On the Generalized Hyers-Ulam Stability of an 𝑛-Dimensional Quadratic and Additive Type Functional Equation

Abstract

We investigate the generalized Hyers-Ulam stability of a functional equation 𝑓(∑^𝑛_𝑗=1𝑥_𝑗)+(𝑛−2)∑^𝑛_𝑗=1𝑓(𝑥_𝑗)−∑_{1≤𝑖<𝑗≤𝑛}𝑓(𝑥_𝑖+𝑥_𝑗)=0.

1. Introduction

Throughout this paper, let 𝑋 be a normed space and 𝑌 a Banach space. For a given mapping 𝑓∶𝑋→𝑌, we define (𝑥)𝑦,(𝑥)𝑦,𝐴𝑓𝑥,𝑦∶=𝑓𝑥+𝑦−𝑓−𝑓𝑄𝑓𝑥,𝑦∶=𝑓𝑥+𝑦+𝑓𝑥−𝑦−2𝑓−2𝑓 (1) for all 𝑥,𝑦∈𝑋. A mapping 𝑓∶𝑋→𝑌 is called an additive mapping (a quadratic mapping, resp.) if 𝑓 satisfies the functional equation 𝐴𝑓=0 (𝑄𝑓=0, resp.). If a mapping is represented by sum of an additive mapping and a quadratic mapping, we call the mapping a quadratic-additive mapping. For a functional equation 𝐸𝑓=0 if all of the solutions of 𝐸𝑓=0 are quadratic-additive mappings and all of quadratic-additive mappings are the solutions of 𝐸𝑓=0, then we call the functional equation 𝐸𝑓=0 a quadratic-additive type functional equation.

In 1940, Ulam [1] raised a question concerning the stability of homomorphisms. Hyers [2], Aoki [3], Rassias [4], and Găvruţa [5] made important role to study the stability of the functional equation. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians (see also [6–9]).

In 2006, Jun and Kim [10] obtained the stability of the functional equation 𝑓^𝑛_𝑗=1𝑥_𝑗+(𝑛−2)^𝑛_𝑗=1𝑓𝑥_𝑗−_{1≤𝑖<𝑗≤𝑛}𝑓𝑥_𝑖+𝑥_𝑗=0, (2) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 (𝑛>2) (see also [11–15]). The functional equation (2) is a quadratic-additive type functional equation (see Theorem 2.6 in [16]). For the case 𝑛=3, Jung [17] proved the stability of the functional equation (2) (see also [18–20]) and, for the case 𝑛=4, Chang et al. [21] proved the stability of the functional equation (2) (see also [22–25]).

In this paper, we will generalize the previous results of the stability problem of the functional equation (2) on the punctured domain. In particular, we will show the superstability (if 𝑝<0) of the functional equation (2) in the sense of Rassias.

2. Stability of the Functional Equation (2) (𝑛 Is Even)

Let (𝑠,𝑡) be a fixed element in {(1,1),(1,−1),(−1,−1)} and let 𝜑∶(𝑋⧵{0})^𝑛→[0,∞) be a function satisfying the conditions: ^∞_𝑗=04^−𝑠𝑗𝜑2^𝑠𝑗𝑥₁,2^𝑠𝑗𝑥₂,…,2^𝑠𝑗𝑥_𝑛<∞,_∞𝑗=02^−𝑡𝑗𝜑2^𝑡𝑗𝑥₁,2^𝑡𝑗𝑥₂,…,2^𝑡𝑗𝑥_𝑛<∞, (3)(4) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0}, where 𝑛 is a fixed even integer greater than 2 in this section. For convenience, we use the following abbreviations in this section for a given mapping 𝑓∶𝑋→𝑌: 𝐷𝑓𝑥₁,𝑥₂,…,𝑥_𝑛∶=𝑓^𝑛_𝑗=1𝑥_𝑗+(𝑛−2)^𝑛_𝑗=1𝑓𝑥_𝑗−_{1≤𝑖<𝑗≤𝑛}𝑓𝑥_𝑖+𝑥_𝑗,𝐽_𝑚𝑓(𝑥)=124^−𝑠𝑚𝑓2^𝑠𝑚𝑥+𝑓−2^𝑠𝑚𝑥−2(𝑛+2)𝑓(0)3𝑛+2^−𝑡𝑚𝑓2^𝑡𝑚𝑥−𝑓−2^𝑡𝑚𝑥,𝑥∶=𝑥,…^𝑛/2+1,𝑥,−𝑥,…^𝑛/2−1,−𝑥, (5) for all 𝑥,𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋. From these, we get the equality 𝐽_𝑚𝑓(𝑥)−𝐽_𝑚+1𝑓(𝑥)=2⋅4^{𝜏−𝑠,𝑚}𝑛(𝑛−2)𝐷𝑓2^{𝜏𝑠,𝑚}𝑥+𝐷𝑓−2^{𝜏𝑠,𝑚}𝑥𝑠+2^{𝜏−𝑡,𝑚−1}𝑛−2𝐷𝑓2^{𝜏𝑡,𝑚}𝑥−𝐷𝑓−2^{𝜏𝑡,𝑚}𝑥𝑡 (6) for all 𝑥∈𝑋⧵{0} and all nonnegative integers 𝑚, where 𝜏_𝑘,𝑚 are the integers defined by 𝜏_𝑘,𝑚1=𝑘𝑚+2−12 (7) for 𝑘∈{−1,1}.

Lemma 1. If 𝑓∶𝑋→𝑌 is a mapping such that 𝐷𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=0, (8) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0}, then 𝐽_𝑚𝑓(𝑥)=𝑓(𝑥)−𝑛+2𝑓(0)3𝑛 (9) for all 𝑥∈𝑋⧵{0} and all nonnegative integers 𝑚.

Proof. We can easily get 𝑓(𝑥)−𝑛+2𝑓(0)3𝑛−𝐽_𝑚𝑓(𝑥)=^𝑚−1_𝑗=02⋅4^{𝜏−𝑠,𝑗}𝑛(𝑛−2)𝐷𝑓2^{𝜏𝑠,𝑗}𝑥+𝐷𝑓−2^{𝜏𝑠,𝑗}𝑥𝑠+2^{𝜏−𝑡,𝑗−1}𝑛−2𝐷𝑓2^{𝜏𝑡,𝑗}𝑥−𝐷𝑓−2^{𝜏𝑡,𝑗}𝑥𝑡=0 (10) for all 𝑥∈𝑋⧵{0} and all nonnegative integers 𝑚.

Theorem 2. Suppose that 𝑓∶𝑋→𝑌 is a mapping such that 𝐷𝑓𝑥₁,𝑥₂,…,𝑥_𝑛≤𝜑𝑥₁,𝑥₂,…,𝑥_𝑛 (11) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} with lim_𝑚→∞𝐽_𝑚𝑓(0)=0. Then, there exists a unique mapping 𝐹∶𝑋→𝑌 satisfying (8) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and 𝑓(𝑥)−𝑛+2𝑓(0)(𝑥≤3𝑛−𝐹)^∞_𝑗=0Φ_𝑗(𝑥) (12) for all 𝑥∈𝑋⧵{0} with 𝐹(0)=0, where Φ_𝑗 are the mappings defined by Φ_𝑗(𝑥)∶=2⋅4^{𝜏−𝑠,𝑗}𝑛()𝜑𝑛−22^{𝜏𝑠,𝑗}𝑥+𝜑−2^{𝜏𝑠,𝑗}𝑥+2^{𝜏−𝑡,𝑗−1}𝜑𝑛−22^{𝜏𝑡,𝑗}𝑥+𝜑−2^{𝜏𝑡,𝑗}𝑥 (13) for all 𝑥∈𝑋⧵{0}.

Proof. It follows from (6) and (11) that 𝐽_𝑚𝑓(𝑥)−𝐽_𝑚+𝑚′𝑓(𝑥=_{𝑚+𝑚′−1𝑗=𝑚})2⋅4^{𝜏−𝑠,𝑗}𝑛(𝑛−2)𝐷𝑓2^{𝜏𝑠,𝑗}𝑥+𝐷𝑓−2^{𝜏𝑠,𝑗}𝑥𝑠+2^{𝜏−𝑡,𝑗−1}𝑛−2𝐷𝑓2^{𝜏𝑡,𝑗}𝑥−𝐷𝑓−2^{𝜏𝑡,𝑗}𝑥𝑡≤^{𝑚+𝑚′−1}_𝑗=𝑚Φ_𝑗(𝑥) (14) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and all nonnegative integers 𝑚, 𝑚^′ with 𝑚^′>0. From (3), (4), and (14), it follows that the sequence {𝐽_𝑚𝑓(𝑥)} is Cauchy for all 𝑥∈𝑋⧵{0}. Since 𝑌 is complete, the sequence {𝐽_𝑚𝑓(𝑥)} converges. From this and lim_𝑚→∞𝐽_𝑚𝑓(0)=0, we can define the mapping 𝐹∶𝑋→𝑌 by 𝐹(𝑥)∶=lim_𝑚→∞𝐽_𝑚𝑓(𝑥) (15) for all 𝑥∈𝑋. Moreover, letting 𝑚=0 and taking the limit as 𝑚^′→∞ in (14), we get the inequality (12). Notice that lim_𝑚→∞𝐽_𝑚𝑓(0)=0, lim_𝑚→∞4^−𝑠𝑚𝜑(2^𝑠𝑚𝑥₁,…,2^𝑠𝑚𝑥_𝑛)=0, and lim_𝑚→∞2^−𝑡𝑚𝜑(2^𝑡𝑚𝑥₁,…,2^𝑡𝑚𝑥_𝑛)=0 for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0}. Hence, it follows from (11) and the definition of 𝐹 that 𝐷𝐹𝑥₁,𝑥₂,…,𝑥_𝑛=lim_𝑚→∞124^−𝑠𝑚𝐷𝑓2^𝑠𝑚𝑥₁,…,2^𝑠𝑚𝑥_𝑛+𝐷𝑓−2^𝑠𝑚𝑥₁,…,−2^𝑠𝑚𝑥_𝑛+2^−𝑡𝑚𝐷𝑓2^𝑡𝑚𝑥₁,…,2^𝑡𝑚𝑥_𝑛−𝐷𝑓−2^𝑡𝑚𝑥₁,…,−2^𝑡𝑚𝑥_𝑛=0 (16) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0}.
Now, let 𝐹^′∶𝑋→𝑌 be another mapping satisfying (8) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and (12) with 𝐹^′(0)=0. Using Lemma 1, (12), and 𝐹^′(0)=0=𝐹(0), we obtain 𝐹(𝑥)−𝐹^′(𝑥≤𝐽_𝑚)𝐹(𝑥)−𝐽_𝑚𝐹^′(𝑥≤4_−𝑠𝑚)2𝑓−𝐹2^𝑠𝑚𝑥−𝑛+2𝑓(0+3𝑛)𝑓−𝐹^′2^𝑠𝑚𝑥−𝑛+2𝑓(0+3𝑛)𝑓−𝐹−2^𝑠𝑚𝑥−𝑛+2𝑓(0+3𝑛)𝑓−𝐹^′−2^𝑠𝑚𝑥−𝑛+2𝑓(0+23𝑛_−𝑡𝑚)2𝑓−𝐹2^𝑡𝑚𝑥−𝑛+2𝑓(0+3𝑛)𝑓−𝐹^′2^𝑡𝑚𝑥−𝑛+2𝑓(0+3𝑛)𝑓−𝐹−2^𝑡𝑚𝑥−𝑛+2𝑓(0+3𝑛)𝑓−𝐹^′−2^𝑡𝑚𝑥−𝑛+2𝑓(03𝑛) (17) for all 𝑥∈𝑋⧵{0} and all positive integers 𝑚. It follows from (12) and (17) that 𝐹(𝑥)−𝐹^′(𝑥≤_∞𝑗=0)24^−𝑠𝑚Φ_𝑗2^𝑠𝑚𝑥+2^−𝑡𝑚Φ_𝑗2^𝑡𝑚𝑥 (18) for all 𝑥∈𝑋⧵{0} and all positive integers 𝑚. We can easily show that the terms on the right-hand side of the inequality (18) tend to 0 as 𝑚→∞ for the cases (𝑠,𝑡)=(1,1) and (𝑠,𝑡)=(−1,−1). For the case (𝑠,𝑡)=(1,−1), we have 𝑛(𝑛−2)^∞_𝑗=04^−𝑚Φ_𝑗2^𝑚𝑥+2^𝑚Φ_𝑗2^−𝑚𝑥=^∞_𝑗=02𝜑2^𝑗+𝑚𝑥+2𝜑−2^𝑗+𝑚𝑥4^𝑗+𝑚+1+2^𝑗𝑛2^{2𝑚+1𝑚/2−1}_𝑗=0+^𝑚−1_𝑗=𝑚/2+^∞_𝑗=𝑚×𝜑2^𝑚𝑥2^𝑗+1+𝜑−2^𝑚𝑥2^𝑗+1+2^𝑚2^{2𝑗+1𝑚/2−1}_𝑗=0+^𝑚−1_𝑗=𝑚/2+^∞_𝑗=𝑚×𝜑2^𝑗𝑥2^𝑚+𝜑−2^𝑗𝑥2^𝑚+^∞_𝑗=02^𝑗+𝑚𝑛2𝜑2^{−𝑗−𝑚−1}𝑥+𝜑−2^{−𝑗−𝑚−1}𝑥≤12^∞_𝑗=𝑚+𝑛^𝑚−1_𝑗=𝑚/2+𝑛2^{𝑚/2𝑚/2−1}_𝑗=0×𝜑2^𝑗𝑥+𝜑−2^𝑗𝑥4^𝑗+𝑛2^𝑚∞_𝑗=1+^𝑚_{𝑗=𝑚/2+1}+12^𝑚/2𝑚/2_𝑗=1×2^𝑗𝜑𝑥2^𝑗+𝜑−𝑥2^𝑗+12^𝑚∞_𝑗=0𝜑2^𝑗𝑥+𝜑−2^𝑗𝑥4^𝑗+𝑛4^∞_𝑗=𝑚+12^𝑗𝜑𝑥2^𝑗+𝜑−𝑥2^𝑗 (19) for all 𝑥∈𝑋⧵{0} and all positive even integers 𝑚. So, we also show that the terms on the right-hand side of the inequality (18) tend to 0 as 𝑚→∞ for the cases (𝑠,𝑡)=(1,−1). Using the equality 𝐹(0)=0=𝐹^′(0), we can conclude that 𝐹(𝑥)=𝐹^′(𝑥) for all 𝑥∈𝑋. This proves the uniqueness of 𝐹.

Corollary 3. Let 𝑝≠1,2 be a real number. Suppose that 𝑓∶𝑋→𝑌 is a mapping such that 𝐷𝑓𝑥₁,𝑥₂,…,𝑥_𝑛≤𝑥₁^𝑝+𝑥₂^𝑝+⋯+𝑥_𝑛^𝑝 (20) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} (with 𝑓(0)=0 if 𝑝>2). Then, there exists a unique mapping 𝐹 satisfying (8) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and 𝑓(𝑥)(𝑥≤4−𝐹)2^𝑝+𝑛−42^𝑝‖𝑥‖−2^𝑝𝑓(𝑥)−𝑛−2,𝑖𝑓𝑝>0,𝑛+2𝑓(0)(𝑥≤43𝑛−𝐹)2^𝑝+𝑛−42^𝑝1−2𝑛−2,𝑖𝑓𝑝=0,𝑓(𝑥)=𝐹(𝑥),𝑖𝑓𝑝<0 (21)(22) for all 𝑥∈𝑋⧵{0} with 𝐹(0)=0.

Proof. Put 𝜑(𝑥₁,𝑥₂,…,𝑥_𝑛)=‖𝑥₁‖^𝑝+‖𝑥₂‖^𝑝+⋯+‖𝑥_𝑛‖^𝑝 for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0}. By Theorem 2, there exists a unique mapping 𝐹 satisfying (8) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and 𝑓(𝑥)−𝑛+2𝑓(0)(𝑥≤43𝑛−𝐹)2^𝑝+𝑛−42^𝑝‖𝑥‖−2𝑛−^𝑝2 (23) for all 𝑥∈𝑋⧵{0} with 𝐹(0)=0. From these, we get the inequalities (𝑛−1)𝑛²−4𝑓(0≤(+()−6𝑛)𝐷𝑓−𝐷𝐹𝑘𝑥,𝑘𝑥,…,𝑘𝑥)𝐹−𝑓𝑛𝑘𝑥𝑛+2𝑓(0(()−3𝑛)+𝑛𝑛−2)𝐹−𝑓𝑘𝑥𝑛+2𝑓(0+𝑛()3𝑛)𝑛−12𝑓−𝐹(2𝑘𝑥)−𝑛+2𝑓(0≤𝑛3𝑛)𝑛+^𝑝()+𝑛()2+𝑛𝑛−2𝑛−1^𝑝2×42^𝑝+𝑛−42^𝑝𝑘−2^𝑝‖𝑥‖^𝑝 (24) for all 𝑥∈𝑋⧵{0} and all positive real numbers 𝑘. Taking the limit as 𝑘→∞ or 𝑘→0 in the above inequality, we have 𝑓(0)=0 if 𝑝≠0. Hence, if 𝑝≠0,1,2, then the inequality 𝑓(𝑥)(𝑥≤4−𝐹)2^𝑝+𝑛−42^𝑝‖𝑥‖−2𝑛−^𝑝2 (25) for all 𝑥∈𝑋⧵{0} follows from (23). If 𝑝<0, then we get the inequalities 𝑓(𝑥)(𝑥≤1−𝐹))+))𝑥+((+()𝑥+()𝑛−1𝐷𝑓−𝐷𝐹((−𝑘+1𝑥,𝑘𝑥,…,𝑘𝑥)𝑓−𝐹(((𝑛−2𝑘+1)𝑛−1)(𝑛−2)𝑓−𝐹𝑘𝑥)𝑛−2)𝑓−𝐹((−𝑘+1)𝑛−1)(𝑛−22(≤𝑘𝑓−𝐹2𝑘𝑥)_𝑝+(2𝑘)^𝑝2+(−𝑘+1)^𝑝+))𝑛−1((𝑛−2𝑘+1^𝑝()×4𝑛−1)(𝑛−22^𝑝+𝑛−42^𝑝‖𝑥‖−2^𝑝+(−𝑘+1)^𝑝𝑛−1+𝑘^𝑝‖𝑥‖^𝑝 (26) for all 𝑥∈𝑋⧵{0} and all positive integers 𝑘. Taking the limit as 𝑘→∞ in the above inequality, we get 𝐹(𝑥)=𝑓(𝑥) for all 𝑥∈𝑋⧵{0}. Since 𝑓(0)=0=𝐹(0), the equality 𝑓(𝑥)=𝐹(𝑥) holds for all 𝑥∈𝑋. The result follows from this, (23), and (25).

Lemma 4. If 𝑓∶𝑋→𝑌 is a mapping satisfying (8) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} with 𝑓(0)=0 and 𝑓(𝑡𝑥) is continuous in 𝑡 for each fixed 𝑥, then 𝑓 is represented by 𝑓()=𝑓(𝑥)()𝑟𝑥+𝑓−𝑥2𝑟²+𝑓(𝑥)−𝑓(−𝑥)2𝑟 (27) for all 𝑥∈𝑋 and all 𝑟∈ℝ.

Proof. We will prove the equality 𝑓()=𝑓(𝑥)()𝑚𝑥+𝑓−𝑥2𝑚²+𝑓(𝑥)−𝑓(−𝑥)2𝑚 (28) for all integers 𝑚. First, we will use the induction on 𝑚 to prove the equality (28) for all nonnegative integers 𝑚. Note that 𝑓(0)=0. We can easily prove it for the cases 𝑚=0,1. For the case 𝑚=2, we can show that 𝑓()22𝑥=−𝑛()(𝑛−2𝐷𝑓𝑥)+𝐷𝑓()−1−𝑥2()(𝑛−2𝐷𝑓𝑥)−𝐷𝑓()(𝑥)()=𝑓(𝑥)()−𝑥+3𝑓+𝑓−𝑥+𝑓−𝑥22²+𝑓(𝑥)−𝑓(−𝑥)22 (29) for all 𝑥∈𝑋. Assume that (28) holds for all 𝑥∈𝑋 and all nonnegative integers 𝑘(≤𝑚). Then, we obtain 𝑓((𝑚+1)𝑥)=−𝐷𝑓𝑚𝑥,𝑥,…^𝑛/2,𝑥,^𝑛/2−1())𝑥)−−𝑥,…,−𝑥+2𝑓𝑚𝑥−𝑓((𝑚−1𝑛−44𝑓()−𝑛−2𝑥4𝑓()(𝑥)+()𝑓()=2𝑥+𝑛𝑓𝑛−2−𝑥2𝑚²−(𝑚−1)²−𝑛−44⋅(−2)²−𝑛4⋅2²+𝑛⋅1²+(𝑛−2)⋅(−1)²×𝑓(𝑥)+𝑓(−𝑥)2+𝑓(𝑥)−𝑓(−𝑥)2×2𝑚−(𝑚−1)−𝑛−44⋅()−𝑛−24()=𝑓(𝑥)()⋅2+𝑛⋅1+𝑛−2)(−1+𝑓−𝑥2(𝑚+1)²+𝑓(𝑥)−𝑓(−𝑥)2(𝑚+1) (30) which completes (28) for all nonnegative integers 𝑚. Using the similar method, we also can prove the equality (28) for all negative integers 𝑚. By (28), we get the equalities 𝑓(𝑚𝑥)+𝑓(−𝑚𝑥)2=𝑓(𝑥)+𝑓(−𝑥)2𝑚²,𝑓(𝑚𝑥)−𝑓(−𝑚𝑥)2=𝑓(𝑥)−𝑓(−𝑥)2𝑓()()𝑚,𝑥/𝑚+𝑓−𝑥/𝑚2=𝑓(𝑥)+𝑓(−𝑥)21𝑚²,𝑓(𝑥/𝑚)−𝑓(−𝑥/𝑚)2=𝑓(𝑥)−𝑓(−𝑥)2𝑚 (31) for all 𝑥∈𝑋 and all integers 𝑚≠0. Hence, 𝑓𝑝𝑞𝑥=𝑓𝑝/𝑞𝑥+𝑓−𝑝/𝑞𝑥2+𝑓𝑝/𝑞𝑥−𝑓−𝑝/𝑞𝑥2=𝑓𝑥/𝑞+𝑓−𝑥/𝑞2𝑝²+𝑓𝑥/𝑞−𝑓−𝑥/𝑞2𝑝=𝑓(𝑥)+𝑓(−𝑥)2𝑝²𝑞²+𝑓(𝑥)−𝑓(−𝑥)2𝑝𝑞 (32) for all 𝑥∈𝑋 and all integers 𝑝,𝑞(≠0). If 𝑟∈ℝ, then there exists a rational sequence {𝑟_𝑚} satisfying lim_𝑚→∞𝑟_𝑚=𝑟. Since 𝑓(𝑡𝑥) is continuous in 𝑡 for each fixed 𝑥, we have 𝑓(𝑟𝑥)=lim_𝑚→∞𝑓𝑟_𝑚𝑥=lim_𝑚→∞𝑓(𝑥)+𝑓(−𝑥)2𝑟²_𝑚+𝑓(𝑥)−𝑓(−𝑥)2𝑟_𝑚=𝑓(𝑥)+𝑓(−𝑥)2𝑟²+𝑓(𝑥)−𝑓(−𝑥)2𝑟 (33) for all 𝑥∈𝑋.

3. Stability of the Functional Equation (2) (𝑛 Is Odd)

Let (𝑠,𝑡), 𝜑, 𝐷𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛), and 𝜏_𝑘,𝑚 be as in Section 2. In this section, let 𝑛 be an odd integer greater than 2. For convenience, we use the following abbreviations in this section for a given mapping 𝑓∶𝑋→𝑌: 𝐽_𝑚𝑓(𝑥)=124^−𝑠𝑚𝑓2^𝑠𝑚𝑥+𝑓−2^𝑠𝑚𝑥−2(𝑛+1)3()𝑓(0)𝑛−1+2^−𝑡𝑚𝑓2^𝑡𝑚𝑥−𝑓−2^𝑡𝑚𝑥,𝑥=𝑥,…,^()𝑛+1/2𝑥,−𝑥,…,^()𝑛−1/2−𝑥, (34) for 𝑥∈𝑋. From these, we get 𝐽_𝑚𝑓(𝑥)−𝐽_𝑚+1𝑓(𝑥)=4^{𝜏−𝑠,𝑚+1}2(𝑛−1)(𝑛−1)𝐷𝑓2^{𝜏𝑠,𝑚}𝑥+𝐷𝑓−2^{𝜏𝑠,𝑚}𝑥𝑠+2^{𝜏−𝑡,𝑚}𝑛−1𝐷𝑓2^{𝜏𝑡,𝑚}𝑥−𝐷𝑓−2^{𝜏𝑡,𝑚}𝑥𝑡 (35) for all 𝑥∈𝑋.

Using (35) and a similar method in the proof of Lemma 1, we get the following lemma.

Lemma 5. If 𝑓∶𝑋→𝑌 is a mapping satisfying (8) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0}, then 𝐽_𝑚𝑓(𝑥)=𝑓(𝑥)−𝑛+13()𝑓(0)𝑛−1 (36) for all 𝑥∈𝑋⧵{0}.

From (35), Lemma 5, and similar methods used in Theorem 2, we get the following theorem.

Theorem 6. If 𝑓∶𝑋→𝑌 is a unique mapping satisfying (11) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} with lim_𝑚→∞𝐽_𝑚𝑓(0)=0, then there exists a unique mapping 𝐹∶𝑋→𝑌 satisfying (8) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and 𝑓(𝑥)−𝑛+13()𝑓(0)(𝑥≤𝑛−1−𝐹)^∞_𝑗=0Φ_𝑗(𝑥) (37) for all 𝑥∈𝑋⧵{0} with 𝐹(0)=0, where Φ_𝑗 are the mappings defined by Φ_𝑗(𝑥)4∶=^{𝜏−𝑠,𝑚+1}2()𝜑𝑛−1)(𝑛−12^{𝜏𝑠,𝑚}𝑥+𝜑−2^{𝜏𝑠,𝑚}𝑥+2^{𝜏−𝑡,𝑚}𝜑𝑛−12^{𝜏𝑡,𝑚}𝑥+𝜑−2^{𝜏𝑡,𝑚}𝑥 (38) for all 𝑥∈𝑋⧵{0}.

From Theorem 6 and similar methods used in Corollary 3, we get the following corollary.

Corollary 7. Let 𝑝≠1,2 be a real number. Suppose that 𝑓∶𝑋→𝑌 is a mapping satisfying (20) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} (with 𝑓(0)=0 if 𝑝>2). Then, there exists a unique mapping 𝐹 satisfying (8) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and 𝑓(𝑥)(𝑥≤4−𝐹)(𝑛−1)2^𝑝+2−42^𝑝−2𝑛‖𝑥‖^𝑝,𝑓(𝑥)−𝑛−1𝑖𝑓𝑝>0,𝑛+13()𝑓(0)(𝑥≤4𝑛−1−𝐹)(𝑛−1)2^𝑝+2−42^𝑝𝑛−2,𝑓(𝑥)(𝑥)𝑛−1𝑖𝑓𝑝=0,=𝐹,𝑖𝑓𝑝<0, (39) for all 𝑥∈𝑋⧵{0} with 𝐹(0)=0.

Proof. Put 𝜑(𝑥₁,𝑥₂,…,𝑥_𝑛)=‖𝑥₁‖^𝑝+‖𝑥₂‖^𝑝+⋯+‖𝑥_𝑛‖^𝑝 for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0}. By Theorem 6, there exists a unique mapping 𝐹 satisfying (8) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} and 𝑓(𝑥)−𝑛+13()𝑓(0)(𝑥≤4𝑛−1−𝐹)(𝑛−1)2^𝑝+2−42^𝑝𝑛‖𝑥‖−2𝑛−1^𝑝 (40) for all 𝑥∈𝑋⧵{0} with 𝐹(0)=0. From these, we get the inequalities (𝑛+1)(𝑛−2)6𝑓(0≤()−)𝐹−𝑓𝑛𝑘𝑥𝑛+13()𝑓(0(()−𝑛−1)+𝑛𝑛−2)𝐹−𝑓𝑘𝑥𝑛+13()𝑓(0+𝑛()𝑛−1)𝑛−12𝑓−𝐹(2𝑘𝑥)−𝑛+13()𝑓(0+(≤𝑛𝑛−1)𝐷𝑓−𝐷𝐹𝑘𝑥,𝑘𝑥,…,𝑘𝑥)_𝑝()+𝑛()2+𝑛𝑛−2𝑛−1^𝑝2×4(𝑛−1)2^𝑝+2−42^𝑝𝑛−2𝑘𝑛−1+𝑛^𝑝‖𝑥‖^𝑝 (41) for all 𝑥∈𝑋⧵{0} and all positive real numbers 𝑘. Taking the limit as 𝑘→∞ or 𝑘→0 in the above inequality, we have 𝑓(0)=0 if 𝑝≠0. Hence, if 𝑝≠0,1,2, then the inequality 𝑓(𝑥)(𝑥≤4−𝐹)(𝑛−1)2^𝑝+2−42^𝑝𝑛‖𝑥‖−2𝑛−1^𝑝 (42) for all 𝑥∈𝑋⧵{0} follows from (40). If 𝑝<0, then we get the inequalities 𝑓(𝑥)(𝑥≤1−𝐹))+))𝑥+((+()𝑥+()𝑛−1𝐷𝑓−𝐷𝐹((−𝑘+1𝑥,𝑘𝑥,…,𝑘𝑥)𝑓−𝐹(((𝑛−2𝑘+1)𝑛−1)(𝑛−2)𝑓−𝐹𝑘𝑥)𝑛−2)𝑓−𝐹((−𝑘+1)𝑛−1)(𝑛−22(≤()𝑘𝑓−𝐹2𝑘𝑥)𝑛−2^𝑝+(2𝑘)^𝑝2+(−𝑘+1)^𝑝+))𝑛−1((𝑛−2𝑘+1^𝑝()×4𝑛−1)(𝑛−22^𝑝)+2−4(𝑛−12^𝑝𝑛‖𝑥‖−2^𝑝+()𝑛−1−𝑘+1^𝑝𝑛−1+𝑘^𝑝‖𝑥‖^𝑝 (43) for all 𝑥∈𝑋⧵{0} and all positive integers 𝑘. Taking the limit as 𝑘→∞ in the above inequality, we get 𝐹(𝑥)=𝑓(𝑥) for all 𝑥∈𝑋⧵{0}. Since 𝑓(0)=0=𝐹(0), the equality 𝑓(𝑥)=𝐹(𝑥) holds for all 𝑥∈𝑋. The result follows from this, (40), and (42).

From similar methods used in Lemma 4, we get the following lemma.

Lemma 8. If 𝑓∶𝑋→𝑌 is a mapping satisfying (8) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0} with 𝑓(0)=0 and 𝑓(𝑡𝑥) is continuous in 𝑡 for each fixed 𝑥, then 𝑓 is represented by 𝑓()=𝑓(𝑥)()𝑟𝑥+𝑓−𝑥2𝑟²+𝑓(𝑥)−𝑓(−𝑥)2𝑟 (44) for all 𝑥∈𝑋 and all 𝑟∈ℝ.

Proof. We will use the induction on 𝑚 to prove (44) for all nonnegative integers 𝑚. Note that 𝑓(0)=0. We can easily prove it for the cases 𝑚=0,1. For the case 𝑚=2, we can show that 𝑓()22𝑥=−(𝑛−1)²𝐷𝑓(𝑥)+𝐷𝑓()−1−𝑥(𝑛−1𝐷𝑓𝑥)−𝐷𝑓()(𝑥)()=𝑓(𝑥)()−𝑥+3𝑓+𝑓−𝑥+𝑓−𝑥22²+𝑓(𝑥)−𝑓(−𝑥)22 (45) for all 𝑥∈𝑋. Assume that (44) holds for all 𝑥∈𝑋 and all nonnegative integers 𝑘(≤𝑚). Then, we obtain 𝑓)𝑥)2((𝑚+1=−𝑛−1𝐷𝑓𝑚𝑥,𝑥,…,^()𝑛−1/2𝑥,^()𝑛−1/2())𝑥)−−𝑥,…,−𝑥+2𝑓𝑚𝑥−𝑓((𝑚−1𝑛−34𝑓(−2𝑥)−𝑛−34𝑓()+()𝑓(𝑥)+()𝑓()=2𝑥𝑛−2𝑛−2−𝑥2𝑚²−(𝑚−1)²−𝑛−34⋅(−2)²+2²+(𝑛−2)1+(−1)²×𝑓(𝑥)+𝑓(−𝑥)2+𝑓(𝑥)−𝑓(−𝑥)2×2𝑚−(𝑚−1)−𝑛−34×))+((=𝑓(𝑥)()((−2+2𝑛−2)(1+−1))+𝑓−𝑥2(𝑚+1)²+𝑓(𝑥)−𝑓(−𝑥)2(𝑚+1) (46) which completes the proof of (44). The remainder of the proof is the same in the proof of Lemma 4.

Corollary 9. If 𝑓∶𝑋→𝑌 is a mapping satisfying (8) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0}, then 𝑓(0)=0.

Proof. Put 𝑝=−1. Then, we have 𝐷𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=0≤𝑥₁^𝑝+𝑥₂^𝑝+⋯+𝑥_𝑛^𝑝 (47) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋⧵{0}. By Corollaries 3 and 7, 𝑓(𝑥)=𝐹(𝑥) for all 𝑥∈𝑋 with 𝐹(0)=0. So, we get the desired result.

Corollary 10. Let 𝑝<0 be a real number and 𝑛>2 an integer. Suppose that 𝑓∶ℝ→ℝ is a mapping satisfying 𝐷𝑓𝑥₁,𝑥₂,…,𝑥_𝑛≤𝑥₁^𝑝+𝑥₂^𝑝+⋯+𝑥_𝑛^𝑝 (48) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈ℝ⧵{0} and 𝑓 is continuous. Then, 𝑓 is represented by 𝑓(𝑥)=𝑓(1)+𝑓(−1)2𝑥²+𝑓(1)−𝑓(−1)2𝑥 (49) for all 𝑥∈ℝ and 𝐷𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛)=0 for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈ℝ.

Proof. If 𝑛 is even, then the equality (49) follows from Corollary 3 and Lemma 4. If 𝑛 is odd, then the equality (49) follows from Corollary 7 and Lemma 8. And we can easily show that the function defined by (49) satisfies the functional equation 𝐷𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛)=0 for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈ℝ.

4. Another Proof for the Stability of the Functional Equation (2)

Let (𝑠,𝑡), 𝐷𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛), 𝜏_𝑘,𝑚 be as in Section 2. In this section, Let 𝑛 be a fixed integer greater than 2 and let 𝜑∶𝑋^𝑛→[0,∞) be a function satisfying the conditions (3) and (4) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋. For convenience, we use the following abbreviations in this section for a given mapping 𝑓∶𝑋→𝑌: 𝐽_𝑚𝑓(𝑥)=124^−𝑠𝑚𝑓2^𝑠𝑚𝑥+𝑓−2^𝑠𝑚𝑥+(𝑛−4)(𝑛+1)3𝑓(0)+2^−𝑡𝑚𝑓2^𝑡𝑚𝑥−𝑓−2^𝑡𝑚𝑥,𝑥=(𝑥,−𝑥,𝑥,0,…,0), (50) for all 𝑥∈𝑋. From these, we get 𝐽_𝑚𝑓(𝑥)−𝐽_𝑚+1𝑓(𝑥)=4^{𝜏−𝑠,𝑚}2𝐷𝑓2^{𝜏𝑠,𝑚}𝑥+𝐷𝑓−2^{𝜏𝑠,𝑚}𝑥𝑠+2^{𝜏−𝑡,𝑚}2𝐷𝑓2^{𝜏𝑡,𝑚}𝑥−𝐷𝑓−2^{𝜏𝑡,𝑚}𝑥𝑡 (51) for all 𝑥∈𝑋. Using (51) and a similar method in the proof of Lemma 1, we get the following lemma.

Lemma 11. If 𝑓∶𝑋→𝑌 is a quadratic-additive mapping, then 𝐽_𝑚𝑓(𝑥)=𝑓(𝑥) (52) for all 𝑥∈𝑋.

Theorem 12 (compare with Theorem 3.1 in [15]). Suppose that 𝑓∶𝑋→𝑌 is a mapping such that 𝐷𝑓𝑥₁,𝑥₂,…,𝑥_𝑛≤𝜑𝑥₁,𝑥₂,…,𝑥_𝑛 (53) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋. Then, there exists a unique quadratic-additive mapping 𝐹∶𝑋→𝑌 such that 𝑓(𝑥)+(𝑛−4)(𝑛+1)6𝑓(0)−𝐹(𝑥)≤^∞_𝑗=0Φ_𝑗(𝑥) (54) for all 𝑥∈𝑋, where Φ_𝑗 are the mappings defined by Φ_𝑗(𝑥)4∶=^{𝜏−𝑠,𝑗}2𝜑2^{𝜏𝑠,𝑗}𝑥+𝜑−2^{𝜏𝑠,𝑗}𝑥+2^{𝜏−𝑡,𝑗}2𝜑2^{𝜏𝑡,𝑗}𝑥+𝜑−2^{𝜏𝑡,𝑗}𝑥 (55) for all 𝑥∈𝑋.

Proof. Note that 𝐽_𝑚𝑓(0=4)^−𝑠𝑚(𝑛−1)(𝑛−2)6𝑓(0=4_−𝑠𝑚)3(≤4𝐷𝑓0,0,…,0)_−𝑠𝑚3𝜑(0,0,…,0) (56) for all positive integers 𝑚. It follows from (3) that lim_𝑚→∞𝐽_𝑚𝑓(0)=0. From this, (51), Lemma 11, and similar methods used in Theorem 2, we obtain this theorem.

Corollary 13 (compare with Corollary 3.3 in [15]). Let 𝑝≠1,2 be a positive real number. Suppose that 𝑓∶𝑋→𝑌 is a mapping such that 𝐷𝑓𝑥₁,𝑥₂,…,𝑥_𝑛≤𝑥₁^𝑝+𝑥₂^𝑝+⋯+𝑥_𝑛^𝑝 (57) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋. Then, there exists a unique quadratic-additive mapping 𝐹 such that 𝑓(𝑥)(𝑥≤3−𝐹)2^𝑝+3−42^𝑝‖𝑥‖−2^𝑝 (58) for all 𝑥∈𝑋.

Proof. Since ‖𝑓(0)‖=(2/(𝑛−1)(𝑛−2))‖𝐷𝑓(0,0,…,0)‖≤0, we get 𝑓(0)=0. From Theorem 12 and similar methods used in Corollary 3, we obtain this corollary.

From Theorem 12 and similar methods used in Corollary 3, we get the following corollary.

Corollary 14 (compare with Corollary 3.2 in [15]). Suppose that 𝑓∶𝑋→𝑌 is a mapping such that 𝐷𝑓𝑥₁,𝑥₂,…,𝑥_𝑛≤𝜀 (59) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋. Then, there exists a unique quadratic-additive mapping 𝐹 such that 𝑓(𝑥)+(𝑛−4)(𝑛+1)6𝑓(0)−𝐹(𝑥)≤4𝜀3 (60) for all 𝑥∈𝑋.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2012R1A1A4A01002971).