Fuzzy Stability of a Quadratic-Additive Functional Equation

Abstract

We investigate a fuzzy version of stability for the functional equation 𝑓(𝑥+𝑦+𝑧+𝑤)+2𝑓(𝑥)+2𝑓(𝑦)+2𝑓(𝑧)+2𝑓(𝑤)−𝑓(𝑥+𝑦)−𝑓(𝑥+𝑧)−𝑓(𝑥+𝑤)−𝑓(𝑦+𝑧)−𝑓(𝑦+𝑤)−𝑓(𝑧+𝑤)=0.

1. Introduction

A classical question in the theory of functional equations is “when is it true that a mapping, which approximately satisfies a functional equation, must be somehow close to an exact solution of the equation?” Such a problem, called a stability problem of the functional equation, was formulated by Ulam [1] in 1940. In the next year, Hyers [2] gave a partial solution of Ulam's problem for the case of approximate additive mappings. Subsequently, his result was generalized by Aoki [3] for additive mappings, and by Rassias [4] for linear mappings, to considering the stability problem with the unbounded Cauchy differences. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians, see [5–16].

In 1984, Katsaras [17] defined a fuzzy norm on a linear space to construct a fuzzy structure on the space. Since then, some mathematicians have introduced several types of fuzzy norm in different points of view. In particular, Bag and Samanta [18], following Cheng and Mordeson [19], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of the Kramosil and Michálek type [20]. In 2008, Mirmostafaee and Moslehian [21] introduced for the first time the notion of fuzzy Hyers-Ulam-Rassias stability. They obtained a fuzzy version of stability for the Cauchy functional equation 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦=0 (1.1) whose solution is called an additive mapping. In the same year, they [22] proved a fuzzy version of stability for the quadratic functional equation 𝑓𝑥+𝑦+𝑓𝑥−𝑦−2𝑓(𝑥)−2𝑓𝑦=0 (1.2) whose solution is called a quadratic mapping. Now we consider the quadratic-additive functional equation 𝑓(𝑥)𝑦(𝑧)(𝑤)()()()𝑥+𝑦+𝑧+𝑤+2𝑓+2𝑓+2𝑓+2𝑓−𝑓𝑥+𝑦−𝑓𝑥+𝑧−𝑓𝑥+𝑤−𝑓𝑦+𝑧−𝑓𝑦+𝑤−𝑓𝑧+𝑤=0 (1.3) whose solution is called a quadratic-additive mapping. In [23], Chang et al. obtained a stability of the quadratic-additive functional equation by taking and composing an additive mapping 𝐴 and a quadratic mapping 𝑄 to prove the existence of a quadratic-additive mapping 𝐹 which is close to the given mapping 𝑓. In their processing, 𝐴 is approximate to the odd part 𝑓((𝑥)−𝑓(−𝑥))/2 of 𝑓 and 𝑄 is close to the even part (𝑓(𝑥)+𝑓(−𝑥))/2 of it, respectively.

In this paper, we get a general stability result of the quadratic-additive functional equation in the fuzzy normed linear space. To do it, we introduce a Cauchy sequence {𝐽_𝑛𝑓(𝑥)} starting from a given mapping 𝑓, which converges to the desired mapping 𝐹 in the fuzzy sense. As we mentioned before, in previous studies of stability problem of (1.3), Chang et al. attempted to get stability theorems by handling the odd and even part of 𝑓, respectively. According to our proposal in this paper, we can take the desired approximate solution 𝐹 at once.

2. Fuzzy Stability of the Quadratic-Additive Functional Equation

We use the definition of a fuzzy normed space given in [18] to exhibit a reasonable fuzzy version of stability for the quadratic-additive functional equation in the fuzzy normed linear space.

Definition 2.1 (see [18]). Let 𝑋 be a real linear space. A mapping 𝑁∶𝑋×ℝ→[0,1] (the so-called fuzzy subset) is said to be a fuzzy norm on 𝑋 if for all 𝑥,𝑦∈𝑋 and all 𝑠,𝑡∈ℝ, (N1) 𝑁(𝑥,𝑐)=0 for 𝑐≤0, (N2) 𝑥=0 if and only if 𝑁(𝑥,𝑐)=1 for all 𝑐>0, (N3)𝑁(𝑐𝑥,𝑡)=𝑁(𝑥,𝑡/|𝑐|) if 𝑐≠0,(N4)𝑁(𝑥+𝑦,𝑠+𝑡)≥min{𝑁(𝑥,𝑠),𝑁(𝑦,𝑡)}, (N5)𝑁(𝑥,⋅) is a nondecreasing mapping on ℝ and lim_𝑡→∞𝑁(𝑥,𝑡)=1.

The pair (𝑋,𝑁) is called a fuzzy normed linear space. Let (𝑋,𝑁) be a fuzzy normed linear space. Let {𝑥_𝑛} be a sequence in 𝑋. Then {𝑥_𝑛} is said to be convergent if there exists 𝑥∈𝑋 such that lim_𝑛→∞𝑁(𝑥_𝑛−𝑥,𝑡)=1 for all 𝑡>0. In this case, 𝑥 is called the limit of the sequence {𝑥_𝑛}, and we denote it by 𝑁−lim_𝑛→∞𝑥_𝑛=𝑥. A sequence {𝑥_𝑛} in 𝑋 is called Cauchy if for each 𝜀>0 and each 𝑡>0 there exists 𝑛₀ such that for all 𝑛≥𝑛₀ and all 𝑝>0 we have 𝑁(𝑥_𝑛+𝑝−𝑥_𝑛,𝑡)>1−𝜀. It is known that every convergent sequence in a fuzzy normed space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed space is called a fuzzy Banach space.

Let (𝑋,𝑁) be a fuzzy normed space and (𝑌,𝑁^′) a fuzzy Banach space. For a given mapping 𝑓∶𝑋→𝑌, we use the abbreviation(𝑥)𝑦(𝑧)(𝑤)()()()𝐷𝑓𝑥,𝑦,𝑧,𝑤∶=𝑓𝑥+𝑦+𝑧+𝑤+2𝑓+2𝑓+2𝑓+2𝑓−𝑓𝑥+𝑦−𝑓𝑥+𝑧−𝑓𝑥+𝑤−𝑓𝑦+𝑧−𝑓𝑦+𝑤−𝑓𝑧+𝑤 (2.1) for all 𝑥,𝑦,𝑧,𝑤∈𝑋. For given 𝑞>0, the mapping 𝑓 is called a fuzzy 𝑞-almost quadratic-additive mapping if𝑁^′𝐷𝑓𝑥,𝑦,𝑧,𝑤,𝑠+𝑡+𝑢+𝑣≥min𝑁𝑥,𝑠^𝑞,𝑁𝑦,𝑡^𝑞,𝑁𝑧,𝑢^𝑞,𝑁𝑤,𝑣^𝑞 (2.2) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝑠,𝑡,𝑢,𝑣∈(0,∞). Now we get the general stability result in the fuzzy normed linear setting.

Theorem 2.2. Let 𝑞 be a positive real number with 𝑞≠1/2,1. And let 𝑓 be a fuzzy q-almost quadratic-additive mapping from a fuzzy normed space (𝑋,𝑁) into a fuzzy Banach space (𝑌,𝑁′). Then there is a unique quadratic-additive mapping 𝐹∶𝑋→𝑌 such that 𝑁^′𝐹(𝑥)−𝑓(𝑥),𝑡≥sup_𝑡′<𝑡𝑁𝑥,4−2^𝑝2−2^𝑝𝑡^′𝑞24−2^𝑝+2−2^𝑝𝑞 (2.3) for all 𝑥∈𝑋 and 𝑡>0, where 𝑝=1/𝑞.

Proof. It follows from (2.2) and (N4) that 𝑁′𝑓(0),𝑡=𝑁^′𝐷𝑓(0,0,0,0),3𝑡≥𝑁0,3𝑡4^𝑞=1 (2.4) for all 𝑡∈(0,∞). By (N2), we have 𝑓(0)=0. We will prove the theorem in three cases: 𝑞>1, 1/2<𝑞<1, and 0<𝑞<1/2. Case 1. Let 𝑞>1 and let 𝐽_𝑛𝑓∶𝑋→𝑌 be a mapping defined by 𝐽_𝑛𝑓(𝑥)=𝑓2^𝑛𝑥+𝑓−2^𝑛𝑥2⋅4^𝑛+𝑓2^𝑛𝑥−𝑓−2^𝑛𝑥2^𝑛+1 (2.5) for all 𝑥∈𝑋. Notice that 𝐽₀𝑓(𝑥)=𝑓(𝑥) and 𝐽_𝑗𝑓(𝑥)−𝐽_𝑗+1𝑓(𝑥)=𝐷𝑓2^𝑗𝑥,−2^𝑗𝑥,2^𝑗𝑥,−2^𝑗𝑥2⋅4^𝑗+1+𝐷𝑓2^𝑗𝑥,2^𝑗𝑥,2^𝑗𝑥,−2^𝑗𝑥2^𝑗+3−𝐷𝑓−2^𝑗𝑥,−2^𝑗𝑥,−2^𝑗𝑥,2^𝑗𝑥2^𝑗+3 (2.6) for all 𝑥∈𝑋 and 𝑗≥0. Together with (N3), (N4), and (2.2), this equation implies that if 𝑛+𝑚>𝑚≥0, then 𝑁^′𝐽_𝑚𝑓(𝑥)−𝐽_𝑛+𝑚𝑓(𝑥),^{𝑛+𝑚−1}_𝑗=𝑚122^𝑝4^𝑗+2^𝑝2^𝑗𝑡^𝑝≥𝑁′^{𝑛+𝑚−1}_𝑗=𝑚𝐽_𝑗𝑓(𝑥)−𝐽_𝑗+1𝑓(𝑥),^{𝑛+𝑚−1}_𝑗=𝑚1+2^𝑗+12^𝑗𝑝2⋅4^𝑗𝑡^𝑝≥min^{𝑛+𝑚−1}_𝑗=𝑚𝑁^′𝐽_𝑗𝑓(𝑥)−𝐽_𝑗+1𝑓(𝑥),1+2^𝑗+12^𝑗𝑝2⋅4^𝑗𝑡^𝑝≥min^{𝑛+𝑚−1}_𝑗=𝑚2min𝑁′𝐷𝑓^𝑗𝑥,−2^𝑗𝑥,2^𝑗𝑥,−2^𝑗𝑥2⋅4^𝑗+1,2^𝑗𝑝𝑡^𝑝2⋅4^𝑗,2𝑁′𝐷𝑓^𝑗𝑥,2^𝑗𝑥,2^𝑗𝑥,−2^𝑗𝑥2^𝑗+3,2^𝑗𝑝𝑡^𝑝2^𝑗+1,𝑁′−𝐷𝑓−2^𝑗𝑥,−2^𝑗𝑥,−2^𝑗𝑥,2^𝑗𝑥2^𝑗+3,2^𝑗𝑝𝑡^𝑝2^𝑗+1≥min^{𝑛+𝑚−1}_𝑗=𝑚𝑁2^𝑗𝑥,2^𝑗𝑡=𝑁(𝑥,𝑡) (2.7) for all 𝑥∈𝑋 and 𝑡>0. Let 𝜀>0 be given. Since lim_𝑡→∞𝑁(𝑥,𝑡)=1, there is 𝑡₀>0 such that 𝑁𝑥,𝑡₀≥1−𝜀. (2.8) We observe that for some 𝑡̃>𝑡₀, the series ∑^∞_𝑗=0(1+2^𝑗+1)2^𝑗𝑝/2⋅4^𝑗𝑡^𝑝̃ converges for 𝑝=(1/𝑞)<1. It guarantees that, for an arbitrary given 𝑐>0, there exists 𝑛₀≥0 such that ^{𝑛+𝑚−1}_𝑗=𝑚1+2^𝑗+12^𝑗𝑝2⋅4^𝑗𝑡^𝑝̃<𝑐 (2.9) for each 𝑚≥𝑛₀ and 𝑛>0. By (N5) and (2.7), we have 𝑁^′𝐽_𝑚𝑓(𝑥)−𝐽_𝑛+𝑚𝑓(𝑥),𝑐≥𝑁^′𝐽_𝑚𝑓(𝑥)−𝐽_𝑛+𝑚𝑓(𝑥),^{𝑛+𝑚−1}_𝑗=𝑚1+2^𝑗+12^𝑗𝑝2⋅4^𝑗𝑡^𝑝̃≥𝑁𝑥,𝑡̃≥𝑁𝑥,𝑡₀≥1−𝜀 (2.10) for all 𝑥∈𝑋. Hence {𝐽_𝑛𝑓(𝑥)} is a Cauchy sequence in the fuzzy Banach space (𝑌,𝑁^′), and so we can define a mapping 𝐹∶𝑋→𝑌 by 𝐹(𝑥)∶=𝑁^′−lim_𝑛→∞𝐽_𝑛𝑓(𝑥) (2.11) for all 𝑥∈𝑋. Moreover, if we put 𝑚=0 in (2.7), we have 𝑁′𝑓(𝑥)−𝐽_𝑛𝑓(𝑥)𝑡,𝑡≥𝑁𝑥,^𝑞∑^𝑛−1_𝑗=0(1/2)2^𝑝/4^𝑗+2^𝑝/2^𝑗𝑞 (2.12) for all 𝑥∈𝑋. Next we will show that 𝐹 is a desired quadratic-additive mapping. Using (N4), we have 𝑁^′𝑁𝐷𝐹𝑥,𝑦,𝑧,𝑤,𝑡≥min^′𝐹𝑥+𝑦+𝑧+𝑤−𝐽_𝑛𝑓,𝑡𝑥+𝑦+𝑧+𝑤,𝑁30_′2𝐹(𝑥)−2𝐽_𝑛𝑓(𝑥),𝑡𝑦15,𝑁′2𝐹−2𝐽_𝑛𝑓𝑦,𝑡,𝑁15_′2𝐹(𝑧)−2𝐽_𝑛𝑓(𝑧),𝑡(𝑤)15,𝑁′2𝐹−2𝐽_𝑛𝑓(𝑤),𝑡,𝑁15_′𝐹𝑥+𝑦−𝐽_𝑛𝑓,𝑡𝑥+𝑦𝐹()30,𝑁′𝑥+𝑧−𝐽_𝑛𝑓(),𝑡𝑥+𝑧,𝑁30_′𝐹(𝑥+𝑤)−𝐽_𝑛𝑓(),𝑡𝑥+𝑤𝐹30,𝑁′𝑦+𝑧−𝐽_𝑛𝑓,𝑡𝑦+𝑧,𝑁30_′𝐹𝑦+𝑤−𝐽_𝑛𝑓,𝑡𝑦+𝑤𝐹()30,𝑁′𝑧+𝑤−𝐽_𝑛𝑓(),𝑡𝑧+𝑤,30𝑁′𝐷𝐽_𝑛𝑓,𝑡𝑥,𝑦,𝑧,𝑤2 (2.13) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝑛∈ℕ. The first eleven terms on the right-hand side of (2.13) tend to 1 as 𝑛→∞ by the definition of 𝐹 and (N2), and the last term holds: 𝑁^′𝐷𝐽_𝑛𝑓,𝑡𝑥,𝑦,𝑧,𝑤22≥min𝑁′𝐷𝑓^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧,2^𝑛𝑤2⋅4^𝑛,𝑡8,𝑁′𝐷𝑓−2^𝑛𝑥,−2^𝑛𝑦,−2^𝑛𝑧,−2^𝑛𝑤2⋅4^𝑛,𝑡8,2𝑁′𝐷𝑓^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧,2^𝑛𝑤2⋅2^𝑛,𝑡8,𝑁′𝐷𝑓−2^𝑛𝑥,−2^𝑛𝑦,−2^𝑛𝑧,−2^𝑛𝑤2⋅2^𝑛,𝑡8 (2.14) for all 𝑥,𝑦,𝑧,𝑤∈𝑋. By (N3) and (2.2), we obtain 𝑁^′𝐷𝑓±2^𝑛𝑥,±2^𝑛𝑦,±2^𝑛𝑧,±2^𝑛𝑤2⋅4^𝑛,𝑡8=𝑁′𝐷𝑓±2^𝑛𝑥,±2^𝑛𝑦,±2^𝑛𝑧,±2^𝑛𝑤,4^𝑛𝑡4≥min𝑁2^𝑛4𝑥,^𝑛𝑡16^𝑞,𝑁2^𝑛4𝑦,^𝑛𝑡16^𝑞,𝑁2^𝑛4𝑧,^𝑛𝑡16^𝑞,𝑁2^𝑛4𝑤,^𝑛𝑡16^q𝑁2≥min𝑥,^{(2𝑞−1)𝑛}2^4𝑞𝑡^𝑞2,𝑁𝑦,^{(2𝑞−1)𝑛}2^4𝑞𝑡^𝑞,𝑁2𝑧,^{(2𝑞−1)𝑛}2^4𝑞𝑡^𝑞2,𝑁𝑤,^{(2𝑞−1)𝑛}2^4𝑞𝑡^𝑞,𝑁^′𝐷𝑓±2^𝑛𝑥,±2^𝑛𝑦,±2^𝑛𝑧,±2^𝑛𝑤2⋅2^𝑛,𝑡8𝑁2≥min𝑥,^{(𝑞−1)𝑛}2^4𝑞𝑡^𝑞2,𝑁𝑦,^{(𝑞−1)𝑛}2^4𝑞𝑡^𝑞,𝑁2𝑧,^{(𝑞−1)𝑛}2^4𝑞𝑡^𝑞2,𝑁𝑤,^{(𝑞−1)𝑛}2^4𝑞𝑡^𝑞 (2.15) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝑛∈ℕ. Since 𝑞>1, together with (N5), we can deduce that the last term of (2.13) also tends to 1 as 𝑛→∞. It follows from (2.13) that 𝑁^′𝐷𝐹𝑥,𝑦,𝑧,𝑤,𝑡=1 (2.16) for each 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝑡>0. By (N2), this means that 𝐷𝐹(𝑥,𝑦,𝑧,𝑤)=0 for all 𝑥,𝑦,𝑧,𝑤∈𝑋. Now we approximate the difference between 𝑓 and 𝐹 in a fuzzy sense. For an arbitrary fixed 𝑥∈𝑋 and 𝑡>0, choose 0<𝜀<1 and 0<𝑡^′<𝑡. Since 𝐹 is the limit of {𝐽_𝑛𝑓(𝑥)}, there is 𝑛∈ℕ such that 𝑁^′(𝐹(𝑥)−𝐽_𝑛𝑓(𝑥),𝑡−𝑡′)≥1−𝜀. By (2.12), we have 𝑁^′𝐹(𝑥)−𝑓(𝑥),𝑡≥min𝑁^′𝐹(𝑥)−𝐽_𝑛𝑓(𝑥),𝑡−𝑡^′,𝑁^′𝐽_𝑛𝑓(𝑥)−𝑓(𝑥),𝑡^′𝑡≥min1−𝜀,𝑁𝑥,^′𝑞∑^𝑛−1_𝑗=0(1/2)2^𝑝/4^𝑗+2^𝑝/2^𝑗𝑞≥min1−𝜀,𝑁𝑥,4−2^𝑝𝑞2−2^𝑝𝑞𝑡^′𝑞24−2^𝑝+2−2^𝑝𝑞 (2.17) since ∑^𝑛−1_𝑗=0((1/2)(2^𝑝/4)^𝑗+(2^𝑝/2)^𝑗)≤2(4−2^𝑝+2−2^𝑝)/((4−2^𝑝)(2−2^𝑝)). Because 0<𝜀<1 is arbitrary, we get inequality (2.3) in this case. Finally, to prove the uniqueness of 𝐹, let 𝐹^′∶𝑋→𝑌 be another quadratic-additive mapping satisfying (2.3). Then by (2.6), we get 𝐹(𝑥)−𝐽_𝑛𝐹(𝑥)=^𝑛−1_𝑗=0𝐽_𝑗𝐹(𝑥)−𝐽_𝑗+1𝐹(𝑥𝐹)=0,_′(𝑥)−𝐽_𝑛𝐹^′(𝑥)=^𝑛−1_𝑗=0𝐽_𝑗𝐹^′(𝑥)−𝐽_𝑗+1𝐹^′(𝑥)=0 (2.18) for all 𝑥∈𝑋 and 𝑛∈ℕ. Together with (N4) and (2.3), this implies that 𝑁^′𝐹(𝑥)−𝐹^′(𝑥),𝑡=𝑁^′𝐽_𝑛𝐹(𝑥)−𝐽_𝑛𝐹^′(𝑥)𝐽,𝑡≥min𝑁′_𝑛𝐹(𝑥)−𝐽_𝑛𝑓(𝑥),𝑡2,𝑁^′𝐽_𝑛𝑓(𝑥)−𝐽_𝑛(𝑥),𝑡𝐹′2≥min𝑁^′𝐹−𝑓2^𝑛𝑥2⋅4^𝑛,𝑡8,𝑁^′𝑓−𝐹′2^𝑛𝑥2⋅4^𝑛,𝑡8,𝑁′𝐹−𝑓−2^𝑛𝑥2⋅4^𝑛,𝑡8,𝑁′𝑓−𝐹^′−2^𝑛𝑥2⋅4^𝑛,𝑡8,2𝑁′𝐹−𝑓^𝑛𝑥2⋅2^𝑛,𝑡8,𝑁′𝑓−𝐹^′2^𝑛𝑥2⋅2^𝑛,𝑡8,𝑁′𝐹−𝑓−2^𝑛𝑥2⋅2^𝑛,𝑡8,𝑁′𝑓−𝐹′−2^𝑛𝑥2⋅2^𝑛,𝑡8≥sup_𝑡′<𝑡𝑁2𝑥,^{(𝑞−1)𝑛−2𝑞}4−2^𝑝𝑞2−2^𝑝𝑞𝑡^′𝑞43−2^𝑝𝑞 (2.19) for all 𝑥∈𝑋 and 𝑛∈ℕ. Observe that, for 𝑞=(1/𝑝)>1, the last term of the above inequality tends to 1 as 𝑛→∞ by (N5). This implies that 𝑁^′(𝐹(𝑥)−𝐹^′(𝑥),𝑡)=1, and so we get 𝐹(𝑥)=𝐹′(𝑥) (2.20) for all 𝑥∈𝑋 by (N2). Case 2. Let (1/2)<𝑞<1 and let 𝐽_𝑛𝑓∶𝑋→𝑌 be a mapping defined by 𝐽_𝑛𝑓(𝑥)=124^−𝑛𝑓2^𝑛𝑥+𝑓−2^𝑛𝑥+2^𝑛𝑓𝑥2^𝑛−𝑥−𝑓2^𝑛 (2.21) for all 𝑥∈𝑋. Then we also have 𝐽₀𝑓(𝑥)=𝑓(𝑥) and 𝐽_𝑗𝑓(𝑥)−𝐽_𝑗+1𝑓(𝑥)=𝐷𝑓2^𝑗𝑥,−2^𝑗𝑥,2^𝑗𝑥,−2^𝑗𝑥2⋅4^𝑗+1−2^𝑗−2𝑥𝐷𝑓2^𝑗+1,𝑥2^𝑗+1,𝑥2^𝑗+1𝑥,−2^𝑗+1+2^𝑗−2−𝑥𝐷𝑓2^𝑗+1𝑥,−2^𝑗+1𝑥,−2^𝑗+1,𝑥2^𝑗+1 (2.22) for all 𝑥∈𝑋 and 𝑗≥0. If 𝑛+𝑚>𝑚≥0, then we have 𝑁^′𝐽_𝑚𝑓(𝑥)−𝐽_𝑛+𝑚𝑓(𝑥),^{𝑛+𝑚−1}_𝑗=𝑚122^𝑝4^𝑗+22^𝑝𝑗+1𝑡^𝑝≥min^{𝑛+𝑚−1}_𝑗=𝑚min𝑁^′𝐷𝑓2^𝑗𝑥,−2^𝑗𝑥,2^𝑗𝑥,−2^𝑗𝑥2⋅4^𝑗+1,2^𝑗𝑝𝑡^𝑝2⋅4^𝑗,𝑁^′−2^𝑗−2𝑥𝐷𝑓2^𝑗+1,𝑥2^𝑗+1,𝑥2^𝑗+1𝑥,−2^𝑗+1,2^𝑗𝑡^𝑝2^(𝑗+1)𝑝,𝑁^′2^𝑗−2−𝑥𝐷𝑓2^𝑗+1𝑥,−2^𝑗+1𝑥,−2^𝑗+1,𝑥2^𝑗+1,2^𝑗𝑡^𝑝2^(𝑗+1)𝑝≥min^{𝑛+𝑚−1}_𝑗=𝑚𝑁2^𝑗𝑥,2^𝑗𝑡𝑥,𝑁2^𝑗+1,𝑡2^𝑗+1=𝑁(𝑥,𝑡) (2.23) for all 𝑥∈𝑋 and 𝑡>0. In a similar argument following (2.7) of the previous case, we can define the limit 𝐹(𝑥)∶=𝑁^′−lim_𝑛→∞𝐽_𝑛𝑓(𝑥) of the Cauchy sequence {𝐽_𝑛𝑓(𝑥)} in the Banach fuzzy space 𝑌. Moreover, putting 𝑚=0 in the above inequality, we have 𝑁^′𝑓(𝑥)−𝐽_𝑛𝑓(𝑥)𝑡,𝑡≥𝑁𝑥,^𝑞∑^𝑛−1_𝑗=0(1/2)2^𝑝/4^𝑗+2/2^{𝑝𝑗+1𝑞} (2.24) for each 𝑥∈𝑋 and 𝑡>0. To prove that 𝐹 is a quadratic-additive mapping, we have enough to show that the last term of (2.13) in Case 1 tends to 1 as 𝑛→∞. By (N3) and (2.2), we get 𝑁^′𝐷𝐽_𝑛𝑓,𝑡𝑥,𝑦,𝑧,𝑤2≥min𝑁^′𝐷𝑓2^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧,2^𝑛𝑤2⋅4^𝑛,𝑡8,𝑁^′𝐷𝑓−2^𝑛𝑥,−2^𝑛𝑦,−2^𝑛𝑧,−2^𝑛𝑤2⋅4^𝑛,𝑡8,𝑁^′2^𝑛−1𝑥𝐷𝑓2^𝑛,𝑦2^𝑛,𝑧2^𝑛,𝑤2^𝑛,𝑡8,𝑁^′2^𝑛−1𝐷𝑓−𝑥2^𝑛,−𝑦2^𝑛,−𝑧2^𝑛,−𝑤2^𝑛,𝑡8≥min𝑁𝑥,2^{(2𝑞−1)𝑛−4𝑞}𝑡^𝑞,𝑁𝑦,2^{(2𝑞−1)𝑛−4𝑞}𝑡^𝑞,𝑁𝑧,2^{(2𝑞−1)𝑛−4𝑞}𝑡^𝑞,𝑁𝑤,2^{(2𝑞−1)𝑛−4𝑞}𝑡^𝑞,𝑁𝑥,2^{(1−𝑞)𝑛−4𝑞}𝑡^𝑞,𝑁𝑦,2^{(1−𝑞)𝑛−4𝑞}𝑡^𝑞,𝑁𝑧,2^{(1−𝑞)𝑛−4𝑞}𝑡^𝑞,𝑁𝑤,2^{(1−𝑞)𝑛−4𝑞}𝑡^𝑞 (2.25) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝑡>0. Observe that all the terms on the right-hand side of the above inequality tend to 1 as 𝑛→∞, since 1/2<𝑞<1. Hence, with arguments similar to those of (2.13)–(2.15), we can say that 𝐷𝐹(𝑥,𝑦,𝑧,𝑤)=0 for all 𝑥,𝑦,𝑧,𝑤∈𝑋. Recall that, in Case 1, inequality (2.3) follows from (2.12). By the same reasoning, we get (2.3) from (2.24) in this case.
Now to prove the uniqueness of 𝐹, let 𝐹^′ be another quadratic-additive mapping satisfying (2.3). Then, together with (N4), (2.3), and (2.18), we have 𝑁^′𝐹(𝑥)−𝐹^′(𝑥),𝑡=𝑁^′𝐽_𝑛𝐹(𝑥)−𝐽_𝑛𝐹^′(𝑥)𝑁,𝑡≥min^′𝐽_𝑛𝐹(𝑥)−𝐽_𝑛𝑓(𝑥),𝑡2,𝑁^′𝐽_𝑛𝑓(𝑥)−𝐽_𝑛𝐹^′(𝑥),𝑡22≥min𝑁′𝐹−𝑓^𝑛𝑥2⋅4^𝑛,𝑡82,𝑁′𝑓−𝐹′^𝑛𝑥2⋅4^𝑛,𝑡8,𝑁′𝐹−𝑓−2^𝑛𝑥2⋅4^𝑛,𝑡8,𝑁′𝑓−𝐹′−2^𝑛𝑥2⋅4^𝑛,𝑡8,𝑁′2^𝑛−1𝑥𝐹−𝑓2^𝑛,𝑡8,𝑁′2^𝑛−1𝑓−𝐹^′𝑥2^𝑛,𝑡8,𝑁′2^𝑛−1𝐹−𝑓−𝑥2^𝑛,𝑡8,𝑁^′2^𝑛−1𝑓−𝐹^′−𝑥2^𝑛,𝑡8≥minsup_𝑡′<𝑡𝑁𝑥,2^{(2𝑞−1)𝑛−4𝑞}4−2^𝑝𝑞2^𝑝−2^𝑞𝑡^′𝑞,sup_𝑡′<𝑡𝑁𝑥,2^{(1−𝑞)𝑛−4𝑞}4−2^𝑝𝑞2^𝑝−2^𝑞𝑡^′𝑞 (2.26) for all 𝑥∈𝑋 and 𝑛∈ℕ. Since lim_𝑛→∞2^{(2𝑞−1)𝑛−4𝑞}=lim_𝑛→∞2^{(1−𝑞)𝑛−4𝑞}=∞ in this case, both terms on the right-hand side of the above inequality tend to 1 as 𝑛→∞ by (N5). This implies that 𝑁′(𝐹(𝑥)−𝐹′(𝑥),𝑡)=1 and so 𝐹(𝑥)=𝐹′(𝑥) for all 𝑥∈𝑋 by (N2).Case 3. Finally, we take 0<𝑞<(1/2) and define 𝐽_𝑛𝑓∶𝑋→𝑌 by 𝐽_𝑛𝑓(𝑥)=124^𝑛𝑓2^−𝑛𝑥+𝑓−2^−𝑛𝑥+2^𝑛𝑓𝑥2^𝑛−𝑥−𝑓2^𝑛 (2.27) for all 𝑥∈𝑋. Then we have 𝐽₀𝑓(𝑥)=𝑓(𝑥) and 𝐽_𝑗𝑓(𝑥)−𝐽_𝑗+1𝑓(𝑥)4=−^𝑗2𝑥𝐷𝑓2^𝑗+1𝑥,−2^𝑗+1,𝑥2^𝑗+1𝑥,−2^𝑗+1−2^𝑗−2𝑥𝐷𝑓2^𝑗+1,𝑥2^𝑗+1,𝑥2^𝑗+1𝑥,−2^𝑗+1+2^𝑗−2−𝑥𝐷𝑓2^𝑗+1𝑥,−2^𝑗+1𝑥,−2^𝑗+1,𝑥2^𝑗+1 (2.28) which implies that if 𝑛+𝑚>𝑚≥0 then 𝑁^′𝐽_𝑚𝑓(𝑥)−𝐽_𝑛+𝑚𝑓(𝑥),22^{𝑝𝑛+𝑚−1}_𝑗=𝑚42^𝑝𝑗+22^𝑝𝑗𝑡^𝑝≥min^{𝑛+𝑚−1}_𝑗=𝑚min𝑁^′−4^𝑗2𝑥𝐷𝑓2^𝑗+1𝑥,−2^𝑗+1,𝑥2^𝑗+1𝑥,−2^𝑗+1,2⋅4^𝑗𝑡^𝑝2^(𝑗+1)𝑝,𝑁^′−2^𝑗−2𝑥𝐷𝑓2^𝑗+1,𝑥2^𝑗+1,𝑥2^𝑗+1𝑥,−2^𝑗+1,2^𝑗𝑡^𝑝2^(𝑗+1)𝑝,𝑁^′2^𝑗−2−𝑥𝐷𝑓2^𝑗+1𝑥,−2^𝑗+1𝑥,−2^𝑗+1,𝑥2^𝑗+1,2^𝑗𝑡^𝑝2^(𝑗+1)𝑝≥min^𝑛+m−1_𝑗=𝑚𝑁𝑥2^𝑗+1,𝑡2^𝑗+1=𝑁(𝑥,𝑡) (2.29) for all 𝑥∈𝑋 and 𝑡>0. Similar to the previous cases, it leads us to define the mapping 𝐹∶𝑋→𝑌 by 𝐹(𝑥)∶=𝑁^′−lim_𝑛→∞𝐽_𝑛𝑓(𝑥). Putting 𝑚=0 in the above inequality, we have 𝑁′𝑓(𝑥)−𝐽_𝑛𝑓(𝑥)𝑡,𝑡≥𝑁𝑥,^𝑞∑^𝑛−1_𝑗=0(1/2)4/2^𝑝𝑗+1+2/2^{𝑝𝑗+1𝑞} (2.30) for all 𝑥∈𝑋 and 𝑡>0. Notice that 𝑁′𝐷𝐽_𝑛𝑓,𝑡𝑥,𝑦,𝑧,𝑤24≥min𝑁′^𝑛2𝑥𝐷𝑓2^𝑛,𝑦2^𝑛,𝑧2^𝑛,𝑤2^𝑛,𝑡84,𝑁′^𝑛2𝐷𝑓−𝑥2^𝑛,−𝑦2^𝑛,−𝑧2^𝑛,−𝑤2^𝑛,𝑡8,𝑁^′2^𝑛−1𝑥𝐷𝑓2^𝑛,𝑦2^𝑛,𝑧2^𝑛,𝑤2^𝑛,𝑡8,𝑁^′2^𝑛−1𝐷𝑓−𝑥2^𝑛,−𝑦2^𝑛,−𝑧2^𝑛,−𝑤2^𝑛,𝑡8≥min𝑁𝑥,2^{(1−2𝑞)𝑛−4𝑞}𝑡^𝑞,𝑁𝑦,2^{(1−2𝑞)𝑛−4𝑞}𝑡^𝑞,𝑁𝑧,2^{(1−2𝑞)𝑛−4𝑞}𝑡^𝑞,𝑁𝑤,2^{(1−2𝑞)𝑛−4𝑞}𝑡^𝑞,𝑁𝑥,2^{(1−𝑞)𝑛−4𝑞}𝑡^𝑞,𝑁𝑦,2^{(1−𝑞)𝑛−4𝑞}𝑡^𝑞,𝑁𝑧,2^{(1−𝑞)𝑛−4𝑞}𝑡^𝑞,𝑁𝑤,2^{(1−𝑞)𝑛−4𝑞}𝑡^𝑞 (2.31) for each 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝑡>0. Since 0<𝑞<(1/2), all terms on the right-hand side tend to 1 as 𝑛→∞, which implies that the last term of (2.13) tends to 1 as 𝑛→∞. Therefore, we can say that 𝐷𝐹≡0. Moreover, using arguments similar to those of (2.13)–(2.15) in Case 1, we get inequality (2.3) from (2.30) in this case. To prove the uniqueness of 𝐹, let 𝐹^′∶𝑋→𝑌 be another quadratic-additive mapping satisfying (2.3). Then by (2.18), we get 𝑁^′𝐹(𝑥)−𝐹^′(𝑥)𝑁,𝑡≥min^′𝐽_𝑛𝐹(𝑥)−𝐽_𝑛𝑓(𝑥),𝑡2,𝑁^′𝐽_𝑛𝑓(𝑥)−𝐽_𝑛𝐹^′(𝑥),𝑡2≥min𝑁^′4^𝑛2𝑥𝐹−𝑓2^𝑛,𝑡84,𝑁′^𝑛2𝑓−𝐹^′𝑥2^𝑛,𝑡8,𝑁^′4^𝑛2−𝑥𝐹−𝑓2^𝑛,𝑡8,𝑁^′4^𝑛2−𝑥𝑓−𝐹′2^𝑛,𝑡8,𝑁^′2^𝑛−1𝑥𝐹−𝑓2^𝑛,𝑡8,𝑁′2^𝑛−1𝑥𝑓−𝐹′2^𝑛,𝑡8,𝑁^′2^𝑛−1𝐹−𝑓−𝑥2^𝑛,𝑡8,𝑁^′2^𝑛−1𝑓−𝐹^′−𝑥2^𝑛,𝑡8≥sup_𝑡′<𝑡𝑁2𝑥,^{(1−2𝑞)𝑛−4𝑞}2^𝑝−42^𝑝−2𝑡^′𝑞2^𝑝−3^𝑞 (2.32) for all 𝑥∈𝑋 and 𝑛∈ℕ. Observe that, for 0<𝑞<(1/2), the last term tends to 1 as 𝑛→∞ by (N5). This implies that 𝑁^′(𝐹(𝑥)−𝐹^′(𝑥),𝑡)=1 and 𝐹(𝑥)=𝐹^′(𝑥) for all 𝑥∈𝑋 by (N2).

Remark 2.3. Consider a mapping 𝑓∶𝑋→𝑌 satisfying (2.2) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and a real number 𝑞<0. Take any 𝑡>0. If we choose a real number 𝑠 with 0<4𝑠<𝑡, then 𝑁^′𝐷𝑓𝑥,𝑦,𝑧,𝑤,𝑡≥𝑁^′𝑁𝐷𝑓𝑥,𝑦,𝑧,𝑤,4𝑠≥min𝑥,𝑠^𝑞,𝑁𝑦,𝑠^𝑞,𝑁𝑧,𝑠^𝑞,𝑁𝑤,𝑠^𝑞 (2.33) for all 𝑥,𝑦,𝑧,𝑤∈𝑋. Since 𝑞<0, we have lim_𝑠→0⁺𝑠^𝑞=∞. This implies that lim_𝑠→0+𝑁𝑥,𝑠^𝑞=lim_𝑠→0+𝑁𝑦,𝑠^𝑞=lim_𝑧→0+𝑁𝑥,𝑠^𝑞=lim_𝑤→0+𝑁𝑤,𝑠^𝑞=1, (2.34) and so 𝑁′𝐷𝑓𝑥,𝑦,𝑧,𝑤,𝑡=1 (2.35) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝑡>0. By (N2), we are allowed to get 𝐷𝑓(𝑥,𝑦,𝑧,𝑤)=0 for all 𝑥,𝑦,𝑧,𝑤∈𝑋. In other words, 𝑓 is itself a quadratic-additive mapping if 𝑓 is a fuzzy 𝑞-almost quadratic-additive mapping for the case 𝑞<0.

Corollary 2.4. Let 𝑓 be an even mapping satisfying all of the conditions of Theorem 2.2. Then there is a unique quadratic mapping 𝐹∶𝑋→𝑌 such that 𝑁^′𝐹(𝑥)−𝑓(𝑥),𝑡≥sup_𝑡′<𝑡𝑁𝑥,4−2^𝑝𝑡^′2^𝑞 (2.36) for all 𝑥∈𝑋 and 𝑡>0, where 𝑝=1/𝑞.

Proof. Let 𝐽_𝑛𝑓 be defined as in Theorem 2.2. Since 𝑓 is an even mapping, we obtain 𝐽_𝑛𝑓(𝑥)=𝑓2^𝑛𝑥+𝑓−2^𝑛𝑥2⋅4^𝑛1if𝑞>2,124^𝑛𝑓2^−𝑛𝑥+𝑓−2^−𝑛𝑥1if0<𝑞<2 (2.37) for all 𝑥∈𝑋. Notice that 𝐽₀f(𝑥)=𝑓(𝑥) and 𝐽_𝑗𝑓(𝑥)−𝐽_𝑗+1𝑓(𝑥)=𝐷𝑓2^𝑗𝑥,−2^𝑗𝑥,2^𝑗𝑥,−2^𝑗𝑥2⋅4^𝑗+11if𝑞>2,−4^𝑗2𝑥𝐷𝑓2^𝑗+1𝑥,−2^𝑗+1,𝑥2^𝑗+1𝑥,−2^𝑗+11if0<𝑞<2 (2.38) for all 𝑥∈𝑋 and 𝑗∈ℕ∪{0}. From these, using the similar method in Theorem 2.2, we obtain the quadratic-additive mapping 𝐹 satisfying (2.36). Notice that 𝐹(𝑥)∶=𝑁′−lim_𝑛→∞𝐽_𝑛𝑓(𝑥) for all 𝑥∈𝑋, 𝐹 is even, 𝐹(2𝑥)=4𝐹(𝑥), and 𝐷𝐹(𝑥,𝑦,𝑧,𝑤)=0 for all 𝑥,𝑦,𝑧,𝑤∈𝑋. Hence, we get 𝐹𝑥+𝑦+𝐹𝑥−𝑦−2𝐹(𝑥)−2𝐹𝑦=−𝐷𝐹𝑥,𝑦,𝑥,−𝑥=0 (2.39) for all 𝑥,𝑦∈𝑋. This means that 𝐹 is a quadratic mapping.

Corollary 2.5. Let 𝑓 be an odd mapping satisfying all of the conditions of Theorem 2.2. Then there is a unique additive mapping 𝐹∶𝑋→𝑌 such that 𝑁^′𝐹(𝑥)−𝑓(𝑥),𝑡≥sup_𝑡′<𝑡𝑁𝑥,2−2^𝑝𝑡^′2^𝑞 (2.40) for all 𝑥∈𝑋 and 𝑡>0, where 𝑝=1/𝑞.

Proof. Let 𝐽_𝑛𝑓 be defined as in Theorem 2.2. Since 𝑓 is an odd mapping, we obtain 𝐽_𝑛𝑓(𝑥)=𝑓2^𝑛𝑥+𝑓−2^𝑛𝑥2^𝑛+12if𝑞>1,_𝑛−1𝑓2^−𝑛𝑥+𝑓−2^−𝑛𝑥if0<𝑞<1 (2.41) for all 𝑥∈𝑋. Notice that 𝐽₀𝑓(𝑥)=𝑓(𝑥) and 𝐽_𝑗𝑓(𝑥)−𝐽_𝑗+1𝑓(𝑥)=𝐷𝑓2^𝑗𝑥,2^𝑗𝑥,2^𝑗𝑥,−2^𝑗𝑥2^𝑗+3−𝐷𝑓−2^𝑗𝑥,−2^𝑗𝑥,−2^𝑗𝑥,2^𝑗𝑥2^𝑗+32if𝑞>1,_𝑗−2−𝑥𝐷𝑓2^𝑗+1𝑥,−2^𝑗+1𝑥,−2^𝑗+1,𝑥2^𝑗+1𝑥−𝐷𝑓2^𝑗+1,𝑥2^𝑗+1,𝑥2^𝑗+1𝑥,−2^𝑗+1if0<𝑞<1 (2.42) for all 𝑥∈𝑋 and 𝑗∈ℕ∪{0}. From these, using the similar method in Theorem 2.2, we obtain the quadratic-additive mapping 𝐹 satisfying (2.40). Notice that 𝐹(𝑥)∶=𝑁′−lim_𝑛→∞𝐽_𝑛𝑓(𝑥) for all 𝑥∈𝑋,𝐹 is odd, 𝐹(2𝑥)=2𝐹(𝑥), and 𝐷𝐹(𝑥,𝑦,𝑧,𝑤)=0 for all 𝑥,𝑦,𝑧,𝑤∈𝑋. Hence, we get 𝐹𝑥+𝑦−𝐹(𝑥)−𝐹𝑦=−𝐷𝐹𝑥−𝑦2,−𝑥−𝑦2,𝑥−𝑦2,−𝑥+𝑦2=0 (2.43) for all 𝑥,𝑦∈𝑋. This means that 𝐹 is an additive mapping.

We can use Theorem 2.2 to get a classical result in the framework of normed spaces. Let (𝑋,‖⋅‖) be a normed linear space. Then we can define a fuzzy norm 𝑁_𝑋 on 𝑋 by 𝑁_𝑋()=‖𝑥‖,‖𝑥‖,𝑥,𝑡0if𝑡≤1if𝑡> (2.44) where 𝑥∈𝑋 and 𝑡∈ℝ, see [22]. Suppose that 𝑓∶𝑋→𝑌 is a mapping into a Banach space (𝑌,|‖⋅‖|) such that 𝐷𝑓𝑥,𝑦,𝑧,𝑤≤‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝+‖𝑤‖^𝑝 (2.45) for all 𝑥,𝑦,𝑧,𝑤∈𝑋, where 𝑝>0 and 𝑝≠1,2. Let 𝑁_𝑌 be a fuzzy norm on 𝑌. Then we get 𝑁_𝑌=,𝐷𝑓𝑥,𝑦,𝑧,𝑤,𝑠+𝑡+𝑢+𝑣0if𝑠+𝑡+𝑢+𝑣≤𝐷𝑓𝑥,𝑦,𝑧,𝑤1if𝑠+𝑡+𝑢+𝑣>𝐷𝑓𝑥,𝑦,𝑧,𝑤 (2.46) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝑠,𝑡,𝑢,𝑣∈ℝ. Consider the case 𝑁_𝑌𝐷𝑓𝑥,𝑦,𝑧,𝑤,𝑠+𝑡+𝑢+𝑣=0. (2.47)

This implies that ‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝+‖𝑤‖^𝑝≥𝐷𝑓𝑥,𝑦,𝑧,𝑤≥𝑠+𝑡+𝑢+𝑣, (2.48) and so ‖𝑥‖^𝑝≥𝑠, ‖𝑦‖^𝑝≥𝑡, ‖𝑧‖^𝑝≥𝑢, or ‖𝑤‖^𝑝≥𝑣 in this case. Hence, for 𝑞=1/𝑝, we have min𝑁_𝑋𝑥,𝑠^𝑞,𝑁_𝑋𝑦,𝑡^𝑞,𝑁_𝑋𝑧,𝑢^𝑞,𝑁_𝑋𝑤,𝑣^𝑞=0 (2.49) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝑠,𝑡,𝑢,𝑣>0. Therefore, in every case, the inequality 𝑁_𝑌𝐷𝑓𝑥,𝑦,𝑧,𝑤,𝑠+𝑡+𝑢+𝑣≥min𝑁_𝑋𝑥,𝑠^𝑞,𝑁_𝑋𝑦,𝑡^𝑞,𝑁_𝑋𝑧,𝑢^𝑞,𝑁_𝑋𝑤,𝑣^𝑞 (2.50) holds. It means that 𝑓 is a fuzzy 𝑞-almost quadratic-additive mapping, and by Theorem 2.2, we get the following stability result.

Corollary 2.6. Let (𝑋,‖⋅‖) be a normed linear space and let (𝑌,|‖⋅‖|) be a Banach space. If 𝑓∶𝑋→𝑌 satisfies 𝐷𝑓𝑥,𝑦,𝑧,𝑤≤‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝+‖𝑤‖^𝑝 (2.51) for all 𝑥,𝑦,𝑧,𝑤∈𝑋, where 𝑝>0 and 𝑝≠1,2, then there is a unique quadratic-additive mapping 𝐹∶𝑋→𝑌 such that 𝐹(𝑥)(𝑥≤2−𝑓)2−2^𝑝+24−2^𝑝‖𝑥‖^𝑝 (2.52) for all 𝑥∈𝑋.

Lemma 2.7. If 𝐹∶𝑋→𝑌 is a mapping such that 𝐷𝑓𝑥,𝑦,𝑧,𝑤=0 (2.53) for all 𝑥,𝑦,𝑧,𝑤∈𝑋⧵{0} with 𝐹(0)=0, then 𝐷𝐹𝑥,𝑦,𝑧,𝑤=0 (2.54) for all 𝑥,𝑦,𝑧,𝑤∈𝑋.

Proof. We easily know that ()()()()𝐷𝐹𝑥,𝑦,𝑧,0=𝐷𝐹2𝑥,𝑦,𝑧,−𝑥−𝐷𝐹2𝑥,𝑦,−𝑥,−𝑥+𝐷𝐹2𝑥,𝑦,𝑥,−𝑥−𝐷𝐹2𝑥,𝑧,−𝑥,−𝑥+𝐷𝐹2𝑥,𝑧,𝑥,−𝑥−2𝐷𝐹2x,𝑥,𝑥,−𝑥+𝐷𝐹𝑥,−𝑥,−𝑥,−𝑥=0 (2.55) for all 𝑥,𝑦,𝑧∈𝑋⧵{0}. Using a similar method, we have 𝐷𝐹(𝑥,𝑦,0,𝑤)=0, 𝐷𝐹(𝑥,0,𝑧,𝑤)=0, 𝐷𝐹(0,𝑦,𝑧,𝑤)=0, 𝐷𝐹(𝑥,𝑦,0,0)=0, 𝐷𝐹(𝑥,0,𝑧,0)=0, 𝐷𝐹(𝑥,0,0,𝑤)=0, 𝐷𝐹(0,𝑦,𝑧,0)=0, 𝐷𝐹(0,0,𝑧,𝑤)=0, 𝐷𝐹(0,𝑦,0,𝑤)=0, 𝐷𝐹(𝑥,0,0,0)=0, 𝐷𝐹(0,𝑦,0,0)=0, 𝐷𝐹(0,0,𝑧,0)=0, 𝐷𝐹(0,0,0,𝑤)=0, and 𝐷𝐹(0,0,0,0)=0 for all 𝑥,𝑦,𝑧,𝑤∈𝑋⧵{0}, as we desired.