Stability of an 𝑛-Dimensional Mixed-Type Additive and Quadratic Functional Equation in Random Normed Spaces

Abstract

We investigate the stability problems for the 𝑛-dimensional mixed-type additive and quadratic functional equation 2𝑓(∑^𝑛𝑗=1𝑥_𝑗)+∑_{1≤𝑖,𝑗≤𝑛,𝑖≠𝑗}𝑓(𝑥_𝑖−𝑥_𝑗)=(𝑛+1)∑^𝑛𝑗=1𝑓(𝑥_𝑗)+(𝑛−1)∑^𝑛𝑗=1𝑓(−𝑥_𝑗) in random normed spaces by applying the fixed point method.

1. Introduction

In 1940, Ulam [1] gave a wide-ranging talk before a mathematical colloquium at the University of Wisconsin, in which he discussed a number of important unsolved problems. Among those was the following question concerning the stability of homomorphisms.

Let 𝐺₁ be a group, and let 𝐺₂ be a metric group with a metric 𝑑(⋅,⋅). Given 𝜀>0, does there exist a 𝛿>0 such that if a function ℎ∶𝐺₁→𝐺₂ satisfies the inequality 𝑑(ℎ(𝑥𝑦),ℎ(𝑥)ℎ(𝑦))<𝛿 for all 𝑥,𝑦∈𝐺₁, then there is a homomorphism 𝐻∶𝐺₁→𝐺₂ with 𝑑(ℎ(𝑥),𝐻(𝑥))<𝜀 for all 𝑥∈𝐺₁?

If the answer is affirmative, we say that the functional equation for homomorphisms is stable. Hyers [2] was the first mathematician to present the result concerning the stability of functional equations. He answered the question of Ulam for the case where 𝐺₁ and 𝐺₂ are assumed to be Banach spaces. This result of Hyers is stated as follows.

Let 𝑓∶𝐸₁→𝐸₂ be a function between Banach spaces such that‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝛿(1.1) for some 𝛿>0 and for all 𝑥,𝑦∈𝐸₁. Then the limit 𝐴(𝑥)=lim_𝑛→∞2^−𝑛𝑓(2^𝑛𝑥) exists for each 𝑥∈𝐸₁, and 𝐴∶𝐸₁→𝐸₂ is the unique additive function such that ‖𝑓(𝑥)−𝐴(𝑥)‖≤𝛿 for every 𝑥∈𝐸₁. Moreover, if 𝑓(𝑡𝑥) is continuous in 𝑡 for each fixed 𝑥∈𝐸₁, then function 𝐴 is linear.

We remark that the additive function 𝐴 is directly constructed from the given function 𝑓, and this method is called the direct method. The direct method is a very powerful method for studying the stability problems of various functional equations. Taking this famous result into consideration, the additive Cauchy equation 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦) is said to have the Hyers-Ulam stability on (𝐸₁,𝐸₂) if for every function 𝑓∶𝐸₁→𝐸₂ satisfying the inequality (1.1) for some 𝛿≥0 and for all 𝑥,𝑦∈𝐸₁, there exists an additive function 𝐴∶𝐸₁→𝐸₂ such that 𝑓−𝐴 is bounded on 𝐸₁.

In 1950, Aoki [3] generalized the theorem of Hyers for additive functions, and in the following year, Bourgin [4] extended the theorem without proof. Unfortunately, it seems that their results failed to receive attention from mathematicians at that time. No one has made use of these results for a long time.

In 1978, Rassias [5] addressed the Hyers’s stability theorem and attempted to weaken the condition for the bound of the norm of Cauchy difference and generalized the theorem of Hyers for linear functions.

Let 𝑓∶𝐸₁→𝐸₂ be a function between Banach spaces. If 𝑓 satisfies the functional inequality‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝜃‖𝑥‖^𝑝+‖𝑦‖^𝑝(1.2)for some 𝜃≥0, 𝑝 with 0≤𝑝<1 and for all 𝑥,𝑦∈𝐸₁, then there exists a unique additive function 𝐴∶𝐸₁→𝐸₂ such that ‖𝑓(𝑥)−𝐴(𝑥)‖≤(2𝜃/(2−2^𝑝))‖𝑥‖^𝑝 for each 𝑥∈𝐸₁. If, in addition, 𝑓(𝑡𝑥) is continuous in 𝑡 for each fixed 𝑥∈𝐸₁, then the function 𝐴 is linear.

This result of Rassias attracted a number of mathematicians who began to be stimulated to investigate the stability problems of functional equations. By regarding a large influence of Ulam, Hyers, and Rassias on the study of stability problems of functional equations, the stability phenomenon proved by Rassias is called the Hyers-Ulam-Rassias stability. For the last thirty years, many results concerning the Hyers-Ulam-Rassias stability of various functional equations have been obtained (see [6–17]).

In this paper, applying the fixed point method, we prove the Hyers-Ulam-Rassias stability of the 𝑛-dimensional mixed-type additive and quadratic functional equation 2𝑓^𝑛_𝑗=1𝑥_𝑗+_{1≤𝑖,𝑗≤𝑛,𝑖≠𝑗}𝑓𝑥_𝑖−𝑥_𝑗=(𝑛+1)^𝑛_𝑗=1𝑓𝑥_𝑗+(𝑛−1)^𝑛_𝑗=1𝑓−𝑥_𝑗(1.3) in random normed spaces. Every solution of (1.3) is called a quadratic-additive function.

Throughout this paper, let 𝑛 be an integer larger than 1.

2. Preliminaries

We introduce some terminologies, notations, and conventions usually used in the theory of random normed spaces (see [18, 19]). The set of all probability distribution functions is denoted byΔ⁺∶={𝐹𝐹(0∶)[=0,∞0,]→[0,1]∣𝐹isleft-continuousandnondecreasingon[0,∞),and𝐹(∞)=1}.(2.1) Let us define 𝐷⁺∶={𝐹∈Δ⁺∣lim_𝑡→∞𝐹(𝑡)=1}. The set Δ⁺ is partially ordered by the usual pointwise ordering of functions, that is, 𝐹≤𝐺 if and only if 𝐹(𝑡)≤𝐺(𝑡) for all 𝑡≥0. The maximal element for Δ⁺ in this order is the distribution function 𝜀₀∶[0,∞]→[0,1] given by𝜀₀(𝑡)=01if𝑡=0,if𝑡>0.(2.2)

Definition 2.1 (See [18]). A function 𝜏∶[0,1]×[0,1]→[0,1] is called a continuous triangular norm (briefly, continuous 𝑡-norm) if 𝜏 satisfies the following conditions:(𝑎)𝜏 is commutative and associative;(𝑏)τ is continuous;(𝑐)𝜏(𝑎,1)=𝑎 for all 𝑎∈[0,1];(𝑑)𝜏(𝑎,𝑏)≤𝜏(𝑐,𝑑) for all 𝑎,𝑏,𝑐,𝑑∈[0,1] with 𝑎≤𝑐 and 𝑏≤𝑑.
Typical examples of continuous 𝑡-norms are 𝜏_𝑃(𝑎,𝑏)=𝑎𝑏, 𝜏_𝑀(𝑎,𝑏)=min{𝑎,𝑏}, and 𝜏_𝐿(𝑎,𝑏)=max{𝑎+𝑏−1,0}.

Definition 2.2 (See [19]). Let 𝑋 be a vector space, 𝜏 a continuous 𝑡-norm, and let Λ∶𝑋→𝐷⁺ be a function satisfying the following conditions:(𝑅₁)Λ_𝑥(𝑡)=𝜀₀(𝑡) for all 𝑡>0 if and only if 𝑥=0;(𝑅₂)Λ_𝛼𝑥(𝑡)=Λ_𝑥(𝑡/|𝛼|) for all 𝑥∈𝑋, 𝛼≠0, and for all 𝑡≥0;(𝑅₃)Λ_𝑥+𝑦(𝑡+𝑠)≥𝜏(Λ_𝑥(𝑡),Λ_𝑦(𝑠)) for all 𝑥,𝑦∈𝑋 and all 𝑡,𝑠≥0. A triple (𝑋,Λ,𝜏) is called a random normed space (briefly, RN-space).
If (𝑋,‖⋅‖) is a normed space, we can define a function Λ∶𝑋→𝐷⁺ by Λ_𝑥(𝑡)=𝑡𝑡+‖𝑥‖(2.3) for all 𝑥∈𝑋 and 𝑡>0. Then (𝑋,Λ,𝜏_𝑀) is a random normed space, which is called the induced random normed space.

Definition 2.3. Let (𝑋,Λ,𝜏) be an 𝑅𝑁-space.(𝑖)A sequence {𝑥_𝑛} in 𝑋 is said to be convergent to a point 𝑥∈𝑋 if, for every 𝑡>0 and 𝜀>0, there exists a positive integer 𝑁 such that Λ_{𝑥𝑛−𝑥}(𝑡)>1−𝜀 whenever 𝑛≥𝑁.(𝑖𝑖)A sequence {𝑥_𝑛} in 𝑋 is called a Cauchy sequence if, for every 𝑡>0 and 𝜀>0, there exists a positive integer 𝑁 such that Λ_{𝑥𝑛−𝑥𝑚}(𝑡)>1−𝜀 whenever 𝑛≥𝑚≥𝑁.(𝑖𝑖𝑖)An RN-space (𝑋,Λ,𝜏) is called complete if and only if every Cauchy sequence in 𝑋 converges to a point in 𝑋.

Definition 2.4. Let 𝑋 be a nonempty set. A function 𝑑∶𝑋²→[0,∞] is called a generalized metric on 𝑋 if and only if 𝑑 satisfies(𝑀₁)𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦;(𝑀₂)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋;(𝑀₃)𝑑(𝑥,𝑧)≤𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋.
We now introduce one of the fundamental results of the fixed point theory. For the proof, we refer to [20] or [21].

Theorem 2.5 (See [20, 21]). Let (𝑋,𝑑) be a complete generalized metric space. Assume that Λ∶𝑋→𝑋 is a strict contraction with the Lipschitz constant 𝐿<1. If there exists a nonnegative integer 𝑛₀ such that 𝑑(Λ^𝑛0+1𝑥,Λ^𝑛0𝑥)<∞ for some 𝑥∈𝑋, then the following statements are true:(𝑖)the sequence {Λ^𝑛𝑥} converges to a fixed point 𝑥^∗ of Λ;(𝑖𝑖)𝑥^∗ is the unique fixed point of Λ in 𝑋^∗={𝑦∈𝑋∣𝑑(Λ^𝑛0𝑥,𝑦)<∞};(𝑖𝑖𝑖)if 𝑦∈𝑋^∗, then𝑑𝑦,𝑥^∗≤11−𝐿𝑑(Λ𝑦,𝑦).(2.4)

In 2003, Radu [22] noticed that many theorems concerning the Hyers-Ulam stability of various functional equations follow from the fixed point alternative (Theorem 2.5). Indeed, he applied the fixed point method to prove the existence of a solution of the inequality (1.1) and investigated the Hyers-Ulam stability of the additive Cauchy equation (see also [23–26]). Furthermore, Miheţ and Radu [27] applied the fixed point method to prove the stability theorems of the additive Cauchy equation in random normed spaces.

In 2009, Towanlong and Nakmahachalasint [28] established the general solution and the stability of the 𝑛-dimensional mixed-type additive and quadratic functional equation (1.3) by using the direct method. According to [28], a function 𝑓∶𝐸₁→𝐸₂ is a quadratic-additive function, where 𝐸₁ and 𝐸₂ are vector spaces, if and only if there exist an additive function 𝑎∶𝐸₁→𝐸₂ and a quadratic function 𝑞∶𝐸₁→𝐸₂ such that 𝑓(𝑥)=𝑎(𝑥)+𝑞(𝑥) for all 𝑥∈𝐸₁.

3. Hyers-Ulam-Rassias Stability

Throughout this paper, let 𝑋 be a real vector space and let (𝑌,Λ,𝜏_𝑀) be a complete RN-space. For a given function 𝑓∶𝑋→𝑌, we use the following abbreviation:𝐷𝑓∶𝑥=₁,2𝑥𝑓₂,…_𝑛,𝑥_𝑛_𝑗=1𝑥_𝑗+_{1≤𝑖,𝑗≤𝑛,𝑖≠𝑗}𝑓𝑥_𝑖−𝑥_𝑗−(𝑛+1)^𝑛_𝑗=1𝑓𝑥_𝑗−(𝑛−1)^𝑛_𝑗=1𝑓−𝑥_𝑗(3.1) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋.

We will now prove the stability of the functional equation (1.3) in random normed spaces by using fixed point method.

Theorem 3.1. Let 𝑋 be a real vector space, (𝑍,Λ′,𝜏_𝑀) an RN-space, (𝑌,Λ,𝜏_𝑀) a complete RN-space, and let 𝜑∶(𝑋⧵{0})^𝑛→𝑍 be a function. Assume that 𝜑 satisfies one of the following conditions:(𝑖)Λ^{𝛼𝜑(𝑥}_{1,𝑥2,…,𝑥𝑛)}(𝑡)≤Λ^{𝜑(𝑛𝑥}_{1,𝑛𝑥2,…,𝑛𝑥𝑛)}(t) for some 0<𝛼<𝑛;(𝑖𝑖)Λ^{𝜑(𝑛𝑥}_{1,𝑛𝑥2,…,𝑛𝑥𝑛)}(𝑡)≤Λ^{𝛼𝜑(𝑥}_{1,𝑥2,…,𝑥𝑛)}(𝑡) for some 𝛼>𝑛² for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and 𝑡>0. If a function 𝑓∶𝑋→𝑌 satisfies 𝑓(0)=0 and Λ_{𝐷𝑓(𝑥1,𝑥2,…,𝑥𝑛)}(𝑡)≥Λ^𝜑(𝑥_{1,𝑥2,…,𝑥𝑛)}(𝑡)(3.2) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and 𝑡>0, then there exists a unique function 𝐹∶𝑋→𝑌 such that 𝐷𝐹𝑥₁,𝑥₂,…,𝑥_𝑛=0(3.3) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and Λ_{𝑓(𝑥)−𝐹(𝑥)}(𝑡)≥𝑀(𝑥,2(𝑛−𝛼)𝑡)𝑀𝑥,2𝛼−𝑛₂if𝜑satisfies(𝑖),𝑡if𝜑satisfies(𝑖𝑖)(3.4) for all 𝑥∈𝑋⧵{0} and 𝑡>0, where 𝑀(𝑥,𝑡)∶=𝜏_𝑀(Λ′_𝜑(𝑥̂₎(𝑡),Λ′_𝜑(−𝑥₎(𝑡)), and 𝑥̂=(𝑥,𝑥,…,𝑥).

Proof. We will first treat the case where 𝜑 satisfies the condition (𝑖). Let 𝑆 be the set of all functions 𝑔∶𝑋→𝑌 with 𝑔(0)=0, and let us define a generalized metric on 𝑆 by 𝑑(𝑔,ℎ)∶=inf𝑢∈[0,∞]∣Λ_{𝑔(𝑥)−ℎ(𝑥)}(𝑢𝑡)≥𝑀(𝑥,𝑡)∀𝑥∈𝑋⧵{0},𝑡>0.(3.5) It is not difficult to show that (𝑆,𝑑) is a complete generalized metric space (see [29] or [30, 31]).
Consider the operator 𝐽∶𝑆→𝑆 defined by 𝐽𝑓(𝑥)∶=𝑓(𝑛𝑥)−𝑓(−𝑛𝑥)2𝑛+𝑓(𝑛𝑥)2+𝑛𝑓²(−𝑛𝑥).(3.6) Then we can apply induction on 𝑚 to prove 𝐽^𝑚𝑓(𝑥)=𝑓(𝑛^𝑚𝑥)−𝑓(−𝑛^𝑚𝑥)2𝑛^𝑚+𝑓(𝑛^𝑚𝑥)+𝑓(−𝑛^𝑚𝑥)2𝑛^2𝑚(3.7) for all 𝑥∈𝑋 and 𝑚∈ℕ.
Let 𝑓,𝑔∈𝑆 and let 𝑢∈[0,∞] be an arbitrary constant with 𝑑(𝑔,𝑓)≤𝑢. For some 0<𝛼<𝑛 satisfying the condition (𝑖), it follows from the definition of 𝑑, (𝑅₂), (𝑅₃), and (𝑖) that Λ_{𝐽𝑔(𝑥)−𝐽𝑓(𝑥)}𝛼𝑢𝑡𝑛≥=𝜏Λ^{((𝑛+1)(𝑔(𝑛𝑥)−𝑓(𝑛𝑥))/2𝑛2)−((𝑛−1)(𝑔(−𝑛𝑥)−𝑓(−𝑛𝑥))/2𝑛2)}_𝑀𝛼𝑛𝑢𝑡ΛΛ^{(𝑛+1)(𝑔(𝑛𝑥)−𝑓(𝑛𝑥))/2𝑛2}_{(𝑛−1)(𝑔(−𝑛𝑥)−𝑓(−𝑛𝑥))/2𝑛2}(𝑛+21𝑛²)𝛼𝑢𝑡,≥𝜏_𝑀(𝑛−21𝑛²)𝛼𝑢𝑡≥𝜏_𝑀Λ_{𝑔(𝑛𝑥)−𝑓(𝑛𝑥)}(𝛼𝑢𝑡),Λ_{𝑔(−𝑛𝑥)−𝑓(−𝑛𝑥)}(𝛼𝑢𝑡)≥𝑀(𝑥Λ,𝑡)^𝜑(𝑛𝑥₎(𝛼𝑡),Λ^𝜑(−𝑛𝑥₎(𝛼𝑡)(3.8) for all 𝑥∈𝑋⧵{0} and 𝑡>0, which implies that 𝑑(𝐽𝑓,𝐽𝑔)≤𝛼𝑛𝑑(𝑓,𝑔).(3.9) That is, 𝐽 is a strict contraction with the Lipschitz constant 0<𝛼/𝑛<1.
Moreover, by (𝑅₂), (𝑅₃), and (3.2), we see that Λ_{𝑓(𝑥)−𝐽𝑓(𝑥)}𝑡2𝑛=≥𝜏Λ^{(−(𝑛+1)𝐷𝑓(}_𝑀𝑥̂_{)+(𝑛−1)𝐷𝑓(}−𝑥_))/4𝑛²2𝑡𝑛≥𝜏_𝑀Λ^{(𝑛+1)𝐷𝑓(}𝑥̂_)/4𝑛²(𝑛4+𝑛²1)𝑡,Λ^{(𝑛−1)𝐷𝑓(}−𝑥_)/4𝑛²(𝑛4−𝑛²1)𝑡≥𝑀(𝑥Λ,𝑡)_𝐷𝑓(𝑥̂₎(𝑡),Λ_𝐷𝑓(−𝑥₎(𝑡)(3.10) for all 𝑥∈𝑋⧵{0} and 𝑡>0. Hence, it follows from the definition of 𝑑 that 𝑑(𝑓,𝐽𝑓)≤12𝑛<∞.(3.11) Now, in view of Theorem 2.5, the sequence {𝐽^𝑚𝑓} converges to the unique “fixed point” 𝐹∶𝑋→𝑌 of 𝐽 in the set 𝑇={𝑔∈𝑆∣𝑑(𝑓,𝑔)<∞} and 𝐹 is represented by 𝐹(𝑥)=lim_𝑚→∞𝑓(𝑛^𝑚𝑥)−𝑓(−𝑛^𝑚𝑥)2𝑛^𝑚+𝑓(𝑛^𝑚𝑥)+𝑓(−𝑛^𝑚𝑥)2𝑛^2𝑚(3.12) for all 𝑥∈𝑋.
By Theorem 2.5, (3.11), and the definition of 𝑑, we have 𝑑(𝑓,𝐹)≤11−𝛼/𝑛𝑑(𝑓,𝐽𝑓)≤2(𝑛1−𝛼),(3.13) that is, the first inequality in (3.4) holds true.
We will now show that 𝐹 is a quadratic-additive function. It follows from (𝑅₃) and the definition of 𝜏_𝑀 that Λ_{𝐷𝐹(𝑥1,𝑥2,…,𝑥𝑛)}(𝑡)≥minΛ_{2(𝐹−𝐽}^𝑚_∑𝑓)(^𝑛𝑗=1_𝑥𝑗)𝑡minΛ5_{(𝐹−𝐽𝑚𝑓)(𝑥𝑖−𝑥𝑗)},minΛ_{(𝑛+1)(𝐽𝑚𝑓−𝐹)(𝑥𝑗)}(5𝑛(𝑛𝑡−1))∣1≤𝑖,𝑗≤𝑛,𝑖≠𝑗,minΛ_{(𝑛−1)(𝐽𝑚𝑓−𝐹)(−𝑥𝑗)}(5𝑡𝑛)∣𝑗=1,…,𝑛,Λ(5𝑡𝑛)∣𝑗=1,…,𝑛_{𝐷𝐽𝑚𝑓(𝑥1,𝑥2,…,𝑥𝑛)},5𝑡(3.14) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0}, 𝑡>0, and 𝑚∈ℕ. Due to the definition of 𝐹, the first four terms on the right-hand side of the above inequality tend to 1 as 𝑚→∞.
By a somewhat tedious manipulation, we have 𝐷𝐽^𝑚𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=12𝑛^2𝑚𝐷𝑓𝑛^𝑚𝑥₁,…,𝑛^𝑚𝑥_𝑛+12𝑛^2𝑚+1𝐷𝑓−𝑛^𝑚𝑥₁,…,−𝑛^𝑚𝑥_𝑛2𝑛^𝑚𝐷𝑓𝑛^𝑚𝑥₁,…,𝑛^𝑚𝑥_𝑛−12𝑛^𝑚𝐷𝑓−𝑛^𝑚𝑥₁,…,−𝑛^𝑚𝑥_𝑛.(3.15) Hence, it follows from (𝑅₂), (𝑅₃), definition of 𝜏_𝑀, (3.2), and (𝑖) that Λ_𝐷𝐽^𝑚_{𝑓(𝑥1,…,𝑥𝑛)}𝑡5≥minΛΛ^{𝐷𝑓(𝑛𝑚𝑥1,…,𝑛𝑚𝑥𝑛)/2𝑛2𝑚}2𝑡0,Λ^{𝐷𝑓(−𝑛𝑚𝑥1,…,−𝑛𝑚𝑥𝑛)/2𝑛2𝑚}2𝑡0_{𝐷𝑓(𝑛𝑚𝑥1,…,𝑛𝑚𝑥𝑛)/2𝑛𝑚},≥minΛ_{𝐷𝑓(𝑛𝑚𝑥1,…,𝑛𝑚𝑥𝑛)}2𝑡0,Λ^{𝐷𝑓(−𝑛𝑚𝑥1,…,−𝑛𝑚𝑥𝑛)/2𝑛𝑚}2𝑡0𝑛^2𝑚𝑡10,Λ^{𝐷𝑓(−𝑛𝑚𝑥1,…,−𝑛𝑚𝑥𝑛)}𝑛^2𝑚𝑡Λ10_{𝐷𝑓(𝑛𝑚𝑥1,…,𝑛𝑚𝑥𝑛)},𝑛^𝑚𝑡10,Λ^{𝐷𝑓(−𝑛𝑚𝑥1,…,−𝑛𝑚𝑥𝑛)}𝑛^𝑚𝑡≥minΛ_{𝜑(𝑥1,…,𝑥𝑛)}10𝑛^2𝑚𝑡(10𝛼^𝑚),Λ^{𝜑(−𝑥1,…,−𝑥𝑛)}𝑛^2𝑚𝑡Λ_{𝜑(𝑥1,…,𝑥𝑛)}(10𝛼^𝑚),𝑛^𝑚𝑡(10𝛼^𝑚),Λ^{𝜑(−𝑥1,…,−𝑥𝑛)}𝑛^𝑚𝑡(10𝛼^𝑚),(3.16) which tends to 1 as 𝑚→∞ for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and 𝑡>0. Therefore, (3.14) implies that Λ_{𝐷𝐹(𝑥1,𝑥2,…,𝑥𝑛)}(𝑡)=1(3.17) for any 𝑥₁,…,𝑥_𝑛∈𝑋⧵{0} and 𝑡>0. By (𝑅₁), this implies that 𝐷𝐹(𝑥₁,…,𝑥_𝑛)=0 for all 𝑥₁,…,𝑥_𝑛∈𝑋⧵{0}, which ends the proof of the first part.
Now, assume that 𝜑 satisfies the condition (𝑖𝑖). Let (𝑆,𝑑) be the same as given in the first part. We now consider the operator 𝐽∶𝑆→𝑆 defined by 𝐽𝑔(𝑥)∶=𝑛2𝑔𝑥𝑛−𝑔−𝑥𝑛+𝑛2²𝑔𝑥𝑛+𝑔−𝑥𝑛(3.18) for all 𝑔∈𝑆 and 𝑥∈𝑋. Notice that 𝐽^𝑚𝑔(𝑥)=𝑛^𝑚2𝑔𝑛𝑥^𝑚−𝑔−𝑛𝑥_𝑚+𝑛2^2𝑚𝑔𝑛𝑥^𝑚+𝑔−𝑛𝑥_𝑚(3.19) for all 𝑥∈𝑋 and 𝑚∈ℕ.
Let 𝑓,𝑔∈𝑆 and let 𝑢∈[0,∞] be an arbitrary constant with 𝑑(𝑔,𝑓)≤𝑢. From (𝑅₂), (𝑅₃), the definition of 𝑑, and (𝑖𝑖), we have Λ_{𝐽𝑔(𝑥)−𝐽𝑓(𝑥)}𝑛²𝑢𝑡𝛼=Λ^{((𝑛2+𝑛)/2)(𝑔(𝑥/𝑛)−𝑓(𝑥/𝑛))+((𝑛2−𝑛)/2)(𝑔(−𝑥/𝑛)−𝑓(−𝑥/𝑛))}𝑛²𝑢𝑡≥𝜏_𝑀𝛼Λ^{((𝑛2+𝑛)/2)(𝑔(𝑥/𝑛)−𝑓(𝑥/𝑛))}𝑛²+𝑛𝑢𝑡Λ(2𝛼)_{((𝑛2−𝑛)/2)(𝑔(−𝑥/𝑛)−𝑓(−𝑥/𝑛))},𝑛²−𝑛𝑢𝑡=𝜏_𝑀(2𝛼)≥𝜏_𝑀Λ^{𝑔(𝑥/𝑛)−𝑓(𝑥/𝑛)}𝑢𝛼𝑡,Λ^{𝑔(−𝑥/𝑛)−𝑓(−𝑥/𝑛)}𝑢𝛼𝑡=𝜏_𝑀𝑀𝑥𝑛,𝛼𝑡,𝑀−𝑥𝑛,𝛼𝑡=𝜏_𝑀Λ^𝜑(𝑥/𝑛₎𝛼𝑡,Λ^𝜑(−𝑥/𝑛₎𝛼𝑡Λ^𝛼𝜑(𝑥/𝑛₎≥𝜏_𝑀(𝑡),Λ^𝛼𝜑(−𝑥/𝑛₎(𝑡)=𝑀(𝑥Λ,𝑡)^𝜑(𝑥̂₎(𝑡),Λ^𝜑(−𝑥₎(𝑡)(3.20) for all 𝑥∈𝑋⧵{0}, 𝑡>0, and for some 𝛼>𝑛² satisfying (𝑖𝑖), which implies that 𝑑(𝐽𝑓,𝐽𝑔)≤𝑛²𝛼𝑑(𝑓,𝑔).(3.21) That is, 𝐽 is a strict contraction with the Lipschitz constant 0<𝑛²/𝛼<1.
Moreover, by (𝑅₂), (3.2), and (𝑖𝑖), we see that Λ_{𝑓(𝑥)−𝐽𝑓(𝑥)}(2𝑡𝛼)=≥ΛΛ^{(1/2)𝐷𝑓(}_𝜑(𝑥/𝑛₎(2𝑡𝛼)=Λ𝑥_𝛼𝜑(/𝑛₎𝛼𝑡≥Λ_𝜑(𝑥/𝑛₎(𝑡)≥𝑀(𝑥𝑥,̂₎𝑡()𝑡)(3.22) for all 𝑥∈𝑋⧵{0} and 𝑡>0. This implies that 𝑑(𝑓,𝐽𝑓)≤1/(2𝛼)<∞ by the definition of 𝑑. Therefore, according to Theorem 2.5, the sequence {𝐽^𝑚𝑓} converges to the unique “fixed point” 𝐹∶𝑋→𝑌 of 𝐽 in the set 𝑇={𝑔∈𝑆∣𝑑(𝑓,𝑔)<∞} and 𝐹 is represented by 𝐹(𝑥)=lim_𝑚→∞𝑛2^𝑚𝑓𝑛𝑥^m−𝑓−𝑛𝑥_𝑚+𝑛2^2𝑚𝑓𝑛𝑥^𝑚+𝑓−𝑛𝑥_𝑚(3.23) for all 𝑥∈𝑋. Since 𝑑(𝑓,𝐹)≤11−𝑛²/𝛼𝑑(𝑓,𝐽𝑓)≤2𝛼1−𝑛²,(3.24) the second inequality in (3.4) holds true.
Next, we will show that 𝐹 is a quadratic-additive function. As we did in the first part, we obtain the inequality (3.14). In view of the definition of 𝐹, the first four terms on the right-hand side of the inequality (3.14) tend to 1 as 𝑚→∞. Furthermore, a long manipulation yields 𝐷𝐽^𝑚𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=𝑛^2𝑚2𝐷𝑓𝑛𝑥^1𝑚,…,𝑥^𝑛𝑛^𝑚+𝑛2^2𝑚𝐷𝑓−𝑛𝑥^1𝑚,…,−𝑥^𝑛+𝑛_𝑚𝑛^𝑚2𝐷𝑓𝑛𝑥^1𝑚,…,𝑥^𝑛𝑛^𝑚−𝑛2^𝑚𝐷𝑓−𝑛𝑥^1𝑚,…,−𝑥^𝑛𝑛^𝑚.(3.25) Thus, it follows from (𝑅₂), (𝑅₃), definition of 𝜏_𝑀, (3.2), and (𝑖𝑖) that Λ_𝐷𝐽^𝑚_{𝑓(𝑥1,…,𝑥𝑛)}𝑡≥minΛ5_{(𝑛2𝑚/2)𝐷𝑓(𝑥1/𝑛𝑚,…,𝑥𝑛/𝑛𝑚)}Λ2𝑡0,Λ^{(𝑛2𝑚/2)𝐷𝑓(−𝑥1/𝑛𝑚,…,−𝑥𝑛/𝑛𝑚)}2𝑡0_{(𝑛𝑚/2)𝐷𝑓(𝑥1/𝑛𝑚,…,𝑥𝑛/𝑛𝑚)},≥minΛ_{𝜑(𝑥1/𝑛𝑚,…,𝑥𝑛/𝑛𝑚)}2𝑡0,Λ^{−(𝑛𝑚/2)𝐷𝑓(−𝑥1/𝑛𝑚,…,−𝑥𝑛/𝑛𝑚)}2𝑡0Λ_{𝜑(𝑥1/𝑛𝑚,…,𝑥𝑛/𝑛𝑚)}10𝑛𝑡^2𝑚,Λ^{𝜑(−𝑥1/𝑛𝑚,…,−𝑥𝑛/𝑛𝑚)}10𝑛𝑡^2𝑚,t≥minΛ_{𝛼−𝑚𝜑(𝑥1,…,𝑥𝑛)}(10𝑛^𝑚),Λ^{𝜑(−𝑥1/𝑛𝑚,…,−𝑥𝑛/𝑛𝑚)}(10𝑡𝑛^𝑚)Λ_{𝛼−𝑚𝜑(𝑥1,…,𝑥𝑛)}10𝑛𝑡^2𝑚,Λ^{𝛼−𝑚𝜑(−𝑥1,…,−𝑥𝑛)}10𝑛𝑡^2𝑚,=minΛ_{𝜑(𝑥1,…,𝑥𝑛)}(10𝑡𝑛^𝑚),Λ^{𝛼−𝑚𝜑(−𝑥1,…,−𝑥𝑛)}(10𝑡𝑛^𝑚)𝛼^𝑚𝑡10𝑛^2𝑚,Λ^{𝜑(−𝑥1,…,−𝑥𝑛)}𝛼^𝑚𝑡Λ_{𝜑(𝑥1,…,𝑥𝑛)}10𝑛^2𝑚,𝛼^𝑚𝑡(10𝑛^𝑚),Λ^{𝜑(−𝑥1,…,−𝑥𝑛)}𝛼^𝑚𝑡(10𝑛^𝑚),(3.26) which tends to 1 as 𝑚→∞ for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and 𝑡>0. Therefore, it follows from (3.14) that Λ_{𝐷𝐹(𝑥1,𝑥2,…,𝑥𝑛)}(𝑡)=1(3.27) for any 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and 𝑡>0. By (𝑅₁), this implies that 𝐷𝐹𝑥₁,𝑥₂,…,𝑥_𝑛=0(3.28) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0}, which ends the proof.

By a similar way presented in the proof of Theorem 3.1, we can also prove the preceding theorem if the domains of relevant functions include 0.

Theorem 3.2. Let 𝑋 be a real vector space, (𝑍,Λ′,𝜏_𝑀) an RN-space, (𝑌,Λ,𝜏_𝑀) a complete RN-space, and let 𝜑∶𝑋^𝑛→𝑍 be a function. Assume that 𝜑 satisfies one of the conditions (𝑖) and (𝑖𝑖) in Theorem 3.1 for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and 𝑡>0. If a function 𝑓∶𝑋→𝑌 satisfies 𝑓(0)=0 and (3.2) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and 𝑡>0, then there exists a unique quadratic-additive function 𝐹∶𝑋→𝑌 satisfying (3.4) for all 𝑥∈𝑋 and 𝑡>0.

Now, we obtain general Hyers-Ulam stability results of (1.3) in normed spaces. If 𝑋 is a normed space, then (𝑋,Λ,𝜏_𝑀) is an induced random normed space. We get the following result.

Corollary 3.3. Let 𝑋 be a real vector space, 𝑌 a complete normed space, and let 𝜑∶(𝑋⧵{0})^𝑛→[0,∞) be a function. Assume that 𝜑 satisfies one of the following conditions:(𝑖𝑖𝑖)𝜑(𝑛𝑥₁,…,𝑛𝑥_𝑛)≤𝛼𝜑(𝑥₁,…,𝑥_𝑛) for some 1<𝛼<𝑛;(𝑖𝑣)𝜑(𝑛𝑥₁,…,𝑛𝑥_𝑛)≥𝛼𝜑(𝑥₁,…,𝑥_𝑛) for some 𝛼>𝑛² for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0}. If a function 𝑓∶𝑋→𝑌 satisfies 𝑓(0)=0 and ‖‖𝐷𝑓𝑥₁,𝑥₂,…,𝑥_𝑛‖‖≤𝜑𝑥₁,𝑥₂,…,𝑥_𝑛(3.29) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0}, then there exists a unique function 𝐹∶𝑋→𝑌 such that 𝐷𝐹𝑥₁,𝑥₂,…,𝑥_𝑛=0(3.30) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and ‖𝑓(𝑥)−𝐹(𝑥)‖≤⎧⎪⎪⎨⎪⎪⎩max𝜑(𝑥̂),𝜑−𝑥2(𝑛−𝛼)max𝜑(𝑥̂),𝜑−𝑥if𝜑satisfies(𝑖𝑖𝑖),2𝛼−𝑛²if𝜑satisfies(𝑖𝑣)(3.31) for all 𝑥∈𝑋⧵{0}.

Proof. Let us put 𝑍∶=ℝ,Λ_𝑥(𝑡)∶=𝑡𝑡+‖𝑥‖,Λ^𝑧(𝑡)∶=𝑡𝑡+|𝑧|(3.32) for all 𝑥,𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0}, 𝑧∈ℝ⧵{0}, and 𝑡≥0. If 𝜑 satisfies the condition (𝑖𝑖𝑖) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and for some 1<𝛼<𝑛, then Λ^{𝛼𝜑(𝑥}_{1,…,𝑥𝑛)}(𝑡)=𝑡𝑡+𝛼𝜑𝑥₁,…,𝑥_𝑛≤𝑡+𝜑𝑛𝑥₁𝑡,…,𝑛𝑥_𝑛=Λ^{𝜑(𝑛𝑥1,…,𝑛𝑥𝑛)}(𝑡)(3.33) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and 𝑡>0, that is, 𝜑 satisfies the condition (𝑖). In a similar way, we can show that if 𝜑 satisfies (𝑖𝑣), then it satisfies the condition (𝑖𝑖).
Moreover, we get Λ_{𝐷𝑓(𝑥1,…,𝑥𝑛)}(𝑡)=𝑡𝑡+‖‖𝐷𝑓𝑥₁,…,𝑥_𝑛‖‖≥𝑡+𝜑𝑥₁𝑡,…,𝑥_𝑛=Λ^{𝜑(𝑥1,…,𝑥𝑛)}(𝑡)(3.34) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and 𝑡>0, that is, 𝑓 satisfies the inequality (3.2) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0}.
According to Theorem 3.1, there exists a unique function 𝐹∶𝑋→𝑌 such that𝐷𝐹𝑥₁,𝑥₂,…,𝑥_𝑛=0(3.35) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and Λ_{𝑓(𝑥)−𝐹(𝑥)}(𝑡)≥⎧⎪⎨⎪⎩𝜏^𝑀𝜏Λ^𝜑(𝑥̂₎(2(𝑛−𝛼)𝑡),Λ^𝜑(−𝑥₎(2(𝑛−𝛼)𝑡)if𝜑satisfies(𝑖𝑖𝑖),_𝑀Λ^𝜑(𝑥̂₎2𝛼−𝑛²𝑡,Λ^𝜑(−𝑥₎2𝛼−𝑛²𝑡if𝜑satisfies(𝑖𝑣)(3.36) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and 𝑡>0, which ends the proof.

We now prove the Hyers-Ulam-Rassias stability of (1.3) in the framework of normed spaces.

Corollary 3.4. Let 𝑋 be a real normed space, 𝑝∈[0,1)∪(2,∞), and let 𝑌 be a complete normed space. If a function 𝑓∶𝑋→𝑌 satisfies 𝑓(0)=0 and ‖‖𝐷𝑓𝑥₁,𝑥₂,…,𝑥_𝑛‖‖≤𝜃‖‖𝑥₁‖‖^𝑝+‖‖𝑥₂‖‖^𝑝+⋯+‖‖𝑥_𝑛‖‖^𝑝(3.37) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and for some 𝜃≥0, then there exists a unique quadratic-additive function 𝐹∶𝑋→𝑌 such that ‖𝑓(𝑥)−𝐹(𝑥)‖≤⎧⎪⎪⎨⎪⎪⎩𝑛𝜃‖𝑥‖^𝑝2(𝑛−𝑛^𝑝)𝑛𝜃‖𝑥‖_𝑝if0≤p<1,2𝑛^𝑝−𝑛²ifp>2(3.38) for all 𝑥∈𝑋.

Proof. If we put 𝜑𝑥₁,𝑥₂,…,𝑥_𝑛∶=𝜃‖‖𝑥₁‖‖^𝑝+‖‖𝑥₂‖‖^𝑝+⋯+‖‖𝑥_𝑛‖‖^𝑝,(3.39) then the induced random normed space (𝑋,Λ_𝑥,𝜏_𝑀) satisfies the conditions stated in Theorem 3.2 with 𝛼=𝑛^𝑝.

Corollary 3.5. Let 𝑋 be a real normed space, 𝑝∈(−∞,0), and let 𝑌 be a complete normed space. If a function 𝑓∶𝑋→𝑌 satisfies 𝑓(0)=0 and ‖‖𝐷𝑓𝑥₁,𝑥₂,…,𝑥_𝑛‖‖≤𝜃_{1≤𝑖≤𝑛,𝑥𝑖≠0}‖‖𝑥_𝑖‖‖^𝑝(3.40) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and for some 𝜃≥0, then there exists a unique quadratic-additive function 𝐹∶𝑋→𝑌 satisfying ‖𝑓(𝑥)−𝐹(𝑥)‖≤⎧⎪⎨⎪⎩2𝑛(𝜃𝑛‖−𝑥‖𝑛^𝑝𝑝)0ifx∈X⧵{0},ifx=0.(3.41)

Proof. If we put 𝑍∶=ℝ, 𝛼∶=𝑛^𝑝, and define Λ_𝑥(𝑡)∶=𝑡𝑡+‖𝑥‖,Λ^𝑧(𝑡)∶=𝑡𝜑𝑥𝑡+|𝑧|₁,,𝑥₂,…,𝑥_𝑛∶=𝜃∑_{1≤𝑖≤𝑛,𝑥𝑖≠0}‖‖𝑥_𝑖‖‖^𝑝(3.42) for all 𝑥,𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and 𝑧∈𝑍, then we have Λ^{𝛼𝜑(𝑥}_{1,𝑥2,…,𝑥𝑛)}(𝑡)=𝑡=𝑡+𝛼𝜑𝑥₁𝑡,…,𝑥_𝑛=Λ𝑡+𝜑𝑛𝑥₁,…,_{𝜑(𝑛𝑥1,𝑛𝑥2,…,𝑛𝑥𝑛)}𝑛𝑥_𝑛(𝑡),(3.43) that is, 𝜑 satisfies condition (𝑖) given in Theorem 3.1 for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and 𝑡>0. We moreover get Λ_{𝐷𝑓(𝑥1,𝑥2,…,𝑥𝑛)}(𝑡)=𝑡≥𝑡+‖‖𝐷𝑓𝑥₁𝑡,…,𝑥_𝑛‖‖𝑡+𝜃∑_{1≤𝑖≤𝑛,𝑥𝑖≠0}‖‖𝑥_𝑖‖‖^𝑝=𝑡=Λ𝑡+𝜑𝑥₁,…_{𝜑(𝑥1,𝑥2,…,𝑥𝑛)},𝑥_𝑛(𝑡),(3.44) that is, 𝑓 satisfies the inequality (3.2) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and 𝑡>0.
According to Theorem 3.2, there exists a unique quadratic-additive function 𝐹∶𝑋→𝑌 satisfying 𝑡𝑡+‖𝑓(𝑥)−𝐹(𝑥)‖=Λ^{𝑓(𝑥)−𝐹(𝑥)}≥𝑀(𝑥,2(𝑛(𝑡−)𝑛_𝑝=⎧⎪⎨⎪⎩2(𝑛−𝑛_𝑝)𝑡))𝑡2(𝑛−𝑛^𝑝)𝑡+𝑛𝜃‖𝑥‖^𝑝1if𝑥∈𝑋⧵{0},if𝑥=0(3.45) for all 𝑡>0, or equivalently ‖𝑓(𝑥)−𝐹(𝑥)‖𝑡≤⎧⎪⎨⎪⎩𝑛𝜃‖𝑥‖^𝑝2(𝑛−𝑛^𝑝)𝑡0if𝑥∈𝑋⧵{0},if𝑥=0(3.46) for all 𝑡>0, which ends the proof.

Acknowledgment

The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2011-0004919).