑖=0ℓ^𝑛−3=2^𝑛−3, we lead to Λ_{𝑓(3𝑥)−4𝑓(2𝑥)+5𝑓(𝑥)}(𝑡)≥𝜉_{𝑥,𝑥,𝑥,0,…,0}2^𝑛−3𝑡 (2.25) for all 𝑥∈𝑋 and 𝑡>0. Putting 𝑥₁=2𝑥,𝑥₂=𝑥₃=𝑥 and 𝑥_𝑖=0(𝑖=4,…,𝑛) in (2.10), we have Λ_{∑𝑛−3ℓ=0ℓ𝑛−3(𝑓(4𝑥)+2𝑓(2𝑥))−2𝑛−2(2𝑓(3𝑥)+2𝑓(𝑥))+2𝑛−1𝑓(2𝑥)}(𝑡)≥𝜉_{2𝑥,𝑥,𝑥,0,…,0}(𝑡) (2.26) for all 𝑥∈𝑋 and 𝑡>0. So Λ_{𝑓(4𝑥)−4𝑓(3𝑥)+6𝑓(2𝑥)−4𝑓(𝑥)}(𝑡)≥𝜉_{2𝑥,𝑥,𝑥,0,…,0}2^𝑛−3𝑡 (2.27) for all 𝑥∈𝑋 and 𝑡>0. It follows from (2.25), (2.27), and (PN₄) that Λ_{𝑓(4𝑥)−10𝑓(2𝑥)+16𝑓(𝑥)}(𝑡)≥𝑇𝜉_{2𝑥,𝑥,𝑥,0,…,0}2^𝑛−4𝑡,𝜉_{𝑥,𝑥,𝑥,0,…,0}2^𝑛−6𝑡 (2.28) for all 𝑥∈𝑋 and 𝑡>0. Let 𝑔∶𝑋→𝑌 be a function defined by 𝑔(𝑥)∶=𝑓(2𝑥)−8𝑓(𝑥) for all 𝑥∈𝑋. From (2.28), we conclude that Λ_{𝑔(2𝑥)/2−𝑔(𝑥)}(𝑡)≥𝑇𝜉_{2𝑥,𝑥,𝑥,0,…,0}2^𝑛−3𝑡,𝜉_{𝑥,𝑥,𝑥,0,…,0}2^𝑛−5𝑡≥𝑇𝜉_{2𝑥,𝑥,𝑥,0,…,0}2^𝑛−4𝑡,𝜉_{𝑥,𝑥,𝑥,0,…,0}2^𝑛−6𝑡 (2.29) for all 𝑥∈𝑋 and 𝑡>0, which implies that Λ_{𝑔(2ℓ+1𝑥)/2ℓ+1−𝑔(2ℓ𝑥)/2ℓ}(𝑡)≥𝑇𝜉_{2ℓ+1𝑥,2ℓ𝑥,2ℓ𝑥,0,…,0}2^ℓ+𝑛−3𝑡,𝜉_{2ℓ𝑥,2ℓ𝑥,2ℓ𝑥,0,…,0}2^ℓ+𝑛−5𝑡 (2.30) for all 𝑥∈𝑋, 𝑡>0 and ℓ∈ℕ. It follows from (2.30) and (PN₄) that Λ_{𝑔(22𝑥)/22−𝑔(𝑥)}(𝑡)≥𝑇Λ_{𝑔(22𝑥)/22−𝑔(2𝑥)/2}𝑡2,Λ_{𝑔(2𝑥)/2−𝑔(𝑥)}𝑡2≥𝑇𝑇𝜉_{22𝑥,2𝑥,2𝑥,0,…,0}2^𝑛−3𝑡,𝜉_{2𝑥,2𝑥,2𝑥,0,…,0}2^𝑛−5𝑡,𝑇𝜉_{2𝑥,𝑥,𝑥,0,…,0}2^𝑛−4𝑡,𝜉_{𝑥,𝑥,𝑥,0,…,0}2^𝑛−6𝑡≥𝑇𝑇𝜉_{22𝑥,2𝑥,2𝑥,0,…,0}2^𝑛−4𝑡,𝜉_{2𝑥,2𝑥,2𝑥,0,…,0}2^𝑛−6𝑡,𝑇𝜉_{2𝑥,𝑥,𝑥,0,…,0}2^𝑛−4𝑡,𝜉_{𝑥,𝑥,𝑥,0,…,0}2^𝑛−6𝑡,Λ_{𝑔(23𝑥)/23−𝑔(𝑥)}(𝑡)≥𝑇Λ_{𝑔(23𝑥)/23−𝑔(2𝑥)/2}𝑡2,Λ_{𝑔(2𝑥)/2−𝑔(𝑥)}𝑡2≥𝑇𝑇Λ_{𝑔(23𝑥)/23−𝑔(22𝑥)/22}𝑡4,Λ_{𝑔(22𝑥)/22−𝑔(2𝑥)/2}𝑡4,Λ_{𝑔(2𝑥)/2−𝑔(𝑥)}𝑡2≥𝑇𝑇𝑇𝜉_{23𝑥,22𝑥,22𝑥,0,…,0}2^𝑛−3𝑡,𝜉_{22𝑥,22𝑥,22𝑥,0,…,0}2^𝑛−5𝑡,𝑇𝜉_{22𝑥,2𝑥,2𝑥,0,…,0}2^𝑛−4𝑡,𝜉_{2𝑥,2𝑥,2𝑥,0,…,0}2^𝑛−6𝑡,𝑇𝜉_{2𝑥,𝑥,𝑥,0,…,0}2^𝑛−4𝑡,𝜉_{𝑥,𝑥,𝑥,0,…,0}2^𝑛−6𝑡≥𝑇𝑇𝑇𝜉_{23𝑥,22𝑥,22𝑥,0,…,0}2^𝑛−4𝑡,𝜉_{22𝑥,22𝑥,22𝑥,0,…,0}2^𝑛−6𝑡,𝑇𝜉_{22𝑥,2𝑥,2𝑥,0,…,0}2^𝑛−4𝑡,𝜉_{2𝑥,2𝑥,2𝑥,0,…,0}2^𝑛−6𝑡,𝑇𝜉_{2𝑥,𝑥,𝑥,0,…,0}2^𝑛−4𝑡,𝜉_{𝑥,𝑥,𝑥,0,…,0}2^𝑛−6𝑡=𝑇𝑇𝑇𝜉_{2𝑥,𝑥,𝑥,0,…,0}2^𝑛−4𝑡,𝜉_{𝑥,𝑥,𝑥,0,…,0}2^𝑛−6𝑡,𝑇𝜉_{22𝑥,2𝑥,2𝑥,0,…,0}2^𝑛−4𝑡,𝜉_{2𝑥,2𝑥,2𝑥,0,…,0}2^𝑛−6𝑡,𝑇𝜉_{23𝑥,22𝑥,22𝑥,0,…,0}2^𝑛−4𝑡,𝜉_{22𝑥,22𝑥,22𝑥,0,…,0}2^𝑛−6𝑡 (2.31) for all 𝑥∈𝑋 and 𝑡>0. Thus, Λ_{𝑔(2𝑚𝑥)/2𝑚−𝑔(𝑥)}(𝑡)≥𝑇^𝑚_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}2^𝑛−4𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}2^𝑛−6𝑡 (2.32) for all 𝑥∈𝑋 and 𝑡>0. In order to prove the convergence of the sequence {𝑔(2^𝑚𝑥)/2^𝑚}, we replace 𝑥 with 2^𝑚′𝑥 in (2.32) to find that Λ_{𝑔(2′𝑚+𝑚𝑥)/2′𝑚+𝑚−𝑔(2𝑚′𝑥)/2𝑚′}(𝑡)≥𝑇^𝑚_ℓ=1𝑇𝜉_{2𝑚′+ℓ𝑥,2𝑚′+ℓ−1𝑥,2𝑚′+ℓ−1𝑥,0,…,0}2^{𝑚′+𝑛−4}𝑡,𝜉_{2𝑚′+ℓ−1𝑥,2𝑚′+ℓ−1𝑥,2𝑚′+ℓ−1𝑥,0,…,0}2^{𝑚′+𝑛−6}𝑡 (2.33) for all 𝑥∈𝑋 and 𝑡>0. Since the right-hand side of the inequality tends to 1 as 𝑚′ and 𝑚 tend to infinity, the sequence {𝑔(2^𝑚𝑥)/2^𝑚} is a Cauchy sequence. Therefore, one can define the function 𝐴∶𝑋→𝑌 by 𝐴(𝑥)∶=lim_𝑚→∞(1/2^𝑚)𝑔(2^𝑚𝑥) for all 𝑥∈𝑋. Now, if we replace 𝑥₁,…,𝑥_𝑛 with 2^𝑚𝑥₁,…,2^𝑚𝑥_𝑛 in (2.10), respectively, it follows that Λ_{Δ𝑔(2𝑚𝑥1,…,2𝑚𝑥𝑛)/2𝑚}(𝑡)=Λ_{Δ𝑓(2𝑚+1𝑥1,…,2𝑚+1𝑥𝑛)/2𝑚−8(Δ𝑓(2𝑚𝑥1,…,2𝑚𝑥𝑛)/2𝑚)}(𝑡)≥𝑇Λ_{Δ𝑓(2𝑚+1𝑥1,…,2𝑚+1𝑥𝑛)/2𝑚}𝑡2,Λ_{Δ𝑓(2𝑚𝑥1,…,2𝑚𝑥𝑛)/2𝑚−3}𝑡2≥𝑇𝜉_{2𝑚+1𝑥1,…,2𝑚+1𝑥𝑛}2^𝑚−1𝑡,𝜉_{2𝑚𝑥1,…,2𝑚𝑥𝑛}2^𝑚−4𝑡 (2.34) for all 𝑥₁,…,𝑥_𝑛∈𝑋 and 𝑡>0. By letting 𝑚→∞ in (2.34), we find that Λ_{Δ𝐴(𝑥1,…,𝑥𝑛)}(𝑡)=1 for all 𝑡>0, which implies Δ𝐴(𝑥₁,…,𝑥_𝑛)=0, thus 𝐴 satisfies (1.8). Hence, by Theorem 2.1 (see [30, Lemma  2.2]) the function 𝐴∶𝑋→𝑌 is additive.
To prove (2.23), take the limit as 𝑚→∞ in (2.32).
Finally, to prove the uniqueness of the additive function 𝐴 subject to (2.23), let us assume that there exists a additive function 𝐴′ which satisfies (2.23). Since 𝐴(2^𝑚𝑥)=2^𝑚𝐴(𝑥) and 𝐴′(2^𝑚𝑥)=2^𝑚𝐴′(𝑥) for all 𝑥∈𝑋 and 𝑚∈ℕ, from (2.23), it follows that Λ_{𝐴(𝑥)−𝐴′(𝑥)}(𝑡)=Λ_{𝐴(2𝑚𝑥)−𝐴′(2𝑚𝑥)}2^𝑚𝑡≥𝑇Λ_{𝐴(2𝑚𝑥)−𝑔(2𝑚𝑥)}2^𝑚−1𝑡,Λ_{𝑔(2𝑚𝑥)−𝐴′(2𝑚𝑥)}2^𝑚−1𝑡≥𝑇𝑇^∞_ℓ=1𝑇𝜉_{2𝑚+ℓ𝑥,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,0,…,0,}2^{𝑚+𝑛−5}𝑡,𝜉_{2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,0,…,0}2^{𝑚+𝑛−7}𝑡,𝑇^∞_ℓ=1𝑇𝜉_{2𝑚+ℓ𝑥,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,0,…,0,}2^{𝑚+𝑛−5}𝑡,𝜉_{2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,0,…,0}2^{𝑚+𝑛−7}𝑡 (2.35) for all 𝑥∈𝑋 and 𝑡>0. By letting 𝑚→∞ in (2.35), we find that 𝐴=𝐴′.

Theorem 2.4. Let 𝜉∶𝑋^𝑛→𝐷⁺ be a function such that lim_𝑚→∞𝑇𝜉_{2𝑚𝑥1,…,2𝑚𝑥𝑛}2^3𝑚𝑡,𝜉_{2𝑚𝑥1,…,2𝑚𝑥𝑛}2^3𝑚−2𝑡=1 (2.36) for all 𝑥₁,…,𝑥_𝑛∈𝑋, 𝑡>0 and lim_𝑚→∞𝑇^∞_ℓ=1𝑇𝜉_{2𝑚+ℓ𝑥,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,0,…,0,}2^{3𝑚+2ℓ+𝑛−5}𝑡,𝜉_{2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,0,…,0}2^{3𝑚+2ℓ+𝑛−7}𝑡=1 (2.37) for all 𝑥∈𝑋 and 𝑡>0. Suppose that an odd function 𝑓∶𝑋→𝑌 satisfies (2.10) for all 𝑥₁,…,𝑥_𝑛∈𝑋 and 𝑡>0. Then, there exists a unique cubic function 𝐶∶𝑋→𝑌 such that Λ_{𝑓(2𝑥)−2𝑓(𝑥)−𝐶(𝑥)}(𝑡)≥𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}2^{2ℓ+𝑛−4}𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}2^{2ℓ+𝑛−6}𝑡 (2.38) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Similar to proof of Theorem 2.3, we obtain (2.28) for all 𝑥∈𝑋 and 𝑡>0. Let ℎ∶𝑋→𝑌 be a function defined by ℎ(𝑥)∶=𝑓(2𝑥)−2𝑓(𝑥) for all 𝑥∈𝑋. Therefore, (2.28) implies that Λ_{ℎ(2𝑥)/23−ℎ(𝑥)}(𝑡)≥𝑇𝜉_{2𝑥,𝑥,𝑥,0,…,0}2^𝑛−1𝑡,𝜉_{𝑥,𝑥,𝑥,0,…,0}2^𝑛−3𝑡≥𝑇𝜉_{2𝑥,𝑥,𝑥,0,…,0}2^𝑛−2𝑡,𝜉_{𝑥,𝑥,𝑥,0,…,0}2^𝑛−4𝑡 (2.39) for all 𝑥∈𝑋 and 𝑡>0, which implies that Λ_{ℎ(2ℓ+1𝑥)/23(ℓ+1)−ℎ(2ℓ𝑥)/23ℓ}(𝑡)≥𝑇𝜉_{2ℓ+1𝑥,2ℓ𝑥,2ℓ𝑥,0,…,0}2^{3ℓ+𝑛−1}𝑡,𝜉_{2ℓ𝑥,2ℓ𝑥,2ℓ𝑥,0,…,0}2^{3ℓ+𝑛−3}𝑡 (2.40) for all 𝑥∈𝑋, 𝑡>0, and ℓ∈ℕ. It follows from (2.40) and (PN₄) that Λ_{ℎ(2𝑚𝑥)/23𝑚−ℎ(𝑥)}(𝑡)≥𝑇^𝑚_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}2^{2ℓ+𝑛−4}𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}2^{2ℓ+𝑛−6}𝑡 (2.41) for all 𝑥∈𝑋 and 𝑡>0. In order to prove the convergence of the sequence {ℎ(2^𝑚𝑥)/2^3𝑚}, we replace 𝑥 with 2^𝑚′𝑥 in (2.41) to find that Λ_{ℎ(2′𝑚+𝑚𝑥)/2′)3(𝑚+𝑚−ℎ(2𝑚′𝑥)/2′3𝑚}(𝑡)≥𝑇^𝑚_ℓ=1𝑇𝜉_{2𝑚′+ℓ𝑥,2𝑚′+ℓ−1𝑥,2𝑚′+ℓ−1𝑥,0,…,0}2^{3𝑚′+2ℓ+𝑛−4}𝑡,𝜉_{2𝑚′+ℓ−1𝑥,2𝑚′+ℓ−1𝑥,2′3𝑚+ℓ−1𝑥,0,…,0}2^{3𝑚′+2ℓ+𝑛−6}𝑡 (2.42) for all 𝑥∈𝑥 and 𝑡>0. Since the right-hand side of the inequality tends to 1 as 𝑚′ and 𝑚 tend to infinity, the sequence {ℎ(2^𝑚𝑥)/2^3𝑚} is a Cauchy sequence. Therefore, one can define the function 𝐶∶𝑋→𝑌 by 𝐶(𝑥)∶=lim_𝑚→∞(1/2^3𝑚)ℎ(2^𝑚𝑥) for all 𝑥∈𝑋. Now, if we replace 𝑥₁,…,𝑥_𝑛 with 2^𝑚𝑥₁,…,2^𝑚𝑥_𝑛 in (2.10), respectively, it follows that Λ_{Δℎ(2𝑚𝑥1,…,2𝑚𝑥𝑛)/23𝑚}(𝑡)≥𝑇𝜉_{2𝑚+1𝑥1,…,2𝑚+1𝑥𝑛}2^3𝑚−1𝑡,𝜉_{2𝑚𝑥1,…,2𝑚𝑥𝑛}2^3𝑚−2𝑡 (2.43) for all 𝑥₁,…,𝑥_𝑛∈𝑥 and 𝑡>0. By letting 𝑚→∞ in (2.43), we find that Λ_{Δ𝐶(𝑥1,…,𝑥𝑛)}(𝑡)=1 for all 𝑡>0, which implies Δ𝐶(𝑥₁,…,𝑥_𝑛)=0, thus 𝐶 satisfies (1.8). Hence, by Theorem 2.1 (see [30, Lemma  2.2]), the function 𝐶∶𝑋→𝑌 is cubic. The rest of the proof is similar to the proof of Theorem 2.3.

Theorem 2.5. Let 𝜉∶𝑋^𝑛→𝐷⁺ be a function satisfies (2.21) for all 𝑥₁,…,𝑥_𝑛∈𝑋, 𝑡>0 and (2.22) for all 𝑥∈𝑋 and 𝑡>0. Suppose that an odd function 𝑓∶𝑋→𝑌 satisfies (2.10) for all 𝑥₁,…,𝑥_𝑛∈𝑋 and 𝑡>0. Then, there exists a unique additive function 𝐴∶𝑋→𝑌 and a unique cubic function 𝐶∶𝑋→𝑌 such thatΛ_{𝑓(𝑥)−𝐴(𝑥)−𝐶(𝑥)}(𝑡)≥𝑇𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^𝑛−4𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^𝑛−6𝑡,𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^{2ℓ+𝑛−4}𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^{2ℓ+𝑛−6}𝑡 (2.44) for all 𝑥∈𝑋 and 𝑡>0.

Proof. By Theorems 2.3 and 2.4, there exist an additive function 𝐴₀∶𝑋→𝑌 and a cubic function 𝐶₀∶𝑋→𝑌 such that Λ_{𝑓(2𝑥)−8𝑓(𝑥)−𝐴0(𝑥)}(𝑡)≥𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}2^𝑛−4𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}2^𝑛−6𝑡,Λ_{𝑓(2𝑥)−2𝑓(𝑥)−𝐶0(𝑥)}(𝑡)≥𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}2^{2ℓ+𝑛−4}𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}2^{2ℓ+𝑛−6}𝑡 (2.45) for all 𝑥∈𝑋 and 𝑡>0. It follows from (2.45) that Λ_{𝑓(𝑥)+(1/6)𝐴0(𝑥)−(1/6)𝐶0(𝑥)}(𝑡)≥𝑇Λ_{𝑓(2𝑥)−8𝑓(𝑥)−𝐴0(𝑥)}(3𝑡),Λ_{𝑓(2𝑥)−2𝑓(𝑥)−𝐶0(𝑥)}(3𝑡) (2.46) for all 𝑥∈𝑋 and 𝑡>0. Thus, we obtain (2.44) by letting 𝐴(𝑥)=−(1/6)𝐴₀(𝑥) and 𝐶(𝑥)=(1/6)𝐶₀(𝑥) for all 𝑥∈𝑋.
To prove the uniqueness property of 𝐴 and 𝐶, let 𝐴^′,𝐶^′∶𝑋→𝑌 be another additive and cubic functions satisfying (2.44). Let 𝐴=𝐴−𝐴^′ and 𝐶=𝐶−𝐶^′. So, Λ_{𝐴(𝑥)+𝐶(𝑥)}(𝑡)≥𝑇Λ_{𝑓(𝑥)−𝐴(𝑥)−𝐶(𝑥)}𝑡2,Λ_{𝑓(𝑥)−𝐴′(𝑥)−𝐶′(𝑥)}𝑡2≥𝑇𝑇𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^𝑛−5𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^𝑛−7𝑡,𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^{2ℓ+𝑛−5}𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^{2ℓ+𝑛−7}𝑡,𝑇𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^𝑛−5𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^𝑛−7𝑡,𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^{2ℓ+𝑛−5}𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^{2ℓ+𝑛−7}𝑡 (2.47) for all 𝑥∈𝑋 and 𝑡>0, then (2.47) implies that Λ_{𝐴(2𝑚𝑥)/23𝑚+𝐶(2𝑚𝑥)/23𝑚}(𝑡)≥𝑇𝑇𝑇^∞_ℓ=1𝑇𝜉_{2𝑚+ℓ𝑥,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,0,…,0}3.2^{3𝑚+𝑛−5}𝑡,𝜉_{2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,0,…,0}3.2^{3𝑚+𝑛−7}𝑡,𝑇^∞_ℓ=1𝑇𝜉_{2𝑚+ℓ𝑥,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,0,…,0}3.2^{3𝑚+2ℓ+𝑛−5}𝑡,𝜉_{2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,0,…,0}3.2^{3𝑚+2ℓ+𝑛−7}𝑡,𝑇𝑇^∞_ℓ=1𝑇𝜉_{2𝑚+ℓ𝑥,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,0,…,0}3.2^{3𝑚+𝑛−5}𝑡,𝜉_{2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,0,…,0}3.2^{3𝑚+𝑛−7}𝑡,𝑇^∞_ℓ=1𝑇𝜉_{2𝑚+ℓ𝑥,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,0,…,0}3.2^{3𝑚+2ℓ+𝑛−5}𝑡,𝜉_{2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,2𝑚+ℓ−1𝑥,0,…,0}3.2^{3𝑚+2ℓ+𝑛−7}𝑡 (2.48) for all 𝑥∈𝑋 and 𝑡>0. Since the right-hand side of the inequality tends to 1 as 𝑚 tends to infinity, thus lim_𝑚→∞𝐴(2^𝑚𝑥)/2^3𝑚+𝐶(2^𝑚𝑥)/2^3𝑚=0 for all 𝑥∈𝑋, which implies that 𝐶=0. So, from (2.47), we lead to 𝐴=0.

Now, we are ready to prove the main theorem concerning the stability results for (1.8), in Menger PN-space.

Theorem 2.6. Let 𝜉∶𝑋^𝑛→𝐷⁺ be a function satisfies (2.8) and (2.21) for all 𝑥₁,…,𝑥_𝑛∈𝑋, 𝑡>0 and satisfies (2.9) and (2.22) for all 𝑥∈𝑋 and 𝑡>0. Suppose that a function 𝑓∶𝑋→𝑌 satisfies (2.10) for all 𝑥₁,…,𝑥_𝑛∈𝑋 and 𝑡>0. Furthermore, assume that 𝑓(0)=0 in (2.10) for the case 𝑓 is even. Then, there exists a unique additive function 𝐴∶𝑋→𝑌, a unique quadratic function 𝑄∶𝑋→𝑌, and a unique cubic function 𝐶∶𝑋→𝑌 satisfying Λ_{𝑓(𝑥)−𝐴(𝑥)−𝑄(𝑥)−𝐶(𝑥)}(𝑡)≥𝑇𝑇𝑇^∞_ℓ=1𝜉_{0,…,0,2ℓ−1𝑥,2ℓ−1𝑥}2^ℓ+𝑛−3𝑡,𝑇^∞_ℓ=1𝜉_{0,…,0,2ℓ−1𝑥,2ℓ−1𝑥}2^ℓ+𝑛−3𝑡,𝑇𝑇𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^𝑛−5𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^𝑛−7𝑡,𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^{2ℓ+𝑛−5}𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^{2ℓ+𝑛−7}𝑡,𝑇𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^𝑛−5𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^𝑛−7𝑡,𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^{2ℓ+𝑛−5}𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^{2ℓ+𝑛−7}𝑡 (2.49) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Let 𝑓_𝑒(𝑥)=(1/2)(𝑓(𝑥)+𝑓(−𝑥)) for all 𝑥∈𝑋. Then, 𝑓_𝑒(0)=0,𝑓_𝑒(−𝑥)=𝑓_𝑒(𝑥) and Λ_{Δ𝑓𝑒(𝑥1,…,𝑥𝑛)}(𝑡)=Λ_{(Δ𝑓(𝑥1,…,𝑥𝑛)+Δ𝑓(−𝑥1,…,−𝑥𝑛))/2}(𝑡)≥𝑇Λ_{Δ𝑓(𝑥1,…,𝑥𝑛)}(𝑡),Λ_{Δ𝑓(−𝑥1,…,−𝑥𝑛)}(𝑡𝜉)≥𝑇_{𝑥1,…,𝑥𝑛}(𝑡),𝜉_{−𝑥1,…,−𝑥𝑛}(𝑡)=𝑇𝜉_{𝑥1,…,𝑥𝑛}(𝑡),𝜉_{𝑥1,…,𝑥𝑛}(𝑡) (2.50) for all 𝑥₁,…,𝑥_𝑛∈𝑋 and 𝑡>0. Hence, in view of Theorem 2.2, there exist a unique quadratic function 𝑄∶𝑋→𝑌 such that Λ_{𝑓𝑒(𝑥)−𝑄(𝑥)}(𝑡)≥𝑇𝑇^∞_ℓ=1𝜉_{0,…,0,2ℓ−1𝑥,2ℓ−1𝑥}2^ℓ+𝑛−2𝑡,𝑇^∞_ℓ=1𝜉_{0,…,0,2ℓ−1𝑥,2ℓ−1𝑥}2^ℓ+𝑛−2𝑡 (2.51) for all 𝑥∈𝑋 and 𝑡>0. On the other hand, let 𝑓_𝑜(𝑥)=(1/2)(𝑓(𝑥)−𝑓(−𝑥)) for all 𝑥∈𝑋. Then, 𝑓_𝑜(0)=0, 𝑓_𝑜(−𝑥)=−𝑓_𝑜(𝑥) and by using the above method, from Theorem 2.5, there exist a unique additive function 𝐴∶𝑋→𝑌 and a unique cubic function 𝐶∶𝑋→𝑌 such that Λ_{𝑓𝑜(𝑥)−𝐴(𝑥)−𝐶(𝑥)}(𝑡)≥𝑇𝑇𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^𝑛−4𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^𝑛−6𝑡,𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^{2ℓ+𝑛−4}𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^{2ℓ+𝑛−6}𝑡,𝑇𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^𝑛−4𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^𝑛−6𝑡,𝑇^∞_ℓ=1𝑇𝜉_{2ℓ𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^{2ℓ+𝑛−4}𝑡,𝜉_{2ℓ−1𝑥,2ℓ−1𝑥,2ℓ−1𝑥,0,…,0}3.2^{2ℓ+𝑛−6}𝑡 (2.52) for all 𝑥∈𝑋 and 𝑡>0. Hence, (2.49) follows from (2.51) and (2.52).

Acknowledgment

The third author of this work was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011-0021253).