Fuzzy Stability of a Functional Equation Deriving from Quadratic and Additive Mappings

Abstract

We investigate a fuzzy version of stability for the functional equation 𝑓(2𝑥+𝑦)+𝑓(2𝑥−𝑦)+ 2𝑓(𝑥)−𝑓(𝑥+𝑦)−𝑓(𝑥−𝑦)−2𝑓(2𝑥)=0 in the sense of Mirmostafaee and Moslehian.

1. Introduction and Preliminaries

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 unbounded Cauchy differences. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians, see [5–15].

In 1984, Katsaras [16] 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 [17], following Cheng and Mordeson [18], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [19]. In 2008, Mirmostafaee and Moslehian [20] obtained a fuzzy version of stability for the Cauchy functional equation 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦=0.(1.1) In the same year, they [21] proved a fuzzy version of stability for the quadratic functional equation 𝑓𝑥+𝑦+𝑓𝑥−𝑦−2𝑓(𝑥)−2𝑓𝑦=0.(1.2) We call a solution of (1.1) an additive mapping and a solution of (1.2) is called a quadratic mapping. Now, we consider the functional equation 𝑓2𝑥+𝑦+𝑓2𝑥−𝑦+2𝑓(𝑥)−𝑓𝑥+𝑦−𝑓𝑥−𝑦−2𝑓(2𝑥)=0,(1.3) which is called a functional equation deriving from quadratic and additive mappings. We call a solution of (1.3) a general quadratic mapping. In 2008, Najati and Moghimi [22] obtained a stability of the functional equation (1.3) by taking and composing an additive mapping 𝐴 and a quadratic mapping 𝑄 to prove the existence of a general quadratic 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−𝑓(0) of it, respectively.

In this paper, we get a general stability result of the functional equation deriving from quadratic and additive mappings (1.3) 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), they 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. Therefore, this idea is a refinement with respect to the simplicity of the proof.

2. Fuzzy Stability of the Functional Equation (1.3)

We use the definition of a fuzzy normed space given in [17] to exhibit a reasonable fuzzy version of stability for the functional equation deriving from quadratic and additive mappings in the fuzzy normed linear space.

Definition 2.1 (see [17]). Let 𝑋 be a real linear space. A function 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 non-decreasing function 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 abbreviation𝐷𝑓𝑥,𝑦∶=𝑓2𝑥+𝑦+𝑓2𝑥−𝑦+2𝑓(𝑥)−𝑓𝑥+𝑦−𝑓𝑥−𝑦−2𝑓(2𝑥),(2.1) for all 𝑥,𝑦∈𝑋. Recall 𝐷𝑓≡0 means that 𝑓 is a general quadratic mapping. For given 𝑞>0, the mapping 𝑓 is called a fuzzy 𝑞-almost general quadratic mapping if𝑁^′𝐷𝑓𝑥,𝑦,𝑡+𝑠≥min𝑁𝑥,𝑠^𝑞,𝑁𝑦,𝑡^𝑞,(2.2) for all 𝑥,𝑦∈𝑋⧵{0} and all 𝑠,𝑡∈[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 general quadratic mapping from a fuzzy normed space (𝑋,𝑁) into a fuzzy Banach space (𝑌,𝑁^′). Then, there is a unique general quadratic mapping 𝐹∶𝑋→𝑌 such that 𝑁^′𝐹(𝑥)−𝑓(𝑥),𝑡≥sup_{0<𝑡′<𝑡}𝑁𝑡𝑥,^′𝑞7+2^𝑝+3^𝑝+4^𝑝/4−2^𝑝3^𝑝+(5+2⋅2^𝑝+3^𝑝)/22−2^𝑝𝑞,(2.3) for each 𝑥∈𝑋 and 𝑡>0, where 𝑝=1/𝑞.

Proof. We will prove the theorem in three cases, 𝑞>1, 1/2<𝑞<1, and 0<𝑞<1/2.
Case 1. Let 𝑞>1. We define a mapping 𝐽_𝑛𝑓∶𝑋→𝑌 by 𝐽_𝑛𝑓(𝑥)=124^−𝑛𝑓2^𝑛𝑥+𝑓−2^𝑛𝑥−2𝑓(0)+2^−𝑛𝑓2^𝑛𝑥−𝑓−2^𝑛𝑥+𝑓(0),(2.4) for all 𝑥∈𝑋. Then, 𝐽₀𝑓(𝑥)=𝑓(𝑥), 𝐽_𝑗𝑓(0)=𝑓(0), and 𝐽_𝑗𝑓(𝑥)−𝐽_𝑗+1𝑓(𝑥)=𝐷𝑓2^𝑗𝑥/3,2^𝑗𝑥/34^𝑗+1−𝐷𝑓2^𝑗𝑥/3,2^𝑗+1𝑥/32⋅4^𝑗+1−𝐷𝑓2^𝑗𝑥/3,2^𝑗𝑥2⋅4^𝑗+1−𝐷𝑓2^𝑗𝑥/3,2^𝑗+2𝑥/32⋅4^𝑗+1+𝐷𝑓−2^𝑗𝑥/3,−2^𝑗𝑥/34^𝑗+1−𝐷𝑓−2^𝑗𝑥/3,−2^𝑗+1𝑥/32⋅4^𝑗+1−𝐷𝑓−2^𝑗𝑥/3,−2^𝑗𝑥2⋅4^𝑗+1−𝐷𝑓−2^𝑗𝑥/3,−2^𝑗+2𝑥/32⋅4^𝑗+1+𝐷𝑓2^𝑗+1𝑥,2^𝑗𝑥2^𝑗+2−𝐷𝑓2^𝑗𝑥,3⋅2^𝑗𝑥2^𝑗+2+𝐷𝑓2^𝑗𝑥,2^𝑗𝑥2^𝑗+2+𝐷𝑓2^𝑗𝑥,−2^𝑗+1𝑥2^𝑗+2,(2.5) for all 𝑥∈𝑋⧵{0} 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=𝑚7+2^𝑝+3^𝑝+4^𝑝4⋅3^𝑝2^𝑝4^𝑗+5+2⋅2^𝑝+3^𝑝42^𝑝2^𝑗𝑡^𝑝≥min^{𝑛+𝑚−1}_𝑗=𝑚𝑁^′𝐽_𝑗𝑓(𝑥)−𝐽_𝑗+1𝑓(𝑥),7+2^𝑝+3^𝑝+4^𝑝4^𝑗+1⋅3^𝑝+5+2⋅2^𝑝+3^𝑝2^𝑗+22^𝑗𝑝𝑡^𝑝≥min^{𝑛+𝑚−1}_𝑗=𝑚min𝑁^′𝐷𝑓2^𝑗𝑥/3,2^𝑗𝑥/34^𝑗+1,2^𝑗𝑝𝑡^𝑝2⋅4^𝑗⋅3^𝑝,𝑁^′−𝐷𝑓2^𝑗𝑥/3,2^𝑗+1𝑥/32⋅4^𝑗+1,2^𝑗𝑝1+2^𝑝𝑡^𝑝2⋅4^𝑗+1⋅3^𝑝,𝑁^′−𝐷𝑓2^𝑗𝑥/3,2^𝑗𝑥2⋅4^𝑗+1,2^𝑗𝑝1+3^𝑝𝑡^𝑝2⋅4^𝑗+1⋅3^𝑝,𝑁^′−𝐷𝑓2^𝑗𝑥/3,2^𝑗+2𝑥/32⋅4^𝑗+1,2^𝑗𝑝1+4^𝑝𝑡^𝑝2⋅4^𝑗+1⋅3^𝑝,𝑁^′𝐷𝑓−2^𝑗𝑥/3,−2^𝑗𝑥/34^𝑗+1,2^𝑗𝑝𝑡^𝑝2⋅4^𝑗⋅3^𝑝,𝑁^′−𝐷𝑓−2^𝑗𝑥/3,−2^𝑗+1𝑥/32⋅4^𝑗+1,2^𝑗𝑝1+2^𝑝𝑡^𝑝2⋅4^𝑗+1⋅3^𝑝,𝑁^′−𝐷𝑓−2^𝑗𝑥/3,−2^𝑗𝑥2⋅4^𝑗+1,2^𝑗𝑝1+3^𝑝𝑡^𝑝2⋅4^𝑗+1⋅3^𝑝,𝑁^′−𝐷𝑓−2^𝑗𝑥/3,−2^𝑗+2𝑥/32⋅4^𝑗+1,2^𝑗𝑝1+4^𝑝𝑡^𝑝2⋅4^𝑗+1⋅3^𝑝,2𝑁′𝐷𝑓^𝑗+1𝑥,2^𝑗𝑥2^𝑗+2,2^𝑗𝑝1+2^𝑝𝑡^𝑝2^𝑗+2,𝑁^′−𝐷𝑓2^𝑗𝑥,3⋅2^𝑗𝑥2^𝑗+2,2^𝑗𝑝1+3^𝑝𝑡^𝑝2^𝑗+2,𝑁^′𝐷𝑓2^𝑗𝑥,2^𝑗𝑥2^𝑗+2,2^𝑗𝑝𝑡^𝑝2^𝑗+1,𝑁^′𝐷𝑓2^𝑗𝑥,−2^𝑗+1𝑥2^𝑗+2,2^j𝑝1+2^𝑝𝑡^𝑝2^𝑗+2≥min^{𝑛+𝑚−1}_𝑗=𝑚min𝑁2^𝑗𝑥,2^𝑗𝑡,𝑁2^𝑗+1𝑥,2^𝑗+1𝑡,𝑁3⋅2^𝑗𝑥,3⋅2^𝑗𝑡,𝑁2^𝑗𝑥3,2^𝑗𝑡32,𝑁^𝑗+1𝑥3,2^𝑗+1𝑡32,𝑁^𝑗+2𝑥3,2^𝑗+2𝑡3=𝑁(𝑥,𝑡),(2.6) for all 𝑥∈𝑋⧵{0} 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.(2.7) We observe that for some 𝑡̃>𝑡₀, the series ∑^∞_𝑗=0((7+2^𝑝+3^𝑝+4^𝑝)/(4^𝑗+1⋅3^𝑝)+(5+2⋅2^𝑝+3^𝑝)/(2^𝑗+2))2^𝑗𝑝𝑡^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=𝑚7+2^𝑝+3^𝑝+4^𝑝4^𝑗+1⋅3^𝑝+5+2⋅2^𝑝+3^𝑝2^𝑗+22^𝑗𝑝𝑡^𝑝̃<𝑐,(2.8) for each 𝑚≥𝑛₀ and 𝑛>0. By (N5) and (2.6) we have 𝑁^′𝐽_𝑚𝑓(𝑥)−𝐽_𝑛+𝑚𝑓(𝑥),𝑐≥𝑁^′𝐽_𝑚𝑓(𝑥)−𝐽_𝑛+𝑚𝑓(𝑥),^{𝑛+𝑚−1}_𝑗=𝑚7+2^𝑝+3^𝑝+4^𝑝4^𝑗+1⋅3^𝑝+5+2⋅2^𝑝+3^𝑝2^𝑗+22^𝑗𝑝𝑡^𝑝̃≥𝑁𝑥,𝑡̃≥𝑁𝑥,𝑡₀≥1−𝜀,(2.9) for all 𝑥∈𝑋⧵{0}. Recall 𝐽_𝑛𝑓(0)=𝑓(0) for all 𝑛>0. Thus, {𝐽_𝑛𝑓(𝑥)} becomes a Cauchy sequence for all 𝑥∈𝑋. Since (𝑌,𝑁^′) is complete, we can define a mapping 𝐹∶𝑋→𝑌 by 𝐹(𝑥)∶=𝑁^′−lim_𝑛→∞𝐽_𝑛𝑓(𝑥),(2.10) for all 𝑥∈𝑋. Moreover, if we put 𝑚=0 in (2.6), we have 𝑁^′𝑓(𝑥)−𝐽_𝑛𝑓(𝑥)𝑡,𝑡≥𝑁𝑥,^𝑞∑^𝑛−1_𝑗=07+2^𝑝+3^𝑝+4^𝑝/4^𝑗+1⋅3^𝑝+5+2⋅2^𝑝+3^𝑝/2^𝑗+22^𝑗𝑝𝑞,(2.11) for all 𝑥∈𝑋. Next, we will show that 𝐹 is a general quadratic mapping. Using (N4), we have 𝑁^′𝑁𝐷𝐹𝑥,𝑦,𝑡≥min^′𝐹2𝑥+𝑦−𝐽_𝑛𝑓,𝑡2𝑥+𝑦,𝑁16_′−𝐹𝑥+𝑦+𝐽_𝑛𝑓,𝑡𝑥+𝑦16,𝑁^′−𝐹𝑥−𝑦+𝐽_𝑛𝑓,𝑡𝑥−𝑦,𝑁16_′2𝐹(𝑥)−2𝐽_𝑛𝑓(𝑥),𝑡8,𝑁^′−2𝐹(2𝑥)+2𝐽_𝑛𝑓(),𝑡2𝑥8,𝑁^′𝐹2𝑥−𝑦+𝐽_𝑛𝑓,𝑡2𝑥−𝑦16,𝑁^′𝐷𝐽_𝑛𝑓,𝑡𝑥,𝑦2,(2.12) for all 𝑥,𝑦∈𝑋⧵{0} and 𝑛∈ℕ. The first six terms on the right hand side of (2.12) tend to 1 as 𝑛→∞ by the definition of 𝐹 and (N2), and the last term holds 𝑁^′𝐷𝐽_𝑛𝑓,𝑡𝑥,𝑦2≥min𝑁^′𝐷𝑓2^𝑛𝑥,2^𝑛𝑦2⋅4^𝑛,𝑡8,𝑁^′𝐷𝑓−2^𝑛𝑥,−2^𝑛𝑦2⋅4^𝑛,𝑡8,𝑁^′𝐷𝑓2^𝑛𝑥,2^𝑛𝑦2⋅2^𝑛,𝑡8,𝑁^′𝐷𝑓−2^𝑛𝑥,−2^𝑛𝑦2⋅2^𝑛,𝑡8,(2.13) for all 𝑥,𝑦∈𝑋⧵{0}. By (N3) and (2.2), we obtain 𝑁^′𝐷𝑓±2^𝑛𝑥,±2^𝑛𝑦2⋅4^𝑛,𝑡8=𝑁^′𝐷𝑓±2^𝑛𝑥,±2^𝑛𝑦,2⋅4^𝑛𝑡8≥min𝑁±2^𝑛4𝑥,^𝑛𝑡8^𝑞,𝑁±2^𝑛4𝑦,^𝑛𝑡8^𝑞≥min𝑁𝑥,2^{(2𝑞−1)𝑛−3𝑞}𝑡^𝑞,𝑁𝑦,2^{(2𝑞−1)𝑛−3𝑞}𝑡^𝑞,𝑁^′𝐷𝑓±2^𝑛𝑥,±2^𝑛𝑦2⋅2^𝑛,𝑡8≥min𝑁𝑥,2^{(𝑞−1)𝑛−3𝑞}𝑡^𝑞,𝑁𝑦,2^{(𝑞−1)𝑛−3𝑞}𝑡^𝑞,(2.14) for all 𝑥,𝑦∈𝑋⧵{0} and 𝑛∈ℕ. Since 𝑞>1, together with (N5), we can deduce that the last term of (2.12) also tends to 1 as 𝑛→∞. It follows from (2.12) that 𝑁^′𝐷𝐹𝑥,𝑦,𝑡=1,(2.15) for all 𝑥,𝑦∈𝑋⧵{0} and 𝑡>0. Since 𝐷𝐹(0,0)=0, 𝐷𝐹(𝑥,0)=0 and 𝐷𝐹(0,𝑦)=0 for all 𝑥,𝑦∈𝑋⧵{0}, this means that 𝐷𝐹(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝑋 by (N2).
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 (2.11), we have 𝑁^′𝐹(𝑥)(𝑥)𝑁−𝑓,𝑡≥min^′𝐹(𝑥)−𝐽_𝑛𝑓(𝑥),𝑡−𝑡^′,𝑁^′𝐽_𝑛𝑓(𝑥)−𝑓(𝑥),𝑡^′≥min1−𝜀,𝑁𝑥,𝑡′^𝑞∑^𝑛−1_𝑗=07+2^𝑝+3^𝑝+4^𝑝/4^𝑗+1⋅3^𝑝+5+2⋅2^𝑝+3^𝑝/2^𝑗+22^𝑗𝑝𝑞≥min1−𝜀,𝑁𝑥,𝑡′^𝑞7+2^𝑝+3^𝑝+4^𝑝/4−2^𝑝3^𝑝+5+2⋅2^𝑝+3^𝑝/22−2^𝑝𝑞.(2.16) Because 0<𝜀<1 is arbitrary and 𝐹(0)=𝑓(0), we get (2.3) in this case.
Finally, to prove the uniqueness of 𝐹, let 𝐹^′∶𝑋→𝑌 be another general quadratic mapping satisfying (2.3). Then, by (2.5), 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,(2.17) 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)𝑁,𝑡≥min^′𝐽_𝑛𝐹(𝑥)−𝐽_𝑛𝑓(𝑥),𝑡2,𝑁^′𝐽_𝑛𝑓(𝑥)−𝐽_𝑛𝐹^′(𝑥),𝑡2≥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,𝑁^′𝐹−𝑓2^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}𝑡′^𝑞7+2^𝑝+3^𝑝+4^𝑝/4−2^𝑝3^𝑝+5+2⋅2^𝑝+3^𝑝/22−2^𝑝𝑞,(2.18) for all 𝑥∈𝑋 and 𝑛∈ℕ. Observe that for 𝑞=1/𝑝, 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),(2.19) 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−2𝑓(0)+2^𝑛𝑓𝑥2^𝑛−𝑥−𝑓2^𝑛+𝑓(0),(2.20) for all 𝑥∈𝑋. Then, we have 𝐽₀𝑓(𝑥)=𝑓(𝑥), 𝐽_𝑗𝑓(0)=𝑓(0), and 𝐽_𝑗𝑓(𝑥)−𝐽_𝑗+1𝑓(𝑥)=𝐷𝑓2^𝑗𝑥/3,2^𝑗𝑥/34^𝑗+1−𝐷𝑓2^𝑗𝑥/3,2^𝑗+1𝑥/32⋅4^𝑗+1−𝐷𝑓2^𝑗𝑥/3,2^𝑗𝑥2⋅4^𝑗+1−𝐷𝑓2^𝑗𝑥/3,2^𝑗+2𝑥/32⋅4^𝑗+1+𝐷𝑓−2^𝑗𝑥/3,−2^𝑗𝑥/34^𝑗+1−𝐷𝑓−2^𝑗𝑥/3,−2^𝑗+1𝑥/32⋅4^𝑗+1−𝐷𝑓−2^𝑗𝑥/3,−2^𝑗𝑥2⋅4^𝑗+1−𝐷𝑓−2^𝑗𝑥/3,−2^𝑗+2𝑥/32⋅4^𝑗+1−2^𝑗−1𝑥𝐷𝑓2^𝑗,𝑥2^𝑗+1𝑥−𝐷𝑓2^𝑗+1,3𝑥2^𝑗+1𝑥+𝐷𝑓2^𝑗+1,𝑥2^𝑗+1𝑥+𝐷𝑓2^𝑗+1,−𝑥2^𝑗,(2.21) 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=𝑚7+2^𝑝+3^𝑝+4^𝑝4⋅3^𝑝2^𝑝4^𝑗+5+2⋅2^𝑝+3^𝑝2⋅2^𝑝22^𝑝𝑗𝑡^𝑝≥min^{𝑛+𝑚−1}_𝑗=𝑚min𝑁^′𝐷𝑓2^𝑗𝑥/3,2^𝑗𝑥/34^𝑗+1,2⋅2^𝑗𝑝𝑡^𝑝4^𝑗+1⋅3^𝑝,𝑁^′𝐷𝑓−2^𝑗𝑥/3,−2^𝑗𝑥/34^𝑗+1,2⋅2^𝑗𝑝𝑡^𝑝4^𝑗+1⋅3^𝑝,𝑁^′−𝐷𝑓2^𝑗𝑥/3,2^𝑗+1𝑥/32⋅4^𝑗+1,2^𝑗𝑝1+2^𝑝𝑡^𝑝2⋅4^𝑗+1⋅3^𝑝,𝑁^′−𝐷𝑓2^𝑗𝑥/3,2^𝑗𝑥2⋅4^𝑗+1,2^𝑗𝑝1+3^𝑝𝑡^𝑝2⋅4^𝑗+1⋅3^𝑝,𝑁^′−𝐷𝑓2^𝑗𝑥/3,2^𝑗+2𝑥/32⋅4^𝑗+1,2^𝑗𝑝1+4^𝑝𝑡^𝑝2⋅4^𝑗+1⋅3^𝑝,𝑁^′−𝐷𝑓−2^𝑗𝑥/3,−2^𝑗+1𝑥/32⋅4^𝑗+1,2^𝑗𝑝1+2^𝑝𝑡^𝑝2⋅4^𝑗+1⋅3^𝑝,𝑁^′−𝐷𝑓−2^𝑗𝑥/3,−2^𝑗𝑥2⋅4^𝑗+1,2^𝑗𝑝1+3^𝑝𝑡^𝑝2⋅4^𝑗+1⋅3^𝑝,𝑁^′−𝐷𝑓−2^𝑗𝑥/3,−2^𝑗+2𝑥/32⋅4^𝑗+1,2^𝑗𝑝1+4^𝑝𝑡^𝑝2⋅4^𝑗+1⋅3^𝑝,𝑁^′−2^𝑗−1𝑥𝐷𝑓2^𝑗,𝑥2^𝑗+1,2^𝑗−11+2^𝑝𝑡^𝑝2^(𝑗+1)𝑝,𝑁^′2^𝑗−1𝑥𝐷𝑓2^𝑗+1,3𝑥2^𝑗+1,2^𝑗−11+3^𝑝𝑡^𝑝2^(𝑗+1)𝑝,𝑁^′−2^𝑗−1𝑥𝐷𝑓2^𝑗+1,𝑥2^𝑗+1,2^𝑗𝑡^𝑝2^(𝑗+1)𝑝,𝑁^′−2^𝑗−1𝑥𝐷𝑓2^𝑗+1𝑥,−2^𝑗,2^𝑗−11+2^𝑝𝑡^𝑝2^(𝑗+1)𝑝≥min^{𝑛+𝑚−1}_𝑗=𝑚𝑁𝑥min2^𝑗+1,𝑡2^𝑗+1𝑥,𝑁2^𝑗,𝑡2^𝑗,𝑁3𝑥2^𝑗+1,3𝑡2^𝑗+1,𝑁2^𝑗𝑥3,2^𝑗𝑡32,𝑁^𝑗+1𝑥3,2^𝑗+1𝑡32,𝑁^𝑗+2𝑥3,2^𝑗+2𝑡3=𝑁(𝑥,𝑡),(2.22) for all 𝑥∈𝑋 and 𝑡>0. In the similar argument following (2.6) 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=07+2^𝑝+3^𝑝+4^𝑝/4⋅3^𝑝2^𝑝/4^𝑗+5+2⋅2^𝑝+3^𝑝/2⋅2^𝑝2/2^𝑝𝑗𝑞,(2.23) for all 𝑥∈𝑋 and 𝑡>0. To prove that 𝐹 is a general quadratic mapping, we have enough to show that the last term of (2.12) in Case 1 tends to 1 as 𝑛→∞. By (N3) and (2.2), we get 𝑁^′𝐷𝐽_𝑛𝑓,𝑡𝑥,𝑦2≥min𝑁^′𝐷𝑓2^𝑛𝑥,2^𝑛𝑦2⋅4^𝑛,𝑡8,𝑁^′𝐷𝑓−2^𝑛𝑥,−2^𝑛𝑦2⋅4^𝑛,𝑡8,𝑁^′2^𝑛−1𝑥𝐷𝑓2^𝑛,𝑦2^𝑛,𝑡8,𝑁^′2^𝑛−1𝐷𝑓−𝑥2^𝑛,−𝑦2^𝑛,𝑡8≥min𝑁𝑥,2^{(2𝑞−1)𝑛−4𝑞}𝑡^𝑞,𝑁𝑦,2^{(2𝑞−1)𝑛−4𝑞}𝑡^𝑞,𝑁𝑥,2^{(1−𝑞)𝑛−4𝑞}𝑡^𝑞,𝑁𝑦,2^{(1−𝑞)𝑛−4𝑞}𝑡^𝑞,(2.24) for all 𝑥,𝑦∈𝑋⧵{0} 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, together with the similar argument after (2.12), we can say that 𝐷𝐹(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝑋. Recall that in Case 1, (2.3) follows from (2.11). By the same reasoning, we get (2.3) from (2.23) in this case. Now, to prove the uniqueness of 𝐹, let 𝐹^′ be another general quadratic mapping satisfying (2.3). Then, together with (N4), (2.3), and (2.17), 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),𝑡2≥min𝑁^′𝐹−𝑓2^𝑛𝑥2⋅4^𝑛,𝑡8,𝑓−𝐹^′2^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,𝑡8𝑁2≥min𝑥,^{(2𝑞−1)𝑛−2𝑞}𝑡′^𝑞7+2^𝑝+3^𝑝+4^𝑝/4−2^𝑝3^𝑝+5+2⋅2^𝑝+3^𝑝/22^𝑝−2^𝑞,𝑁2𝑥,^{(1−𝑞)𝑛−2𝑞}𝑡′^𝑞7+2^𝑝+3^𝑝+4^𝑝/4−2^𝑝3^𝑝+5+2⋅2^𝑝+3^𝑝/22^𝑝−2^𝑞,(2.25) for all 𝑥∈𝑋 and 𝑛∈ℕ. Since lim_𝑛→∞2^{(2𝑞−1)𝑛−2𝑞}=lim_𝑛→∞2^{(1−𝑞)𝑛−2𝑞}=∞ 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−2𝑓(0)+2^𝑛𝑓𝑥2^𝑛−𝑥−𝑓2^𝑛+𝑓(0),(2.26) for all 𝑥∈𝑋. Then, we have 𝐽₀𝑓(𝑥)=𝑓(𝑥), 𝐽_𝑗𝑓(0)=𝑓(0), and 𝐽_𝑗𝑓(𝑥)−𝐽_𝑗+1𝑓(𝑥)=−4^𝑗𝑥𝐷𝑓3⋅2^𝑗+1,𝑥3⋅2^𝑗+1−4^𝑗𝐷𝑓−𝑥3⋅2^𝑗+1,−𝑥3⋅2^𝑗+1+4^𝑗2𝑥𝐷𝑓3⋅2^𝑗+1,𝑥3⋅2^𝑗+4^𝑗2𝑥𝐷𝑓3⋅2^𝑗+1,𝑥2^𝑗+1+4^𝑗2𝑥𝐷𝑓3⋅2^𝑗+1,𝑥3⋅2^𝑗−1+4^𝑗2𝐷𝑓−𝑥3⋅2^𝑗+1,−𝑥3⋅2^𝑗+4^𝑗2𝐷𝑓−𝑥3⋅2^𝑗+1,−𝑥2^𝑗+1+4^𝑗2𝐷𝑓−𝑥3⋅2^𝑗+1,−𝑥3⋅2^𝑗−1−2^𝑗−1𝑥𝐷𝑓2^𝑗,𝑥2^𝑗+1𝑥−𝐷𝑓2^𝑗+1,3𝑥2^𝑗+1𝑥+𝐷𝑓2^𝑗+1,𝑥2^𝑗+1𝑥+𝐷𝑓2^𝑗+1,−𝑥2^𝑗,(2.27) which implies that if 𝑛+𝑚>𝑚≥0, then 𝑁^′𝐽_𝑚𝑓(𝑥)−𝐽_𝑛+𝑚𝑓(𝑥),^{𝑛+𝑚−1}_𝑗=𝑚7+2^𝑝+3^𝑝+4^𝑝2^𝑝⋅3^𝑝42^𝑝𝑗+5+2⋅2^𝑝+3^𝑝2⋅2^𝑝22^𝑝𝑗𝑡^𝑝≥min^{𝑛+𝑚−1}_𝑗=𝑚min𝑁^′−4^𝑗𝑥𝐷𝑓3⋅2^𝑗+1,𝑥3⋅2^𝑗+1,2⋅4^𝑗𝑡^𝑝2^(𝑗+1)𝑝⋅3^𝑝,𝑁^′−4^𝑗𝐷𝑓−𝑥3⋅2^𝑗+1,−𝑥3⋅2^𝑗+1,2⋅4^𝑗𝑡^𝑝2^(𝑗+1)𝑝⋅3^𝑝,𝑁^′4^𝑗𝐷𝑓𝑥/3⋅2^𝑗+1,𝑥/3⋅2^𝑗2,4^𝑗1+2^𝑝𝑡^𝑝2⋅2^(𝑗+1)𝑝⋅3^𝑝,𝑁^′4^𝑗𝐷𝑓𝑥/3⋅2^𝑗+1,𝑥/2^𝑗+12,4^𝑗1+3^𝑝𝑡^𝑝2⋅2^(𝑗+1)𝑝⋅3^𝑝,𝑁^′4^𝑗𝐷𝑓𝑥/3⋅2^𝑗+1,𝑥/3⋅2^𝑗−12,4^𝑗1+4^𝑝𝑡^𝑝2⋅2^(𝑗+1)𝑝⋅3^𝑝,4𝑁′^𝑗𝐷𝑓−𝑥/3⋅2^𝑗+1,−𝑥/3⋅2^𝑗2,4^𝑗1+2^𝑝𝑡^𝑝2⋅2^(𝑗+1)𝑝⋅3^𝑝,𝑁^′4^𝑗𝐷𝑓−𝑥/3⋅2^𝑗+1,−𝑥/2^𝑗+12,4^𝑗1+3^𝑝𝑡^𝑝2⋅2^(𝑗+1)𝑝⋅3^𝑝,4𝑁′^𝑗𝐷𝑓−𝑥/3⋅2^𝑗+1,−𝑥/3⋅2^𝑗−12,4^𝑗1+4^𝑝𝑡^𝑝2⋅2^(𝑗+1)𝑝⋅3^𝑝,𝑁^′−2^𝑗−1𝑥𝐷𝑓2^𝑗,𝑥2^𝑗+1,2^𝑗−11+2^𝑝𝑡^𝑝2^(𝑗+1)𝑝,𝑁^′2^𝑗−1𝑥𝐷𝑓2^𝑗+1,3𝑥2^𝑗+1,2^𝑗−11+3^𝑝𝑡^𝑝2^(𝑗+1)𝑝,𝑁′−2^𝑗−1𝑥𝐷𝑓2^𝑗+1,𝑥2^𝑗+1,2^𝑗𝑡^𝑝2^(𝑗+1)𝑝,𝑁^′−2^𝑗−1𝑥𝐷𝑓2^𝑗+1𝑥,−2^𝑗,2^𝑗−11+2^𝑝𝑡^𝑝2^(𝑗+1)𝑝≥min^{𝑛+𝑚−1}_𝑗=𝑚𝑁𝑥min2^𝑗+1,𝑡2^𝑗+1𝑥,𝑁2^𝑗,𝑡2^𝑗,𝑁3𝑥2^𝑗+1,3𝑡2^𝑗+1𝑥,𝑁2^𝑗+1,𝑡2^𝑗+1,𝑁𝑥3⋅2^𝑗+1,𝑡3⋅2^𝑗+1𝑥,𝑁3⋅2^𝑗,𝑡3⋅2^𝑗𝑥,𝑁3⋅2^𝑗−1,𝑡3⋅2^𝑗−1=𝑁(𝑥,𝑡),(2.28) for all 𝑥∈𝑋⧵{0} 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=07+2^𝑝+3^𝑝+4^𝑝/2^𝑝⋅3^𝑝4/2^𝑝𝑗+5+2⋅2^𝑝+3^𝑝/2⋅2^𝑝2/2^𝑝𝑗𝑞,(2.29) for all 𝑥∈𝑋. Notice that 𝑁^′𝐷𝐽_𝑛𝑓,𝑡𝑥,𝑦2≥min𝑁^′4^𝑛2𝑥𝐷𝑓2^𝑛,𝑦2^𝑛,𝑡8,𝑁^′4^𝑛2𝐷𝑓−𝑥2^𝑛,−𝑦2^𝑛,𝑡8,𝑁^′2^𝑛−1𝑥𝐷𝑓2^𝑛,𝑦2^𝑛,𝑡8,𝑁^′2^𝑛−1𝐷𝑓−𝑥2^𝑛,−𝑦2^𝑛,𝑡8≥min𝑁𝑥,2^{(1−2𝑞)𝑛−3𝑞}𝑡^𝑞,𝑁𝑦,2^{(1−2𝑞)𝑛−3𝑞}𝑡^𝑞,(2.30) for all 𝑥,𝑦∈𝑋⧵{0} and 𝑡>0. Since 0<𝑞<1/2, both terms on the right-hand side tend to 1 as 𝑛→∞, which implies that the last term of (2.12) tends to 1 as 𝑛→∞. Therefore, we can say that 𝐷𝐹≡0. Moreover, using the similar argument after (2.12) in Case 1, we get (2.3) from (2.29) in this case. To prove the uniqueness of 𝐹, let 𝐹^′∶𝑋→𝑌 be another general quadratic mapping satisfying (2.3). Then, by (2.17), 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,𝑡8,𝑁^′4^𝑛2𝑓−𝐹^′𝑥2^𝑛,𝑡8,𝑁^′4^𝑛2−𝑥𝐹−𝑓2^𝑛,𝑡8,𝑁^′4^𝑛2𝑓−𝐹^′−𝑥2^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≥sup_𝑡′<𝑡𝑁2𝑥,^{(1−2𝑞)𝑛−2𝑞}𝑡^′𝑞7+2^𝑝+3^𝑝+4^𝑝/2^𝑝−43^𝑝+5+2⋅2^𝑝+3^𝑝/22^𝑝−2^𝑞,(2.31) 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). This completes the proof.

Remark 2.3. Consider a mapping 𝑓∶𝑋→𝑌 satisfying (2.2) for all 𝑥,𝑦∈𝑋⧵{0} and a real number 𝑞<0. Take any 𝑡>0. If we choose a real number 𝑠 with 0<2𝑠<𝑡, then we have 𝑁^′𝐷𝑓𝑥,𝑦,𝑡≥𝑁^′𝐷𝑓𝑥,𝑦,2𝑠≥min𝑁𝑥,𝑠^𝑞,𝑁𝑦,𝑠^𝑞,(2.32) for all 𝑥,𝑦∈𝑋⧵{0}. 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=1,(2.33) and so 𝑁^′𝐷𝑓𝑥,𝑦,𝑡=1,(2.34) for all 𝑡>0 and 𝑥,𝑦∈𝑋⧵{0}. Since 𝐷𝐹(0,0)=0, 𝐷𝐹(𝑥,0)=0, and 𝐷𝐹(0,𝑦)=0 for all 𝑥,𝑦∈𝑋⧵{0}, this means that 𝐷𝐹(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝑋 by (N2). In other words, 𝑓 is itself a general quadratic mapping if 𝑓 is a fuzzy 𝑞-almost general quadratic mapping for the case 𝑞<0.

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,𝑡0,𝑡≤1,𝑡>(2.35) where 𝑥∈𝑋 and 𝑡∈ℝ [21]. Suppose that 𝑓∶𝑋→𝑌 is a mapping into a Banach space (𝑌,|‖⋅‖|) such that 𝐷𝑓𝑥,𝑦≤‖𝑥‖^𝑝+𝑦^𝑝,(2.36) for all 𝑥,𝑦∈𝑋, where 𝑝>0 and 𝑝≠1,2. Let 𝑁_𝑌 be a fuzzy norm on 𝑌. Then, we get 𝑁_𝑌=,,𝐷𝑓𝑥,𝑦,𝑡+𝑠0,𝑡+𝑠≤𝐷𝑓𝑥,𝑦1,𝑡+𝑠>𝐷𝑓𝑥,𝑦(2.37) for all 𝑥,𝑦∈𝑋 and 𝑠,𝑡∈ℝ. Consider the case 𝑁_𝑌(𝐷𝑓(𝑥,𝑦),𝑡+𝑠)=0. This implies that ‖𝑥‖^𝑝+𝑦^𝑝≥𝐷𝑓𝑥,𝑦≥𝑡+𝑠,(2.38) and so, either ‖𝑥‖^𝑝≥𝑡 or ‖𝑦‖^𝑝≥𝑠 in this case. Hence, for 𝑞=1/𝑝, we have min𝑁_𝑋𝑥,𝑠^𝑞,𝑁_𝑋𝑦,𝑡^𝑞=0,(2.39) for all 𝑥,𝑦∈𝑋 and 𝑠,𝑡>0. Therefore, in every case, 𝑁_𝑌𝐷𝑓𝑥,𝑦,𝑡+𝑠≥min𝑁_𝑋𝑥,𝑠^𝑞,𝑁_𝑋𝑦,𝑡^𝑞(2.40) holds. It means that 𝑓 is a fuzzy 𝑞-almost general quadratic mapping, and by Theorem 2.2, we get the following stability result.

Corollary 2.4. Let (𝑋,‖⋅‖) be a normed linear space, and let (𝑌,|‖⋅‖|) be a Banach space. If 𝑓∶𝑋→𝑌 satisfies 𝐷𝑓𝑥,𝑦≤‖𝑥‖^𝑝+𝑦^𝑝,(2.41) for all 𝑥,𝑦∈𝑋, where 𝑝>0 and 𝑝≠1,2, then there is a unique general quadratic mapping 𝐹∶𝑋→𝑌 such that 𝐹(𝑥)(𝑥≤2−𝑓)7+2^𝑝+3^𝑝+4^𝑝3^𝑝4−2^𝑝+5+2⋅2+3^𝑝2−2^𝑝‖𝑥‖^𝑝,(2.42) for all 𝑥∈𝑋.