Functional Equations Related to Inner Product Spaces

Abstract

Let 𝑉,𝑊 be real vector spaces. It is shown that an odd mapping 𝑓∶𝑉→𝑊 satisfies ∑^2𝑛_𝑖−1𝑓(𝑥_𝑖−1/2𝑛∑^2𝑛_𝑗=1𝑥_𝑗)=∑^2𝑛_𝑖=1𝑓(𝑥_𝑖)−2𝑛𝑓(1/2𝑛∑^2𝑛_𝑖=1𝑥_𝑖) for all 𝑥₁,…,𝑥_2𝑛∈𝑉 if and only if the odd mapping 𝑓∶𝑉→𝑊 is Cauchy additive. Furthermore, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.

1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' Theorem was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [4] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of Th. M. Rassias' theorem was obtained by Găvruta [5] by replacing the unbounded Cauchy difference by a general control function.

The functional equation, 𝑓(𝑥+𝑦)+𝑓(𝑥−𝑦)=2𝑓(𝑥)+2𝑓(𝑦),(1.1) is called a quadratic functional equation . In particular, every solution of the quadratic functional equation is said to be a quadratic mapping . A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [6] for mappings 𝑓∶𝑋→𝑌, where 𝑋 is a normed space and 𝑌 is a Banach space. Cholewa [7] noticed that the theorem of Skof is still true if the relevant domain 𝑋 is replaced by an Abelian group. The generalized Hyers-Ulam stability of the quadratic functional equation has been proved by Czerwik [8], J. M. Rassias [9], Găvruta [10], and others [11]. In [12], Th. M. Rassias proved that the norm defined over a real vector space 𝑉 is induced by an inner product if and only if for a fixed integer 𝑛≥2^𝑛_𝑖=1‖𝑥_𝑖−1𝑛^𝑛_𝑗=1𝑥_𝑗‖²=^𝑛_𝑖=1‖𝑥_𝑖‖²1−𝑛‖𝑛^𝑛_𝑖=1𝑥_𝑖‖²(1.2) holds for all 𝑥₁,…,𝑥_𝑛∈𝑉. An operator extension of this norm equality is presented in [13]. For more information on the recent results on the stability of quadratic functional equation, see [14]. Inner product spaces, Cauchy equation, Euler-Lagrange-Rassias equations, and Ulam-Găvruta-Rassias stability have been studied by several authors (see [15–27]).

In [28], C. Park, Lee, and Shin proved that if an even mapping 𝑓∶𝑉→𝑊 satisfies

^2𝑛_𝑖=1𝑓(𝑥_𝑖−12𝑛^2𝑛_𝑗=1𝑥_𝑗)=^2𝑛_𝑖=1𝑓(𝑥_𝑖1)−2𝑛𝑓(2𝑛^2𝑛_𝑖=1𝑥_𝑖),(1.3) then the even mapping 𝑓∶𝑉→𝑊 is quadratic. Moreover, they proved the generalized Hyers-Ulam stability of the quadratic functional equation (1.3) in real Banach spaces.

Throughout this paper, assume that 𝑛 is a fixed positive integer, 𝑋 and 𝑌 are real normed vector spaces.

In this paper, we investigate the functional equation ^2𝑛_𝑖=1𝑓(𝑥_𝑖−12𝑛^2𝑛_𝑗=1𝑥_𝑗)=^2𝑛_𝑖=1𝑓(𝑥_𝑖1)−2𝑛𝑓(2𝑛^2𝑛_𝑖=1𝑥_𝑖),(1.4) and prove the generalized Hyers-Ulam stability of the functional equation (1.4) in real Banach spaces.

2. Functional Equations Related to Inner Product Spaces

We investigate the functional equation (1.4).

Lemma 2.1. Let 𝑉 and 𝑊 be real vector spaces. An odd mapping 𝑓∶𝑉→𝑊 satisfies ^2𝑛_𝑖=1𝑓(𝑥_𝑖−12𝑛^2𝑛_𝑗=1𝑥_𝑗)=^2𝑛_𝑖=1𝑓(𝑥_𝑖1)−2𝑛𝑓(2𝑛^2𝑛_𝑖=1𝑥_𝑖),(2.1) for all 𝑥₁,…,𝑥_2𝑛∈𝑉 if and only if the odd mapping 𝑓∶𝑉→𝑊 is Cauchy additive, that is, 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦),(2.2) for all 𝑥,𝑦∈𝑉.

Proof. Assume that 𝑓∶𝑉→𝑊 satisfies (2.1).
Letting 𝑥₁=⋯=𝑥_𝑛=𝑥, 𝑥_𝑛+1=⋯=𝑥_2𝑛=𝑦 in (2.1), we get 𝑛𝑓(𝑥−𝑥+𝑦2)+𝑛𝑓(𝑦−𝑥+𝑦2)=𝑛𝑓(𝑥)+𝑛𝑓(𝑦)−2𝑛𝑓(𝑥+𝑦2),(2.3)
for all 𝑥,𝑦∈𝑉. Since 𝑓∶𝑉→𝑊 is odd, 0=𝑛𝑓(𝑥)+𝑛𝑓(𝑦)−2𝑛𝑓(𝑥+𝑦2),(2.4) for all 𝑥,𝑦∈𝑉 and 𝑓(0)=0. So 2𝑓(𝑥+𝑦2)=𝑓(𝑥)+𝑓(𝑦),(2.5) for all 𝑥,𝑦∈𝑉. Letting 𝑦=0 in (2.5), we get 2𝑓(𝑥/2)=𝑓(𝑥) for all 𝑥∈𝑉. Thus 𝑓(𝑥+𝑦)=2𝑓(𝑥+𝑦2)=𝑓(𝑥)+𝑓(𝑦),(2.6) for all 𝑥,𝑦∈𝑉.
It is easy to prove the converse.

For a given mapping 𝑓∶𝑋→𝑌, we define 𝐷𝑓(𝑥₁,…,𝑥_2𝑛)∶=^2𝑛_𝑖=1𝑓(𝑥_𝑖−12𝑛^2𝑛_𝑗=1𝑥_𝑗)−^2𝑛_𝑖=1𝑓(𝑥_𝑖1)+2𝑛𝑓(2𝑛^2𝑛_𝑖=1𝑥_𝑖),(2.7) for all 𝑥₁,…,𝑥_2𝑛∈𝑋.

We are going to prove the generalized Hyers-Ulam stability of the functional equation 𝐷𝑓(𝑥₁,…,𝑥_2𝑛)=0 in real Banach spaces.Theorem 2.2. Let 𝑓∶𝑋→𝑌 be a mapping satisfying 𝑓(0)=0 for which there exists a function 𝜑∶𝑋^2𝑛→[0,∞) such that 𝜑(𝑥₁,…,𝑥_2𝑛)∶=^∞_𝑗=02^𝑗𝑥𝜑(¹2^𝑗𝑥,…,^2𝑛2^𝑗)<∞,(2.8)‖𝐷𝑓(𝑥₁,…,𝑥_2𝑛)‖≤𝜑(𝑥₁,…,𝑥_2𝑛),(2.9) for all 𝑥₁,…,𝑥_2𝑛∈𝑋. Then there exists a unique Cauchy additive mapping 𝐴∶𝑋→𝑌 satisfying (2.1) such that

1‖𝑓(𝑥)−𝑓(−𝑥)−𝐴(𝑥)‖≤𝑛𝜑(𝑥,…,𝑥_𝑛times,0,…,0_𝑛times1)+𝑛𝜑(−𝑥,…,−𝑥_𝑛times,0,…,0_𝑛times),(2.10) for all 𝑥∈𝑋.

Proof. Letting 𝑥₁=⋯=𝑥_𝑛=𝑥 and 𝑥_𝑛+1=⋯=𝑥_2𝑛=0 in (2.9), we get 𝑥‖3𝑛𝑓(2)+𝑛𝑓(−𝑥2)−𝑛𝑓(𝑥)‖≤𝜑(𝑥,…,𝑥_𝑛times,0,…,0_𝑛times),(2.11) for all 𝑥∈𝑋. Replacing 𝑥 by −𝑥 in (2.11), we get ‖3𝑛𝑓(−𝑥2𝑥)+𝑛𝑓(2)−𝑛𝑓(−𝑥)‖≤𝜑(−𝑥,…,−𝑥_𝑛times,0,…,0_𝑛times),(2.12) for all 𝑥∈𝑋. Let 𝑔(𝑥)∶=𝑓(𝑥)−𝑓(−𝑥) for all 𝑥∈𝑋. It follows from (2.11) and (2.12) that 𝑥‖2𝑛𝑔(2)−𝑛𝑔(𝑥)‖≤𝜑(𝑥,…,𝑥_𝑛times,0,…,0_𝑛times)+𝜑(−𝑥,…,−𝑥_𝑛times,0,…,0_𝑛times),(2.13) for all 𝑥∈𝑋. So 𝑥‖𝑔(𝑥)−2𝑔(21)‖≤𝑛𝜑(𝑥,…,𝑥_𝑛times,0,…,0_𝑛times1)+𝑛𝜑(−𝑥,…,−𝑥_𝑛times,0,…,0_𝑛times),(2.14) for all 𝑥∈𝑋. Hence ‖2^𝑙𝑥𝑔(2^𝑙)−2^𝑚𝑥𝑔(2^𝑚)‖≤^𝑚−1_𝑗=𝑙2^𝑗𝑛𝑥𝜑(2^𝑗𝑥,…,^𝑗_𝑛time2_s,0,…,0_𝑛times)+^𝑚−1_𝑗=𝑙2^𝑗𝑛𝑥𝜑(−2^𝑗𝑥,…,−^𝑗_𝑛time2_s,0,…,0_𝑛times),(2.15) for all nonnegative integers 𝑚 and 𝑙 with 𝑚>𝑙 and all 𝑥∈𝑋. It follows from (2.8) and (2.15) that the sequence {2^𝑘𝑔(𝑥/2^𝑘)} is Cauchy for all 𝑥∈𝑋. Since 𝑌 is complete, the sequence {2^𝑘𝑔(𝑥/2^𝑘)} converges. So one can define the mapping 𝐴∶𝑋→𝑌 by 𝐴(𝑥)∶=lim_𝑘→∞2^𝑘𝑥𝑔(2^𝑘),(2.16) for all 𝑥∈𝑋.
By (2.8) and (2.9),
‖𝐷𝐴(𝑥₁,…,𝑥_2𝑛)‖=lim_𝑘→∞2^𝑘𝑥‖𝐷𝑔(¹2^𝑘𝑥,…,^2𝑛2^𝑘)‖≤lim_𝑘→∞2^𝑘𝑥[𝜑(¹2^𝑘𝑥,…,^2𝑛2^𝑘𝑥)+𝜑(−¹2^𝑘𝑥,…,−^2𝑛2^𝑘)]=0,(2.17) for all 𝑥₁,…,𝑥_2𝑛∈𝑋. So 𝐷𝐴(𝑥₁,…,𝑥_2𝑛)=0. By Lemma 2.1, the mapping 𝐴∶𝑋→𝑌 is Cauchy additive. Moreover, letting 𝑙=0 and passing the limit 𝑚→∞ in (2.15), we get (2.10). So there exists a Cauchy additive mapping 𝐴∶𝑋→𝑌 satisfying (2.1) and (2.10).
Now, let 𝐿∶𝑋→𝑌 be another Cauchy additive mapping satisfying (2.1) and (2.10). Then we have
‖𝐴(𝑥)−𝐿(𝑥)‖=2^𝑞𝑥‖𝐴(2^𝑞𝑥)−𝐿(2^𝑞)‖≤2^𝑞𝑥(‖𝐴(2^𝑞𝑥)−𝑓(2^𝑞)+𝑓(−𝑥2^𝑞𝑥)‖+‖𝐿(2^𝑞𝑥)−𝑓(2^𝑞)+𝑓(−𝑥2^𝑞≤)‖)2⋅2^𝑞𝑛𝑥𝜑(2^𝑞𝑥,…,^𝑞_𝑛time2_s,0,…,0_𝑛times)+2⋅2^𝑞𝑛𝜑(−𝑥2^𝑞,…,−𝑥^𝑞_𝑛time2_s,0,…,0_𝑛times),(2.18) which tends to zero as 𝑞→∞ for all 𝑥∈𝑋. So we can conclude that 𝐴(𝑥)=𝐿(𝑥) for all 𝑥∈𝑋. This proves the uniqueness of 𝐴.Corollary 2.3. Let 𝑝>1 and 𝜃 be positive real numbers, and let 𝑓∶𝑋→𝑌 be a mapping such that ‖𝐷𝑓(𝑥₁,…,𝑥_2𝑛)‖≤𝜃^2𝑛_𝑗=1‖𝑥_𝑗‖^𝑝,(2.19) for all 𝑥₁,…,𝑥_2𝑛∈𝑋. Then there exists a unique Cauchy additive mapping 𝐴∶𝑋→𝑌 satisfying (2.1) such that 2‖𝑓(𝑥)−𝑓(−𝑥)−𝐴(𝑥)‖≤^𝑝+1𝜃2^𝑝−2‖𝑥‖^𝑝,(2.20) for all 𝑥∈𝑋.Proof. Define 𝜑(𝑥₁,…,𝑥_2𝑛)=𝜃∑^2𝑛_𝑗=1‖𝑥_𝑗‖^𝑝, and apply Theorem 2.2 to get the desired result.Corollary 2.4. Let 𝑓∶𝑋→𝑌 be an odd mapping for which there exists a function 𝜑∶𝑋^2𝑛→[0,∞) satisfying (2.8) and (2.9). Then there exists a unique Cauchy additive mapping 𝐴∶𝑋→𝑌 satisfying (2.1) such that 1‖2𝑓(𝑥)−𝐴(𝑥)‖≤𝑛𝜑(𝑥,…,𝑥_𝑛times,0,…,0_𝑛times1)+𝑛𝜑(−𝑥,…,−𝑥_𝑛times,0,…,0_𝑛times),(2.21) or (alternative approximation) 1‖𝑓(𝑥)−𝐴(𝑥)‖≤2𝑛𝜑(𝑥,…,𝑥_𝑛times,0,…,0_𝑛times1)+2𝑛𝜑(−𝑥,…,−𝑥_𝑛times,0,…,0_𝑛times),(2.22) for all 𝑥∈𝑋, where 𝜑 is defined in (2.8).Theorem 2.5. Let 𝑓∶𝑋→𝑌 be a mapping satisfying 𝑓(0)=0 for which there exists a function 𝜑∶𝑋^2𝑛→[0,∞) satisfying (2.9) such that 𝜑(𝑥₁,…,𝑥_2𝑛)∶=^∞_𝑗=12^−𝑗𝜑(2^𝑗𝑥₁,…,2^𝑗𝑥_2𝑛)<∞,(2.23) for all 𝑥₁,…,𝑥_2𝑛∈𝑋. Then there exists a unique Cauchy additive mapping 𝐴∶𝑋→𝑌 satisfying (2.1) such that 1‖𝑓(𝑥)−𝑓(−𝑥)−𝐴(𝑥)‖≤𝑛𝜑(𝑥,…,𝑥_𝑛times,0,…,0_𝑛times1)+𝑛𝜑(−𝑥,…,−𝑥_𝑛times,0,…,0_𝑛times),(2.24) for all 𝑥∈𝑋.

Proof. It follows from (2.13) that 1‖𝑔(𝑥)−21𝑔(2𝑥)‖≤2𝑛𝜑(2𝑥,…,2𝑥_𝑛times,0,…,0_𝑛times1)+2𝑛𝜑(−2𝑥,…,−2𝑥_𝑛times,0,…,0_𝑛times),(2.25) for all 𝑥∈𝑋. So ‖12^𝑙𝑔(2^𝑙1𝑥)−2^𝑚𝑔(2^𝑚𝑥)‖≤^𝑚_𝑗=𝑙+112^𝑗𝑛𝜑(2^𝑗𝑥,…,2^𝑗𝑥_𝑛times,0,…,0_𝑛times)+^𝑚_𝑗=𝑙+112^𝑗𝑛𝜑(−2^𝑗𝑥,…,−2^𝑗𝑥_𝑛times,0,…,0_𝑛times),(2.26) for all nonnegative integers 𝑚 and 𝑙 with 𝑚>𝑙 and all 𝑥∈𝑋. It follows from (2.23) and (2.26) that the sequence {(1/2^𝑘)𝑔(2^𝑘𝑥)} is Cauchy for all 𝑥∈𝑋. Since 𝑌 is complete, the sequence {(1/2^𝑘)𝑔(2^𝑘𝑥)} converges. So one can define the mapping 𝐴∶𝑋→𝑌 by 𝐴(𝑥)∶=lim_𝑘→∞12^𝑘𝑔(2^𝑘𝑥),(2.27) for all 𝑥∈𝑋.
By (2.9) and (2.23),
‖𝐷𝐴(𝑥₁,…,𝑥_2𝑛)‖=lim_𝑘→∞12^𝑘‖𝐷𝑔(2^𝑘𝑥₁,…,2^𝑘𝑥_2𝑛)‖≤lim_𝑘→∞12^𝑘(𝜑(2^𝑘𝑥₁,…,2^𝑘𝑥_2𝑛)+𝜑(−2^𝑘𝑥₁,…,−2^𝑘𝑥_2𝑛))=0,(2.28) for all 𝑥₁,…,𝑥_2𝑛∈𝑋. So 𝐷𝐴(𝑥₁,…,𝑥_2𝑛)=0. By Lemma 2.1, the mapping 𝐴∶𝑋→𝑌 is Cauchy additive. Moreover, letting 𝑙=0 and passing the limit 𝑚→∞ in (2.26), we get (2.24). So there exists a Cauchy additive mapping 𝐴∶𝑋→𝑌 satisfying (2.1) and (2.24).
The rest of the proof is similar to the proof of Theorem 2.2.

Corollary 2.6. Let 𝑝<1 and 𝜃 be positive real numbers, and let 𝑓∶𝑋→𝑌 be a mapping satisfying (2.19). Then there exists a unique Cauchy additive mapping 𝐴∶𝑋→𝑌 satisfying (2.1) such that 2‖𝑓(𝑥)−𝑓(−𝑥)−𝐴(𝑥)‖≤^𝑝+1𝜃2−2^𝑝‖𝑥‖^𝑝,(2.29) for all 𝑥∈𝑋.Proof. Define 𝜑(𝑥₁,…,𝑥_2𝑛)=𝜃∑^2𝑛_𝑗=1‖𝑥_𝑗‖^𝑝, and apply Theorem 2.5 to get the desired result.Corollary 2.7. Let 𝑓∶𝑋→𝑌 be an odd mapping for which there exists a function 𝜑∶𝑋^2𝑛→[0,∞) satisfying (2.9) and (2.23). Then there exists a unique Cauchy additive mapping 𝐴∶𝑋→𝑌 satisfying (2.1) such that 1‖2𝑓(𝑥)−𝐴(𝑥)‖≤𝑛𝜑(𝑥,…,𝑥_𝑛times,0,…,0_𝑛times1)+𝑛𝜑(−𝑥,…,−𝑥_𝑛times,0,…,0_𝑛times),(2.30) or (alternative approximation), 1‖𝑓(𝑥)−𝐴(𝑥)‖≤2𝑛𝜑(𝑥,…,𝑥_𝑛times,0,…,0_𝑛times1)+2𝑛𝜑(−𝑥,…,−𝑥_𝑛times,0,…,0_𝑛times),(2.31) for all 𝑥∈𝑋, where 𝜑 is defined in (2.23).The following was proved in [28].Remark 2.8 ([28]). Let 𝑓∶𝑋→𝑌 be a mapping satisfying 𝑓(0)=0 for which there exists a function 𝜑∶𝑋^2𝑛→[0,∞) satisfying (2.9) such that Φ(𝑥₁,…,𝑥_2𝑛)∶=^∞_𝑗=04^𝑗𝑥𝜑(¹2^𝑗𝑥,…,^2𝑛2^𝑗)<∞,(2.32) for all 𝑥₁,…,𝑥_2𝑛∈𝑋. Then there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 satisfying (2.1) such that 1‖𝑓(𝑥)+𝑓(−𝑥)−𝑄(𝑥)‖≤𝑛Φ(𝑥,…,𝑥_𝑛times,0,…,0_𝑛times1)+𝑛Φ(−𝑥,…,−𝑥_𝑛times,0,…,0_𝑛times),(2.33) for all 𝑥∈𝑋.

Note that ^∞_𝑗=02^𝑗𝑥𝜑(¹2^𝑗𝑥,…,^2𝑛2^𝑗)≤^∞_𝑗=04^𝑗𝑥𝜑(¹2^𝑗𝑥,…,^2𝑛2^𝑗).(2.34) Combining Theorem 2.2 and Remark 2.8, we obtain the following result.Theorem 2.9. Let 𝑓∶𝑋→𝑌 be a mapping satisfying 𝑓(0)=0 for which there exists a function 𝜑∶𝑋^2𝑛→[0,∞) satisfying (2.9) and (2.32). Then there exist a unique Cauchy additive mapping 𝐴∶𝑋→𝑌 satisfying (2.1) and a unique quadratic mapping 𝑄∶𝑋→𝑌 satisfying (2.1) such that 1‖2𝑓(𝑥)−𝐴(𝑥)−𝑄(𝑥)‖≤𝑛𝜑(𝑥,…,𝑥_𝑛times,0,…,0_𝑛times1)+𝑛𝜑(−𝑥,…,−𝑥_𝑛times,0,…,0_𝑛times)+1𝑛Φ(𝑥,…,𝑥_𝑛times,0,…,0_𝑛times1)+𝑛Φ(−𝑥,…,−𝑥_𝑛times,0,…,0_𝑛times),(2.35) for all 𝑥∈𝑋, where 𝜑 and Φ are defined in (2.8) and (2.32), respectively.Corollary 2.10. Let 𝑝>2 and 𝜃 be positive real numbers, and let 𝑓∶𝑋→𝑌 be a mapping satisfying (2.19). Then there exist a unique Cauchy additive mapping 𝐴∶𝑋→𝑌 satisfying (2.1) and a unique quadratic mapping 𝑄∶𝑋→𝑌 satisfying (2.1) such that 2‖2𝑓(𝑥)−𝐴(𝑥)−𝑄(𝑥)‖≤(^𝑝+12^𝑝+2−22^𝑝−^𝑝+14)𝜃‖𝑥‖^𝑝,(2.36) for all 𝑥∈𝑋.Proof. Define 𝜑(𝑥₁,…,𝑥_2𝑛)=𝜃∑^2𝑛_𝑗=1‖𝑥_𝑗‖^𝑝, and apply Theorem 2.9 to get the desired result.The following was proved in [28].Remark 2.11 (see [28]). Let 𝑓∶𝑋→𝑌 be a mapping satisfying 𝑓(0)=0 for which there exists a function 𝜑∶𝑋^2𝑛→[0,∞) satisfying (2.9) such that Φ(𝑥₁,…,𝑥_2𝑛)∶=^∞_𝑗=14^−𝑗𝜑(2^𝑗𝑥₁,…,2^𝑗𝑥_2𝑛)<∞,(2.37) for all 𝑥₁,…,𝑥_2𝑛∈𝑋. Then there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 satisfying (2.1) such that 1‖𝑓(𝑥)+𝑓(−𝑥)−𝑄(𝑥)‖≤𝑛Φ(𝑥,…,𝑥_𝑛times,0,…,0_𝑛times1)+𝑛Φ(−𝑥,…,−𝑥_𝑛times,0,…,0_𝑛times),(2.38) for all 𝑥∈𝑋.

Note that ^∞_𝑗=14^−𝑗𝜑(2^𝑗𝑥₁,…,2^𝑗𝑥_2𝑛)≤^∞_𝑗=12^−𝑗𝜑(2^𝑗𝑥₁,…,2^𝑗𝑥_2𝑛).(2.39)

Combining Theorem 2.5 and Remark 2.11, we obtain the following result.Theorem 2.12. Let 𝑓∶𝑋→𝑌 be a mapping satisfying 𝑓(0)=0 for which there exists a function 𝜑∶𝑋^2𝑛→[0,∞) satisfying (2.9) and (2.23). Then there exist a unique Cauchy additive mapping 𝐴∶𝑋→𝑌 satisfying (2.1) and a unique quadratic mapping 𝑄∶𝑋→𝑌 satisfying (2.1) such that 1‖2𝑓(𝑥)−𝐴(𝑥)−𝑄(𝑥)‖≤𝑛𝜑(𝑥,…,𝑥_𝑛times,0,…,0_𝑛times1)+𝑛𝜑(−𝑥,…,−𝑥_𝑛times,0,…,0_𝑛times)+1𝑛Φ(𝑥,…,𝑥_𝑛times,0,…,0_𝑛times1)+𝑛Φ(−𝑥,…,−𝑥_𝑛times,0,…,0_𝑛times),(2.40) for all 𝑥∈𝑋, where 𝜑 and Φ are defined in (2.23) and (2.37), respectively.Corollary 2.13. Let 𝑝<1 and 𝜃 be positive real numbers, and let 𝑓∶𝑋→𝑌 be a mapping satisfying (2.19). Then there exist a unique Cauchy additive mapping 𝐴∶𝑋→𝑌 satisfying (2.1) and a unique quadratic mapping 𝑄∶𝑋→𝑌 satisfying (2.1) such that 2‖2𝑓(𝑥)−𝐴(𝑥)−𝑄(𝑥)‖≤(^𝑝+12−2^𝑝+2^𝑝+14−2^𝑝)𝜃‖𝑥‖^𝑝,(2.41) for all 𝑥∈𝑋.Proof. Define 𝜑(𝑥₁,…,𝑥_2𝑛)=𝜃∑^2𝑛_𝑗=1‖𝑥_𝑗‖^𝑝, and apply Theorem 2.12 to get the desired result.

Acknowledgment

The first author was supported by National Research Foundation of Korea (NRF-2009-0070788).