Solution and Stability of the Multiquadratic Functional Equation

Abstract

We consider the multiquadratic functional equation. We establish its general solution and provide a characterization for this functional equation. Finally, we prove the Hyers-Ulam-Rassias stability of this functional equation.

1. Introduction

In 1940, Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin, in which he discussed a number of unsolved problems. The stability of a functional equation originated from a question raised by Ulam: “when is it true that the solution of an equation differing slightly from a given one must of necessity be close to the solution of the given equation?” This question was solved by Hyers [2] in the case of the approximately additive functions between Banach spaces. In 1978, Rassias [3] provided a generalized version of Hyers’ result by allowing the Cauchy difference to be unbounded. The paper of Rassias [3] has provided a lot of influence in the development of the stability of functional equations, and this new concept is known as generalized Hyers-Ulam-Rassias stability or Hyers-Ulam-Rassias stability. Since then, the stability problems have been widely studied and extensively developed by many authors for a number of functional equations; see, for example, [4–10] and the books [11–14].

The functional equation 𝑓𝑥+𝑦+𝑓𝑥−𝑦=2𝑓(𝑥)+2𝑓𝑦 (1) is called the quadratic functional equation, and every solution of the quadratic functional equation is said to be a quadratic function. It is well known that a quadratic function 𝑓∶𝐸₁→𝐸₂ between vector spaces can be expressed by a symmetric biadditive (i.e., additive for each fixed one variable) function 𝐵∶𝐸₁×𝐸₁→𝐸₂. On the other hand, the stability problem for the quadratic functional equation has been studied by many mathematicians under various degrees of generality imposed on the equation or on the underlying space; see, for example, [15–20] and the references therein.

In [21], Park and Bae obtained the general solution and the generalized Hyers-Ulam-Rassias stability of the biquadratic functional equation. Let 𝑋 and 𝑌 be vector spaces. Recall from [21] that a mapping 𝑓∶𝑋×𝑋→𝑌 is called biquadratic if 𝑓 satisfies the system of equations 𝑓()𝑓()𝑥+𝑦,𝑧+𝑓𝑥−𝑦,𝑧=2𝑓𝑥,𝑧+2𝑓𝑦,𝑧𝑥,𝑦+𝑧+𝑓𝑥,𝑦−𝑧=2𝑓𝑥,𝑦+2𝑓𝑥,𝑧 (2) for all 𝑥,𝑦,𝑧∈𝑋; that is, 𝑓 is quadratic for each fixed one variable.

A general version of the biquadratic functional equation is the multiquadratic functional equation. Recall from [22] that a mapping 𝑓∶𝑉^𝑛→𝑊, where 𝑉 is a commutative group, 𝑊 is a linear space, and 𝑛≥2 is an integer, is called multiquadratic if it is quadratic in each variable. On the other hand, for more details about the multiadditive (resp., the multi-Jensen mappings) (i.e., mappings satisfying Cauchy’s (resp., Jensen’s) functional equation in each variable) and the stability for them, one can see [23–28] and the references given there.

The stability of the multiquadratic functional equation was also studied by some authors. For example, Park [29] proved the stability of the multiquadratic functional equation in Banach spaces. Ciepliński [22] proved the stability of this functional equation in complete non-Archimedean spaces as well as in Banach spaces but using the fixed point method. However, to our knowledge, not many results are known about the solution of this functional equation.

In the present paper, we establish the general solution of the multiquadratic functional equation and provide a sufficient and necessary condition for a mapping to be multiquadratic. Finally, we prove its Hyers-Ulam-Rassias stability.

2. General Solution

Throughout this section, let 𝑉 and 𝑊 be vector spaces, and let 𝑛 be a positive integer. We begin with the following useful proposition.

Proposition 1 (see [11]). A function 𝑓∶𝑉→𝑊 is quadratic if and only if there exists a unique symmetric biadditive function 𝐵∶𝑉×𝑉→𝑊 such that 𝑓(𝑥)=𝐵(𝑥,𝑥) for any 𝑥∈𝑉. The biadditive function 𝐵 is given by 𝐵=1𝑥,𝑦4𝑓𝑥+𝑦−𝑓𝑥−𝑦∀𝑥,𝑦∈𝑉. (3)

In the following, we give the general solution of the multiquadratic functional equation.

Theorem 2. A mapping 𝑓∶𝑉^𝑛→𝑊 is multiquadratic if and only if there exists a multiadditive mapping 𝑀∶𝑉^2𝑛→𝑊 such that 𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=𝑀𝑥₁,𝑥₁,𝑥₂,𝑥₂,…,𝑥_𝑛,𝑥_𝑛 (4) for all 𝑥₁,…,𝑥_𝑛∈𝑉, and 𝑀 satisfies the following symmetric condition 𝑀𝑥₁₁,𝑥₁₂,…,𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑛1,𝑥_𝑛2=𝑀𝑥₁₁,𝑥₁₂,…,𝑥_𝑖2,𝑥_𝑖1,…,𝑥_𝑛1,𝑥_𝑛2 (5) for all 𝑥_𝑖𝑗∈𝑉, where 𝑖∈{1,2,…,𝑛} and 𝑗∈{1,2}. Moreover, the mapping 𝑀 is given by 𝑀𝑥₁₁,𝑥₁₂,𝑥₂₁,𝑥₂₂,…,𝑥_𝑛1,𝑥_𝑛2=14^𝑛_{𝑖1,…,𝑖𝑛∈{}1,−1}𝑖₁⋯𝑖_𝑛𝑓𝑥₁₁+𝑖₁𝑥₁₂,…,𝑥_𝑛1+𝑖_𝑛𝑥_𝑛2, (6) where 𝑥_𝑖𝑗∈𝑉, 𝑖∈{1,2,…,𝑛}, 𝑗∈{1,2}.

Proof. We prove this theorem by using induction on 𝑛. Clearly, Theorem 2 is true for 𝑛=1 thanks to Proposition 1. Now, we assume that the present theorem is true for some 𝑛≥2, and we consider the case for 𝑛+1.
We first assume that there exists a multiadditive mapping 𝑀∶𝑉^2(𝑛+1)→𝑊 such that 𝑓𝑥₁,𝑥₂,…,𝑥_𝑛,𝑥_𝑛+1=𝑀𝑥₁,𝑥₁,𝑥₂,𝑥₂,…,𝑥_𝑛,𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1 (7) for all 𝑥₁,…,𝑥_𝑛,𝑥_𝑛+1∈𝑉, and 𝑀 satisfies the following symmetric condition: 𝑀𝑥₁₁,𝑥₁₂,…,𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑛1,𝑥_𝑛2,𝑥_𝑛+1,1,𝑥_𝑛+1,2=𝑀𝑥₁₁,𝑥₁₂,…,𝑥_𝑖2,𝑥_𝑖1,…,𝑥_𝑛1,𝑥_𝑛2,𝑥_𝑛+1,1,𝑥_𝑛+1,2 (8) for all 𝑥_𝑖𝑗∈𝑉, where 𝑖∈{1,2,…,𝑛,𝑛+1} and 𝑗∈{1,2}. Then, for each 𝑖∈{1,2,…,𝑛,𝑛+1}, we have that 𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖+𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛,𝑥_𝑛+1+𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖−𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛,𝑥_𝑛+1=𝑀𝑥₁,𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖−1,𝑥_𝑖+𝑥^′_𝑖,𝑥_𝑖+𝑥^′_𝑖,𝑥_𝑖+1,𝑥_𝑖+1,…,𝑥_𝑛,𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1+𝑀𝑥₁,𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖−1,𝑥_𝑖−𝑥^′_𝑖,𝑥_𝑖−𝑥^′_𝑖,𝑥_𝑖+1,𝑥_𝑖+1,…,𝑥_𝑛,𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=2𝑀𝑥₁,𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖−1,𝑥_𝑖,𝑥_𝑖,𝑥_𝑖+1,𝑥_𝑖+1,…,𝑥_𝑛,𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1+2𝑀𝑥₁,𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖−1,𝑥^′_𝑖,𝑥^′_𝑖,𝑥_𝑖+1,𝑥_𝑖+1,…,𝑥_𝑛,𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=2𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛,𝑥_𝑛+1+2𝑓𝑥₁,…,𝑥_𝑖−1,𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛,𝑥_𝑛+1 (9)
for all 𝑥₁,𝑥₂,…,𝑥_𝑛,𝑥_𝑛+1,𝑥^′_𝑖∈𝑉. Thus, 𝑓 is multiquadratic.
Conversely, we assume that 𝑓∶𝑉^𝑛+1→𝑊 is a multiquadratic function. We need to find the desired multiadditive function 𝑀∶𝑉^2(𝑛+1)→𝑊. For this, we give the following notations.
For each fixed 𝑧∈𝑉, define the mapping 𝑔_𝑧∶𝑉^𝑛→𝑊 by 𝑔_𝑧𝑥₁,…,𝑥_𝑛∶=𝑓𝑥₁,…,𝑥_𝑛,𝑧∀𝑥₁,…,𝑥_𝑛∈𝑉. (10)
Then 𝑔_𝑧 is a multiquadratic mapping (as 𝑓∶𝑉^𝑛+1→𝑊 is multiquadratic). By induction, we let 𝑀_𝑧∶𝑉^2𝑛→𝑊 denote the corresponding multiadditive mapping for 𝑔_𝑧; that is, 𝑀_𝑧 satisfies the symmetric condition (5) and 𝑔_𝑧𝑥₁,…,𝑥_𝑛=𝑀_𝑧𝑥₁,𝑥₁,𝑥₂,𝑥₂,…,𝑥_𝑛,𝑥_𝑛 (11) for all 𝑥₁,…,𝑥_𝑛∈𝑉. Moreover, the mapping 𝑀_𝑧∶𝑉^2𝑛→𝑊 is given by 𝑀_𝑧𝑥₁₁,𝑥₁₂,𝑥₂₁,𝑥₂₂,…,𝑥_𝑛1,𝑥_𝑛2=14^𝑛_{𝑖1,…,𝑖𝑛∈{}1,−1}𝑖₁⋯𝑖_𝑛𝑔_𝑧𝑥₁₁+𝑖₁𝑥₁₂,…,𝑥_𝑛1+𝑖_𝑛𝑥_𝑛2 (12) for all 𝑥_𝑖𝑗∈𝑉, where 𝑖∈{1,2,…,𝑛} and 𝑗∈{1,2}.
On the other hand, for any fixed elements 𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2∈𝑉, define ℎ_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}∶𝑉→𝑊 by ℎ_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}(𝑥)=𝑀_𝑥𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2 (13)
for all 𝑥∈𝑉. It can be verified that ℎ_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2} is a quadratic mapping. Thus, it follows from Proposition 1 that there exists a symmetric biadditive mapping 𝑇_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}∶𝑉×𝑉→𝑊 such that ℎ_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}(𝑥)=𝑇_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}(𝑥,𝑥) (14)
for all 𝑥∈𝑉. The mapping 𝑇_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2} is given by 𝑇_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}=1𝑥,𝑦4ℎ_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}𝑥+𝑦−ℎ_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}𝑥−𝑦 (15)
for all 𝑥,𝑦∈𝑉.
Now, we define the mapping 𝑀∶𝑉^2(𝑛+1)→𝑊 by 𝑀𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2,𝑥_𝑛+1,1,𝑥_𝑛+1,21∶=4𝑀_{𝑥𝑛+1,1+𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2−𝑀_{𝑥𝑛+1,1−𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2 (16) for all 𝑥_𝑖𝑗∈𝑉, 𝑖∈{1,2,…,𝑛+1}, 𝑗∈{1,2}. In the following, we will show that 𝑀 is the desired function for 𝑓∶𝑉^𝑛+1→𝑊. First, we show that 𝑀 is multiadditive. Indeed, by the definition of 𝑀 (see (16)) and noting that for any 𝑧∈𝑉 the function 𝑀_𝑧 is multiadditive, one can obtain that for each 𝑖∈{1,2,…,𝑛}, 𝑀𝑥₁₁,𝑥₁₂,…,𝑥_𝑖1+𝑥^′_𝑖1,𝑥_𝑖2,…,𝑥_𝑛1,𝑥_𝑛2,𝑥_𝑛+1,1,𝑥_𝑛+1,2=14𝑀_{𝑥𝑛+1,1+𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥_𝑖1+𝑥^′_𝑖1,𝑥_𝑖2,…,𝑥_𝑛1,𝑥_𝑛2−𝑀_{𝑥𝑛+1,1−𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥_𝑖1+𝑥^′_𝑖1,𝑥_𝑖2,…,𝑥_𝑛1,𝑥_𝑛2=14𝑀_{𝑥𝑛+1,1+𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑛1,𝑥_𝑛2+𝑀_{𝑥𝑛+1,1+𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥^′_𝑖1,𝑥_𝑖2,…,𝑥_𝑛1,𝑥_𝑛2−𝑀_{𝑥𝑛+1,1−𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑛1,𝑥_𝑛2−𝑀_{𝑥𝑛+1,1−𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥^′_𝑖1,𝑥_𝑖2,…,𝑥_𝑛1,𝑥_𝑛2=14𝑀_{𝑥𝑛+1,1+𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑛1,𝑥_𝑛2−𝑀_{𝑥𝑛+1,1−𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑛1,𝑥_𝑛2+14𝑀_{𝑥𝑛+1,1+𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥^′_𝑖1,𝑥_𝑖2,…,𝑥_𝑛1,𝑥_𝑛2−𝑀_{𝑥𝑛+1,1−𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥^′_𝑖1,𝑥_𝑖2,…,𝑥_𝑛1,𝑥_𝑛2=𝑀𝑥₁₁,𝑥₁₂,…,𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑛1,𝑥_𝑛2,𝑥_𝑛+1,1,𝑥_𝑛+1,2+𝑀𝑥₁₁,𝑥₁₂,…,𝑥^′_𝑖1,𝑥_𝑖2,…,𝑥_𝑛1,𝑥_𝑛2,𝑥_𝑛+1,1,𝑥_𝑛+1,2 (17) for all 𝑥^′_𝑖1,𝑥₁₁,𝑥₁₂,…,𝑥_𝑛+1,1,𝑥_𝑛+1,2∈𝑉. Moreover, by the definition of 𝑀 in (16) and the notations we gave in (13) and (15), we have that 𝑀𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2,𝑥_𝑛+1,1+𝑥^′_𝑛+1,1,𝑥_𝑛+1,2=14𝑀_{𝑥𝑛+1,1+𝑥′𝑛+1,1+𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2−𝑀_{𝑥𝑛+1,1+𝑥′𝑛+1,1−𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2=14ℎ_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}𝑥_𝑛+1,1+𝑥^′_𝑛+1,1+𝑥_𝑛+1,2−ℎ_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}𝑥_𝑛+1,1+𝑥^′_𝑛+1,1−𝑥_𝑛+1,2=𝑇_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}𝑥_𝑛+1,1+𝑥^′_𝑛+1,1,𝑥_𝑛+1,2=𝑇_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}𝑥_𝑛+1,1,𝑥_𝑛+1,2+𝑇_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}𝑥^′_𝑛+1,1,𝑥_𝑛+1,2=14ℎ_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}𝑥_𝑛+1,1+𝑥_𝑛+1,2−ℎ_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}𝑥_𝑛+1,1−𝑥_𝑛+1,2+14ℎ_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}𝑥^′_𝑛+1,1+𝑥_𝑛+1,2−ℎ_{𝑥11,𝑥12,…,𝑥𝑛1,𝑥𝑛2}𝑥^′_𝑛+1,1−𝑥_𝑛+1,2=14𝑀_{𝑥𝑛+1,1+𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2−𝑀_{𝑥𝑛+1,1−𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2+14𝑀_{𝑥′𝑛+1,1+𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2−𝑀_{𝑥′𝑛+1,1−𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2=𝑀𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2,𝑥_𝑛+1,1,𝑥_𝑛+1,2+𝑀𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2,𝑥^′_𝑛+1,1,𝑥_𝑛+1,2 (18) for all 𝑥^′_𝑛+1,1,𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2,𝑥_𝑛+1,1,𝑥_𝑛+1,2∈𝑉. Similarly, we can see that 𝑀 is additive in the other variables. Thus, we have shown that 𝑀 is multiadditive.
Furthermore, since 𝑓 is multiquadratic, we obtain that 𝑓𝑥₁,𝑥₂,…,𝑥_𝑛𝑓𝑥,0=0,₁,…,𝑥_𝑛,2𝑥_𝑛+1=4𝑓𝑥₁,…,𝑥_𝑛,𝑥_𝑛+1 (19) for all 𝑥₁,…,𝑥_𝑛,𝑥_𝑛+1∈𝑉. Thus, by the definition of 𝑀 in (16) and the notations we gave in (10) and (11), one has 𝑓𝑥₁,𝑥₂,…,𝑥_𝑛,𝑥_𝑛+1=14𝑓𝑥₁,𝑥₂,…,𝑥_𝑛,2𝑥_𝑛+1−𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=1,04𝑔_2𝑥𝑛+1𝑥₁,𝑥₂,…,𝑥_𝑛−𝑔₀𝑥₁,𝑥₂,…,𝑥_𝑛=14𝑀_2𝑥𝑛+1𝑥₁,𝑥₁,𝑥₂,𝑥₂,…,𝑥_𝑛,𝑥_𝑛−𝑀₀𝑥₁,𝑥₁,𝑥₂,𝑥₂,…,𝑥_𝑛,𝑥_𝑛=𝑀𝑥₁,𝑥₁,𝑥₂,𝑥₂,…,𝑥_𝑛,𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1 (20) for all 𝑥₁,…,𝑥_𝑛,𝑥_𝑛+1∈𝑉.
Now, we verify the expression of the mapping 𝑀. By the definition of 𝑀 again and the notations we gave in (10) and (12), also noting that 𝑓 is multiquadratic, one can obtain that 𝑀𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2,𝑥_𝑛+1,1,𝑥_𝑛+1,2=14𝑀_{𝑥𝑛+1,1+𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2−𝑀_{𝑥𝑛+1,1−𝑥𝑛+1,2}𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2=1414^𝑛_{𝑖1,…,𝑖𝑛∈{}1,−1}𝑖₁⋯𝑖_𝑛𝑔_{𝑥𝑛+1,1+𝑥𝑛+1,2}𝑥₁₁+𝑖₁𝑥₁₂𝑥,…,_𝑛1+𝑖_𝑛𝑥_𝑛2−14^𝑛_{𝑖1,…,𝑖𝑛∈{}1,−1}𝑖₁⋯𝑖_𝑛𝑔_{𝑥𝑛+1,1−𝑥𝑛+1,2}𝑥₁₁+𝑖₁𝑥₁₂𝑥,…,_𝑛1+𝑖_𝑛𝑥_𝑛2=14^𝑛+1_{𝑖1,…,𝑖𝑛∈{}1,−1}𝑖₁⋯𝑖_𝑛×𝑓𝑥₁₁+𝑖₁𝑥₁₂,…,𝑥_𝑛1+𝑖_𝑛𝑥_𝑛2,𝑥_𝑛+1,1+𝑥_𝑛+1,2−_{𝑖1,…,𝑖𝑛∈{}1,−1}𝑖₁⋯𝑖_𝑛×𝑓𝑥₁₁+𝑖₁𝑥₁₂,…,𝑥_𝑛1+𝑖_𝑛𝑥_𝑛2,𝑥_𝑛+1,1−𝑥_𝑛+1,2=14^𝑛+1_{𝑖1,…,𝑖𝑛,𝑖𝑛+1∈{}1,−1}𝑖₁⋯𝑖_𝑛⋅𝑖_𝑛+1×𝑓𝑥₁₁+𝑖₁𝑥₁₂,…,𝑥_𝑛1+𝑖_𝑛𝑥_𝑛2,𝑥_𝑛+1,1+𝑖_𝑛+1𝑥_𝑛+1,2 (21) for all 𝑥_𝑖𝑗∈𝑉, 𝑖∈{1,2,…,𝑛+1}, 𝑗∈{1,2}.
Finally, we check the symmetric property of 𝑀. Fix any 𝑥_𝑖𝑗∈𝑉, where 𝑖∈{1,2,…,𝑛+1} and 𝑗∈{1,2}. Since 𝑓 is multiquadratic, it follows that 𝑓 is an even mapping in each variable. Then by (21), it is easy to verify that 𝑀𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2,𝑥_𝑛+1,1,𝑥_𝑛+1,2=𝑀𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2,𝑥_𝑛+1,2,𝑥_𝑛+1,1. (22) Moreover, due to the symmetric property of 𝑀_{𝑥𝑛+1,1+𝑥𝑛+1,2} and 𝑀_{𝑥𝑛+1,1−𝑥𝑛+1,2} and from the definition of 𝑀 (see (16)) we can get 𝑀𝑥₁₁,𝑥₁₂,…,𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑛1,𝑥_𝑛2,𝑥_𝑛+1,1,𝑥_𝑛+1,2=𝑀𝑥₁₁,𝑥₁₂,…,𝑥_𝑖2,𝑥_𝑖1,…,𝑥_𝑛1,𝑥_𝑛2,𝑥_𝑛+1,1,𝑥_𝑛+1,2 (23) for each 𝑖∈{1,2,…,𝑛}. So the desired symmetric property of 𝑀 is proved. Thus, we have shown that 𝑀∶𝑉^2(𝑛+1)→𝑊 is the desired multiadditive mapping for the multiquadratic mapping 𝑓∶𝑉^𝑛→𝑊. The proof is complete.

3. A Characterization for the Multiquadratic Functional Equation

The following theorem provides a sufficient and necessary condition for a mapping to be multiquadratic.

Theorem 3. Let 𝑉 be a commutative semigroup with the identity element 0, and let 𝑊 be a linear space. A mapping 𝑓∶𝑉^𝑛→𝑊 is multiquadratic if and only if _{𝑖1,…,𝑖𝑛∈{}1,−1}𝑓𝑥₁₁+𝑖₁𝑥₁₂,…,𝑥_𝑛1+𝑖_𝑛𝑥_𝑛2=2^𝑛_{𝑗1,…,𝑗𝑛∈{}1,2}𝑓𝑥_1𝑗1,…,𝑥_𝑛𝑗𝑛 (24) for all (𝑥₁₁,…,𝑥_𝑛1), (𝑥₁₂,…,𝑥_𝑛2)∈𝑉^𝑛.

Proof. Assume that 𝑓∶𝑉^𝑛→𝑊 satisfies (24). Putting 𝑥₁₁,…,𝑥_𝑛1=𝑥₁₂,…,𝑥_𝑛2=(0,…,0) (25) in (24) we get 2^𝑛𝑓(0,…,0)=2^2𝑛𝑓(0,…,0), and consequently, we have 𝑓(0,…,0)=0. Next, fix 𝑗∈{1,…,𝑛}, 𝑥_𝑗1∈𝑉, and put 𝑥_𝑗2=𝑥_𝑘𝑖𝑘=0, where 𝑖_𝑘∈{1,2}, for 𝑘∈{1,…,𝑛}⧵{𝑗}. Then, by (24), 2^𝑛𝑓0,…,0,𝑥_𝑗1,0,…,0=2^𝑛2^𝑛−1𝑓0,…,0,𝑥_𝑗1,0,…,0, (26) and thus 𝑓(0,…,0,𝑥_𝑗1,0,…,0)=0. Continuing in this fashion, we obtain that 𝑓(𝑥)=0 for any 𝑥∈𝑉^𝑛 with at least one component which is equal to 0.
Now, fix 𝑗∈{1,…,𝑛}, 𝑥₁₁,…,𝑥_𝑛1,𝑥_𝑗2∈𝑉 and put 𝑥_𝑘2=0 for 𝑘∈{1,…,𝑛}⧵{𝑗} in (24). Then 2^𝑛−1𝑓𝑥₁₁,…,𝑥_𝑗1+𝑥_𝑗2,…,𝑥_𝑛1+2^𝑛−1𝑓𝑥₁₁,…,𝑥_𝑗1−𝑥_𝑗2,…,𝑥_𝑛1=2^𝑛𝑓𝑥₁₁,…,𝑥_𝑗1,…,𝑥_𝑛1+𝑓𝑥₁₁,…,𝑥_𝑗2,…,𝑥_𝑛1, (27) and thus 𝑓𝑥₁₁,…,𝑥_𝑗1+𝑥_𝑗2,…,𝑥_𝑛1+𝑓𝑥₁₁,…,𝑥_𝑗1−𝑥_𝑗2,…,𝑥_𝑛1=2𝑓𝑥₁₁,…,𝑥_𝑗1,…,𝑥_𝑛1+2𝑓𝑥₁₁,…,𝑥_𝑗2,…,𝑥_𝑛1, (28) which proves that 𝑓 is multiquadratic.
Conversely, we assume that 𝑓 is multiquadratic, and we prove (24) by mathematical induction. If 𝑓∶𝑉→𝑊 is quadratic, then 𝑓(𝑥₁₁+𝑥₁₂)+𝑓(𝑥₁₁−𝑥₁₂)=2𝑓(𝑥₁₁)+2𝑓(𝑥₁₂) for all 𝑥₁₁,𝑥₁₂∈𝑉. So, (24) holds for 𝑛=1. It is easy to verify that (24) holds for 𝑛=2. Indeed, 𝑓𝑥₁₁+𝑥₁₂,𝑥₂₁+𝑥₂₂+𝑓𝑥₁₁+𝑥₁₂,𝑥₂₁−𝑥₂₂+𝑓𝑥₁₁−𝑥₁₂,𝑥₂₁+𝑥₂₂+𝑓𝑥₁₁−𝑥₁₂,𝑥₂₁−𝑥₂₂=2𝑓𝑥₁₁+𝑥₁₂,𝑥₂₁+2𝑓𝑥₁₁+𝑥₁₂,𝑥₂₂+2𝑓𝑥₁₁−𝑥₁₂,𝑥₂₁+2𝑓𝑥₁₁−𝑥₁₂,𝑥₂₂=4𝑓𝑥₁₁,𝑥₂₁+𝑓𝑥₁₂,𝑥₂₁+𝑓𝑥₁₁,𝑥₂₂+𝑓𝑥₁₂,𝑥₂₂ (29) for all 𝑥₁₁,𝑥₁₂,𝑥₂₁,𝑥₂₂∈𝑉. Assume that (24) holds for some positive integer 𝑛>2. Then, _{𝑖1,…,𝑖𝑛+1∈{}1,−1}𝑓𝑥₁₁+𝑖₁𝑥₁₂,…,𝑥_𝑛1+𝑖_𝑛𝑥_𝑛2,𝑥_𝑛+1,1+𝑖_𝑛+1𝑥_𝑛+1,2=_{𝑖1,…,𝑖𝑛∈{}1,−1}𝑓𝑥₁₁+𝑖₁𝑥₁₂,…,𝑥_𝑛1+𝑖_𝑛𝑥_𝑛2,𝑥_𝑛+1,1+𝑥_𝑛+1,2+𝑓𝑥₁₁+𝑖₁𝑥₁₂,…,𝑥_𝑛1+𝑖_𝑛𝑥_𝑛2,𝑥_𝑛+1,1−𝑥_𝑛+1,2=2^𝑛_{𝑗1,…,𝑗𝑛∈{1,2}}𝑓𝑥_1𝑗1,…,𝑥_𝑛𝑗𝑛,𝑥_𝑛+1,1+𝑥_𝑛+1,2+𝑓𝑥_1𝑗1,…,𝑥_𝑛𝑗𝑛,𝑥_𝑛+1,1−𝑥_𝑛+1,2=2^𝑛_{𝑗1,…,𝑗𝑛∈{1,2}}2𝑓𝑥_1𝑗1,…,𝑥_𝑛𝑗𝑛,𝑥_𝑛+1,1+2𝑓𝑥_1𝑗1,…,𝑥_𝑛𝑗𝑛,𝑥_𝑛+1,2=2^𝑛+1_{𝑗1,…,𝑗𝑛,𝑗𝑛+1∈{}1,2}𝑓𝑥_1𝑗1,…,𝑥_𝑛𝑗𝑛,𝑥_{𝑛+1𝑗𝑛+1}. (30) Thus, (24) holds for 𝑛+1, and this completes the proof.

4. Stability

In this section, we give two results on the stability of the multiquadratic functional equation. Throughout this section, let 𝑉 be a commutative semigroup with the identity element 0, and let 𝑊 be a Banach space.

Theorem 4. Assume that for every 𝑖∈{1,…,𝑛}, 𝜑_𝑖∶𝑉^𝑛+1→[0,∞) is a mapping such that for any (𝑥₁,…,𝑥_𝑛+1)∈𝑉^𝑛+1𝜑_𝑖𝑥₁,…,𝑥_𝑛+1∶=^∞_𝑗=014^𝑗𝜑_𝑖2^𝑗𝑥₁,𝑥₂,…,𝑥_𝑛+1+⋯+𝜑_𝑖𝑥₁,…,𝑥_𝑖−2,2^𝑗𝑥_𝑖−1,𝑥_𝑖,…,𝑥_𝑛+1+14𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,2^𝑗𝑥_𝑖,2^𝑗𝑥_𝑖+1,𝑥_𝑖+2,…,𝑥_𝑛+1+𝜑_𝑖𝑥₁,…,𝑥_𝑖,𝑥_𝑖+1,2^𝑗𝑥_𝑖+2,𝑥_𝑖+3,…,𝑥_𝑛+1+⋯+𝜑_𝑖𝑥₁,…,𝑥_𝑛,2^𝑗𝑥_𝑛+1<∞. (31) If 𝑓∶𝑉^𝑛→𝑊 is a function satisfying 𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖+𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛+𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖−𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛−2𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛−2𝑓𝑥₁,…,𝑥_𝑖−1,𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛≤𝜑_𝑖𝑥₁,…,𝑥_𝑖,𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛 (32) for all (𝑥₁,…,𝑥_𝑖,𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛)∈𝑉^𝑛+1, 𝑖∈{1,…,𝑛}, then for every 𝑖∈{1,…,𝑛} there exists a multiquadratic mapping 𝐹_𝑖∶𝑉^𝑛→𝑊 such that for any (𝑥₁,…,𝑥_𝑛)∈𝑉^𝑛 one has 𝑓𝑥₁,…,𝑥_𝑛−13𝑓𝑥₁,…,𝑥_𝑖−1,0,𝑥_𝑖+1,…,𝑥_𝑛−𝐹_𝑖𝑥₁,…,𝑥_𝑛≤𝜑_𝑖𝑥₁,…,𝑥_𝑖,𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛. (33) For every 𝑖∈{1,…,𝑛} the function 𝐹_𝑖 is given by 𝐹_𝑖𝑥₁,…,𝑥_𝑛∶=lim_𝑗→∞14^𝑗𝑓𝑥₁,…,𝑥_𝑖−1,2^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛 (34) for all (𝑥₁,…,𝑥_𝑛)∈𝑉^𝑛.

Proof. Fix 𝑥₁,…,𝑥_𝑛∈𝑉, 𝑗∈𝑁∪{0} (where 𝑁 denotes the set of the positive integers) and 𝑖∈{1,…,𝑛}. Putting 𝑥^′_𝑖∶=𝑥_𝑖 in (32), we get 𝑓𝑥₁,…,𝑥_𝑖−1,2𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛+𝑓𝑥₁,…,𝑥_𝑖−1,0,𝑥_𝑖+1,…,𝑥_𝑛−4𝑓𝑥₁,…,𝑥_𝑛≤𝜑_𝑖𝑥₁,…,𝑥_𝑖,𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛. (35) Hence 𝑓𝑥₁,…,𝑥_𝑛−14𝑓𝑥₁,…,𝑥_𝑖−1,2𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛−14𝑓𝑥₁,…,𝑥_𝑖−1,0,𝑥_𝑖+1,…,𝑥_𝑛≤14𝜑_𝑖𝑥₁,…,𝑥_𝑖,𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛. (36) Dividing both sides of the above inequality by 4^𝑗 and replacing 𝑥_𝑖 by 2^𝑗𝑥_𝑖, we obtain 14^𝑗𝑓𝑥₁,…,𝑥_𝑖−1,2^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛−14^𝑗+1𝑓𝑥₁,…,𝑥_𝑖−1,2^𝑗+1𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛−14^𝑗+1𝑓𝑥₁,…,𝑥_𝑖−1,0,𝑥_𝑖+1,…,𝑥_𝑛≤14^𝑗+1𝜑_𝑖𝑥₁,…,2^𝑗𝑥_𝑖,2^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛, (37) and consequently for any nonnegative integers 𝑙 and 𝑚 with 𝑙<𝑚, we obtain 14^𝑙𝑓𝑥₁,…,𝑥_𝑖−1,2^𝑙𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛−14^𝑚𝑓𝑥₁,…,𝑥_𝑖−1,2^𝑚𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛−^𝑚−1_𝑗=𝑙14^𝑗+1𝑓𝑥₁,…,𝑥_𝑖−1,0,𝑥_𝑖+1,…,𝑥_𝑛≤^𝑚−1_𝑗=𝑙14^𝑗+1𝜑_𝑖𝑥₁,…,2^𝑗𝑥_𝑖,2^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛. (38) Therefore, it follows from (31) that {1/4^𝑗𝑓(𝑥₁,…,𝑥_𝑖−1,2^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛)}_𝑗∈𝑁 is a Cauchy sequence. Since the space 𝑊 is complete, this sequence is convergent, and we define 𝐹_𝑖∶𝑉^𝑛→𝑊 by (34). Putting 𝑙=0, letting 𝑚→∞ in (38), and using (31), we see that (33) holds.
Finally, fix also 𝑥^′_𝑖∈𝑉, 𝑗∈𝑁, and notice that according to (32) we have 14^𝑗𝑓𝑥₁,…,𝑥_𝑖−1,2^𝑗𝑥_𝑖+𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛+14^𝑗𝑓𝑥₁,…,𝑥_𝑖−1,2^𝑗𝑥_𝑖−𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛−24^𝑗𝑓𝑥₁,…,𝑥_𝑖−1,2^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛−24^𝑗𝑓𝑥₁,…,𝑥_𝑖−1,2^𝑗𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛≤14^𝑗𝜑_𝑖𝑥₁,…,2^𝑗𝑥_𝑖,2^𝑗𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛. (39) Next, fix 𝑘∈{1,…,𝑛}⧵{𝑖}, 𝑥^′_𝑘∈𝑉, and assume that 𝑘<𝑖 (the same arguments apply to the case where 𝑘>𝑖). Then, it follows from (32) that 14^𝑗𝑓𝑥₁,…,𝑥_𝑘−1,𝑥_𝑘+𝑥^′_𝑘,𝑥_𝑘+1,…,𝑥_𝑖−1,2^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛+14^𝑗𝑓𝑥₁,…,𝑥_𝑘−1,𝑥_𝑘−𝑥^′_𝑘,𝑥_𝑘+1,…,𝑥_𝑖−1,2^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛−24^𝑗𝑓𝑥₁,…,𝑥_𝑖−1,2^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛−24^𝑗𝑓𝑥₁,…,𝑥_𝑘−1,𝑥^′_𝑘,𝑥_𝑘+1𝑥,…,_𝑖−1,2^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛≤14^𝑗𝜑_𝑘𝑥₁,…,𝑥_𝑘,𝑥^′_𝑘,𝑥_𝑘+1,…,𝑥_𝑖−1,2^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛. (40) Letting 𝑗→∞ in the above two inequalities and using (31), we see that the mapping 𝐹_𝑖 is multiquadratic.

Theorem 5. Assume that 𝜑∶𝑉^2𝑛→[0,∞) is a mapping such that 𝜑𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2∶=^∞_𝑗=014^𝑛(𝑗+1)𝜑2^𝑗𝑥₁₁,2^𝑗𝑥₁₂,…,2^𝑗𝑥_𝑛1,2^𝑗𝑥_𝑛2<∞ (41) for all (𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2)∈𝑉^2𝑛. If 𝑓∶𝑉^𝑛→𝑊 is a function satisfying _{𝑖1,…,𝑖𝑛∈{}1,−1}𝑓𝑥₁₁+𝑖₁𝑥₁₂,…,𝑥_𝑛1+𝑖_𝑛𝑥_𝑛2−2^𝑛_{𝑗1,…,𝑗𝑛∈{}1,2}𝑓𝑥_1𝑗1,…,𝑥_𝑛𝑗𝑛≤𝜑𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2 (42) for all (𝑥₁₁,𝑥₁₂,…,𝑥_𝑛1,𝑥_𝑛2)∈𝑉^2𝑛 and letting 𝑓(𝑥)=0 for any 𝑥∈𝑉^𝑛 with one component which is equal to 0, then there exists a unique multiquadratic mapping 𝐹∶𝑉^𝑛→𝑊 such that 𝑓𝑥₁₁,…,𝑥_𝑛1−𝐹𝑥₁₁,…,𝑥_𝑛1≤𝜑𝑥₁₁,𝑥₁₁,…,𝑥_𝑛1,𝑥_𝑛1 (43) for all (𝑥₁₁,…,𝑥_𝑛1)∈𝑉^𝑛. The function 𝐹 is given by 𝐹𝑥₁₁,…,𝑥_𝑛1∶=lim_𝑗→∞14^𝑛𝑗𝑓2^𝑗𝑥₁₁,…,2^𝑗𝑥_𝑛1 (44) for all (𝑥₁₁,…,𝑥_𝑛1)∈𝑉^𝑛.

Proof. Fix (𝑥₁₁,…,𝑥_𝑛1)∈𝑉^𝑛 and 𝑗∈𝑁∪{0}. Putting 𝑥_𝑖2∶=𝑥_𝑖1 for 𝑖∈{1,…,𝑛} in (42), we get 𝑓2𝑥₁₁,…,2𝑥_𝑛1−4^𝑛𝑓𝑥₁₁,…,𝑥_𝑛1≤𝜑𝑥₁₁,𝑥₁₁,…,𝑥_𝑛1,𝑥_𝑛1. (45) Dividing both sides of the above inequality by 4^𝑛(𝑗+1) and replacing 𝑥_𝑖1 by 2^𝑗𝑥_𝑖1 for 𝑖∈{1,…,𝑛}, we see that 14^𝑛(𝑗+1)𝑓2^𝑗+1𝑥₁₁,…,2^𝑗+1𝑥_𝑛1−14^𝑛𝑗𝑓2^𝑗𝑥₁₁,…,2^𝑗𝑥_𝑛1≤14^𝑛(𝑗+1)𝜑2^𝑗𝑥₁₁,2^𝑗𝑥₁₁,…,2^𝑗𝑥_𝑛1,2^𝑗𝑥_𝑛1, (46) and consequently for any nonnegative integers 𝑙 and 𝑚 with 𝑙<𝑚 we obtain 14^𝑛𝑚𝑓2^𝑚𝑥₁₁,…,2^𝑚𝑥_𝑛1−14^𝑛𝑙𝑓2^𝑙𝑥₁₁,…,2^𝑙𝑥_𝑛1≤^𝑚−1_𝑗=𝑙14^𝑛(𝑗+1)𝜑2^𝑗𝑥₁₁,2^𝑗𝑥₁₁,…,2^𝑗𝑥_𝑛1,2^𝑗𝑥_𝑛1. (47) Therefore, it follows from (41) that {1/4^𝑛𝑗𝑓(2^𝑗𝑥₁₁,…,2^𝑗𝑥_𝑛1)}_𝑗∈𝑁 is a Cauchy sequence. Since the space 𝑊 is complete, this sequence is convergent, and we define 𝐹∶𝑉^𝑛→𝑊 by (44). Putting 𝑙=0, taking 𝑚→∞ in (47), and using (41), we can see that the inequality (43) holds.
Next, fix also (𝑥₁₂,…,𝑥_𝑛2)∈𝑉^𝑛, and note that according to (42) we have 14^𝑛𝑗_{𝑖1,…,𝑖𝑛∈{}1,−1}𝑓2^𝑗𝑥₁₁+𝑖₁𝑥₁₂,…,2^𝑗𝑥_𝑛1+𝑖_𝑛𝑥_𝑛2−2^𝑛_{𝑗1,…,𝑗𝑛∈{}1,2}14^𝑛𝑗𝑓2^𝑗𝑥_1𝑗1,…,2^𝑗𝑥_𝑛𝑗𝑛≤14^𝑛𝑗𝜑2^𝑗𝑥₁₁,2^𝑗𝑥₁₂,…,2^𝑗𝑥_𝑛1,2^𝑗𝑥_𝑛2. (48) Letting 𝑗→∞ in the above inequality and using (41), we see that 𝐹 satisfies (24). By Theorem 3, we obtain that 𝐹 is multiquadratic.
Finally, assume that 𝐹^′∶𝑉^𝑛→𝑊 is another multiquadratic mapping satisfying (43). Fix 𝑘∈𝑁∪{0}. Since 𝐹 and 𝐹^′ are multiquadratic mappings, it is easy to verify that 𝐹2^𝑘𝑥₁₁,…,2^𝑘𝑥_𝑛1=4^𝑛𝑘𝐹𝑥₁₁,…,𝑥_𝑛1,𝐹^′2^𝑘𝑥₁₁,…,2^𝑘𝑥_𝑛1=4^𝑛𝑘𝐹^′𝑥₁₁,…,𝑥_𝑛1. (49) Then, using (41) and (43), we have 𝐹𝑥₁₁,…,𝑥_𝑛1−𝐹^′𝑥₁₁,…,𝑥_𝑛1=14^𝑛𝑘𝐹2^𝑘𝑥₁₁,…,2^𝑘𝑥_𝑛1−14^𝑛𝑘𝐹^′2^𝑘𝑥₁₁,…,2^𝑘𝑥_𝑛1≤14^𝑛𝑘𝐹2^𝑘𝑥₁₁,…,2^𝑘𝑥_𝑛1−14^𝑛𝑘𝑓2^𝑘𝑥₁₁,…,2^𝑘𝑥_𝑛1+14^𝑛𝑘𝑓2^𝑘𝑥₁₁,…,2^𝑘𝑥_𝑛1−14^𝑛𝑘𝐹^′2^𝑘𝑥₁₁,…,2^𝑘𝑥_𝑛1≤24^𝑛𝑘𝜑2^𝑘𝑥₁₁,2^𝑘𝑥₁₁,…,2^𝑘𝑥_𝑛1,2^𝑘𝑥_𝑛1=2^∞_𝑗=𝑘14^𝑛(𝑗+1)𝜑2^𝑗𝑥₁₁,2^𝑗𝑥₁₁,…,2^𝑗𝑥_𝑛1,2^𝑗𝑥_𝑛1; (50) hence letting 𝑘→∞ we obtain 𝐹=𝐹^′.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.