Fuzzy Stability of an n-Dimensional Quadratic and Additive Functional Equation

Abstract

We will investigate a fuzzy version of stability for the functional equation 2𝑓(∑^𝑛_𝑖=1𝑥_𝑖)+∑_{1≤𝑖,𝑗≤𝑛,𝑖≠𝑗}𝑓(𝑥_𝑖−𝑥_𝑗)=(𝑛+1)∑^𝑛_𝑖=1𝑓(𝑥_𝑖)+(𝑛−1)∑^𝑛_𝑖=1𝑓(−𝑥_𝑖) in the sense of Mirmostafaee and Moslehian.

1. Introduction

In 1940, Ulam [1] gave a wide-ranging talk before a Mathematical Colloquium at the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the following question concerning the stability of homomorphisms.

Let 𝐺₁ be a group and let 𝐺₂ be a metric group with a metric 𝑑(⋅,⋅). Given 𝜀>0, does there exist a 𝛿>0 such that if a function ℎ∶𝐺₁→𝐺₂ satisfies the inequality 𝑑(ℎ(𝑥𝑦),ℎ(𝑥)ℎ(𝑦))<𝛿 for all 𝑥, 𝑦∈𝐺₁, then there is a homomorphism 𝐻∶𝐺₁→𝐺₂ with 𝑑(ℎ(𝑥),𝐻(𝑥))<𝜀 for all 𝑥∈𝐺₁?

If the answer is affirmative, we say that the functional equation for homomorphisms is stable. Hyers [2] was the first mathematician to present the result concerning the stability of functional equations. He answered the question of Ulam for the case where 𝐺₁ and 𝐺₂ are assumed to be Banach spaces. This result of Hyers is stated as follows.

Let 𝑓∶𝐸₁→𝐸₂ be a function between Banach spaces such that𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝛿 (1)for some 𝛿>0 and for all 𝑥, 𝑦∈𝐸₁. Then the limit 𝐴(𝑥)=𝑙𝑖𝑚_𝑛→∞2^−𝑛𝑓(2^𝑛𝑥) exists for each 𝑥∈𝐸₁, and 𝐴∶𝐸₁→𝐸₂ is the unique additive function such that ‖𝑓(𝑥)−𝐴(𝑥)‖≤𝛿 for every 𝑥∈𝐸₁. Moreover, if 𝑓(𝑡𝑥) is continuous in 𝑡 for each fixed 𝑥∈𝐸₁, then the function 𝐴 is linear.

We remark that the additive function 𝐴 is directly constructed from the given function 𝑓, and this method is called the direct method. The direct method is a very powerful method for studying the stability problems of various functional equations. Taking this famous result into consideration, the additive Cauchy equation 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦) is said to have the Hyers-Ulam stability on (𝐸₁,𝐸₂) if for every function 𝑓∶𝐸₁→𝐸₂ satisfying (1) for some 𝛿≥0 and for all 𝑥, 𝑦∈𝐸₁, there exists an additive function 𝐴∶𝐸₁→𝐸₂ such that 𝑓−𝐴 is bounded on 𝐸₁.

In 1950, Aoki [3] generalized the theorem of Hyers for additive functions, and in the following year, Bourgin [4] extended the theorem without proof. Unfortunately, it seems that their results failed to receive attention from mathematicians at that time. No one has made use of these results for a long time.

In 1978, Rassias [5] addressed the Hyers’s stability theorem and attempted to weaken the condition for the bound of the norm of Cauchy difference and generalized the theorem of Hyers for linear functions.

Let 𝑓∶𝐸₁→𝐸₂ be a function between Banach spaces. If 𝑓 satisfies the functional inequality𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜃‖𝑥‖^𝑝+𝑦^𝑝 (2)for some 𝜃≥0, 𝑝 with 0≤𝑝<1 and for all 𝑥, 𝑦∈𝐸₁, then there exists a unique additive function 𝐴∶𝐸₁→𝐸₂ such that ‖𝑓(𝑥)−𝐴(𝑥)‖≤(2𝜃/(2−2^𝑝))‖𝑥‖^𝑝 for each 𝑥∈𝐸₁. If, in addition, 𝑓(𝑡𝑥) is continuous in 𝑡 for each fixed 𝑥∈𝐸₁, then the function 𝐴 is linear.

This result of Rassias attracted a number of mathematicians who began to be stimulated to investigate the stability problems of functional equations. By regarding a large influence of Ulam, Hyers, and Rassias on the study of stability problems of functional equations, the stability phenomenon proved by Rassias is called the Hyers-Ulam-Rassias stability. For the last thirty years many results concerning the Hyers-Ulam-Rassias stability of various functional equations have been obtained (see [6–17]).

In this paper, we investigate a general stability of the 𝑛-dimensional quadratic and additive functional equation 2𝑓^𝑛_𝑖=1𝑥_𝑖+_{1≤𝑖,𝑗≤𝑛,𝑖≠𝑗}𝑓𝑥_𝑖−𝑥_𝑗=(𝑛+1)^𝑛_𝑖=1𝑓𝑥_𝑖+(𝑛−1)^𝑛_𝑖=1𝑓−𝑥_𝑖 (3) in the fuzzy normed space. We will call each solution of (3) a quadratic-additive function.

In 2006, Jun and Kim [10] proved the stability of (3) by constructing an additive function 𝐴 and a quadratic function 𝑄 separately and by approximating the given function 𝑓 with a quadratic-additive function 𝐹=𝐴+𝑄. In their approach, 𝐴 and 𝑄 approximate the odd part and the even part of 𝑓, respectively. However, their method is not efficient in comparison with our method which approximates the function 𝑓 with the quadratic-additive function 𝐹 simultaneously. Indeed, we introduce a Cauchy sequence {𝐽^𝑛𝑓(𝑥)} by making use of the given function 𝑓, and the sequence {𝐽^𝑛𝑓(𝑥)} converges to a quadratic-additive function 𝐹 which approximates the function 𝑓 in the fuzzy sense. Our idea seems to be a refinement of previous studies.

2. Preliminaries

In 1984, Katsaras [18] defined a fuzzy norm on a linear space to construct a fuzzy structure on the space. Since then, mathematicians have introduced several types of fuzzy norm in different points of view. In particular, adhering to the point of view of Cheng and Mordeson, Bag and Samanta suggested an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type (see [19–21]).

We introduce the definition of a fuzzy normed space to establish a reasonable fuzzy version of stability for the 𝑛-dimensional quadratic and additive functional equation (3) in the fuzzy normed space (cf. [19]).

Definition 1. Let 𝑋 be a real linear space. A function 𝑁∶𝑋×ℝ→[0,1] is said to be a fuzzy norm on 𝑋 if the following conditions are true for all 𝑥, 𝑦∈𝑋 and all 𝑠, 𝑡∈ℝ:(𝐹₁)𝑁(𝑥,𝑡)=0for𝑡≤0; (𝐹₂)𝑥=0  if and only if 𝑁(𝑥,𝑡)=1  for all 𝑡>0;(𝐹₃)𝑁(𝑐𝑥,𝑡)=𝑁(𝑥,𝑡/|𝑐|) if 𝑐≠0;(𝐹₄)𝑁(𝑥+𝑦,𝑠+𝑡)≥min{𝑁(𝑥,𝑠),𝑁(𝑦,𝑡)};(𝐹₅)𝑁(𝑥,⋅) is a nondecreasing function on ℝ and lim_𝑡→∞𝑁(𝑥,𝑡)=1.

The pair (𝑋,𝑁) is called a fuzzy normed space. Let (𝑋,𝑁) be a fuzzy normed space. A sequence {𝑥_𝑛} in 𝑋 is said to be convergent if there exists an 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 write 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 an 𝑛₀∈ℕ such that 𝑁(𝑥_𝑛+𝑝−𝑥_𝑛,𝑡)>1−𝜀 for all 𝑛≥𝑛₀ and all 𝑝∈ℕ. It is known that every convergent sequence in a fuzzy normed space is Cauchy. If every Cauchy sequence in 𝑋 converges in 𝑋, then the fuzzy norm is said to be complete, and the fuzzy normed space is called a fuzzy Banach space.

In 2008, Mirmostafaee and Moslehian first dealt with the fuzzy stability problems for the additive Cauchy equation, Jensen’s functional equation, and the quadratic functional equation (see [22–24]).

3. Fuzzy Stability of (3)

Let (𝑋,𝑁₁) and (𝑌,𝑁₂) be a fuzzy normed space and a fuzzy Banach space, respectively. Assume that 𝑛 is a fixed integer greater than 1. For a given function 𝑓∶𝑋→𝑌, we use the abbreviation𝐷𝑓𝑥₁,𝑥₂,…,𝑥_𝑛∶=2𝑓^𝑛_𝑖=1𝑥_𝑖+_{1≤𝑖,𝑗≤𝑛,𝑖≠𝑗}𝑓𝑥_𝑖−𝑥_𝑗−(𝑛+1)^𝑛_𝑖=1𝑓𝑥_𝑖−(𝑛−1)^𝑛_𝑖=1𝑓−𝑥_𝑖 (4) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋. For a given 𝑞>0, the function 𝑓∶𝑋→𝑌 is called a fuzzy 𝑞-almost quadratic-additive function if𝑁₂𝐷𝑓𝑥₁,…,𝑥_𝑛,𝑡₁+⋯+𝑡_𝑛≥min𝑁₁𝑥₁,𝑡^𝑞₁,…,𝑁₁𝑥_𝑛,𝑡^𝑞_𝑛 (5) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and all 𝑡₁,𝑡₂,…,𝑡_𝑛≥0.

The following theorem gives a fuzzy version of the stability of the 𝑛-dimensional quadratic and additive functional equation 𝐷𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛)=0.

Theorem 2. Let 𝑞 be a positive real number with 𝑞∉{1/2,1}. If a function 𝑓∶𝑋→𝑌 is a fuzzy 𝑞-almost quadratic-additive function from a fuzzy normed space (𝑋,𝑁₁) into a fuzzy Banach space (𝑌,𝑁₂), then there exists a unique quadratic-additive function 𝐹∶𝑋→𝑌 such that 𝑁₂𝐹(𝑥)(𝑥)≥−𝑓,𝑡sup_{0<𝑡′<𝑡}𝑁₁𝑥,2^𝑞𝑛^−𝑞𝑛−𝑛^𝑝𝑞𝑡^′𝑞if𝑞>1,sup_{0<𝑡′<𝑡}𝑁₁2𝑥,^𝑞𝑡^′𝑞𝑛/𝑛²−𝑛^𝑝+𝑛/(𝑛^𝑝−𝑛)^𝑞if1/2<𝑞<1,sup_{0<𝑡′<𝑡}𝑁₁𝑥,2^𝑞𝑛^𝑝−𝑛^2𝑞𝑛^−𝑞𝑡^′𝑞if0<𝑞<1/2 (6) for each 𝑥∈𝑋 and 𝑡>0, where 𝑝=1/𝑞.

Proof. It follows from (𝐹₂), (𝐹₃), and (5) that 𝑁₂𝑓(0),𝑡=𝑁₂𝐷𝑓(0,…,0),(𝑛−1)(𝑛+2)𝑡≥𝑁₁()𝑡0,𝑛−1)(𝑛+2𝑛^𝑞=1 (7) for all 𝑡>0. Thus, it follows from (𝐹₂) that 𝑓(0)=0.
We will split the proof into three parts according to the value of 𝑞, namely, 𝑞>1, 1/2<𝑞<1, or 0<𝑞<1/2.
Case 1. Assume that 𝑞>1 and let 𝐽^𝑚𝑓∶𝑋→𝑌 be a function defined by 𝐽^𝑚𝑓(𝑥)=12𝑛^2𝑚𝑓𝑛^𝑚𝑥+𝑓−𝑛^𝑚𝑥+12𝑛^𝑚𝑓𝑛^𝑚𝑥−𝑓−𝑛^𝑚𝑥 (8) for all 𝑥∈𝑋 and 𝑚∈ℕ₀. Then, 𝐽⁰𝑓(𝑥)=𝑓(𝑥) and by a long manipulation, we get 𝐽^𝑖𝑓(𝑥)−𝐽^𝑖+1𝑓(𝑥)𝑛=−^𝑖+1+1𝐷𝑓𝑛^𝑖𝑥,…,𝑛^𝑖𝑥4𝑛^2𝑖+2+𝑛^𝑖+1−1𝐷𝑓−𝑛^𝑖𝑥,…,−𝑛^𝑖𝑥4𝑛^2𝑖+2 (9) for all 𝑥∈𝑋 and for any 𝑖∈ℕ₀. Together with (𝐹₃), (𝐹₄), and (5), this equation implies that if 𝑚′+𝑚>𝑚≥0 then 𝑁₂𝐽^𝑚𝑓(𝑥)−𝐽^𝑚′+𝑚𝑓(𝑥),^{𝑚′+𝑚−1}_𝑖=𝑚𝑛^𝑝𝑛^𝑖𝑡^𝑝2=𝑁₂^{𝑚′+𝑚−1}_𝑖=𝑚𝐽^𝑖𝑓(𝑥)−𝐽^𝑖+1𝑓(𝑥),^{𝑚′+𝑚−1}_𝑖=𝑚𝑛^𝑝𝑛^𝑖𝑡^𝑝2≥min_{𝑚≤𝑖<𝑚′+𝑚}𝑁₂𝐽^𝑖𝑓(𝑥)−𝐽^𝑖+1𝑓(𝑥),𝑛^𝑝𝑛^𝑖𝑡^𝑝2≥min_{𝑚≤𝑖<𝑚′+𝑚}min𝑁₂−𝑛^𝑖+1+1𝐷𝑓𝑛^𝑖𝑥,…,𝑛^𝑖𝑥4𝑛^2𝑖+2,𝑛^𝑖+1+1𝑛^𝑖𝑝𝑡^𝑝4𝑛^2𝑖+1,𝑁₂𝑛^𝑖+1−1𝐷𝑓−𝑛^𝑖𝑥,…,−𝑛^𝑖𝑥4𝑛^2𝑖+2,𝑛^𝑖+1−1𝑛^𝑖𝑝𝑡^𝑝4𝑛^2𝑖+1≥min_{𝑚≤𝑖<𝑚′+𝑚}𝑁₁𝑛^𝑖𝑥,𝑛^𝑖𝑡=𝑁₁(𝑥,𝑡) (10) for all 𝑥∈𝑋 and for all 𝑡>0.
Let 𝜀>0 be given. Since lim_𝑡→∞𝑁₁(𝑥,𝑡)=1, there exists an 𝑡₀>0 such that 𝑁₁𝑥,𝑡₀≥1−𝜀. (11)We observe that for some 𝑡̃>𝑡₀, the series ∑^∞_𝑖=0(𝑛^𝑝/𝑛)^𝑖(𝑡^𝑝̃/2) converges for 𝑝=1/𝑞<1. Hence, for an arbitrarily given 𝑐>0, there exists an 𝑚₀≥0 such that ^{𝑚′+𝑚−1}_𝑖=𝑚𝑛^𝑝𝑛^𝑖𝑡^𝑝̃2<𝑐 (12) for each 𝑚≥𝑚₀ and 𝑚′>0. By (𝐹₅) and (10), we have 𝑁₂𝐽^𝑚𝑓(𝑥)−𝐽^𝑚′+𝑚𝑓(𝑥),𝑐≥𝑁₂𝐽^𝑚𝑓(𝑥)−𝐽^𝑚′+𝑚𝑓(𝑥),^{𝑚′+𝑚−1}_𝑖=𝑚𝑛^𝑝𝑛^𝑖𝑡^𝑝̃2≥𝑁₁𝑥,𝑡̃≥𝑁₁𝑥,𝑡₀≥1−𝜀 (13)for all 𝑥∈𝑋 and for each 𝑐>0. Hence, {𝐽^𝑚𝑓(𝑥)} is a Cauchy sequence in the fuzzy Banach space (𝑌,𝑁₂), and, thus, we can define a function 𝐹∶𝑋→𝑌 by 𝐹(𝑥)∶=𝑁₂−lim_𝑚→∞𝐽^𝑚𝑓(𝑥) (14)for all 𝑥∈𝑋. Moreover, if we put 𝑚=0 and replace 𝑡 with 2^𝑞𝑡^𝑞/(∑^𝑚′−1_𝑖=0(𝑛^𝑝/𝑛)^𝑖)^𝑞 in (10), then we have 𝑁₂𝑓(𝑥)−𝐽^𝑚′𝑓(𝑥),𝑡≥𝑁₁2𝑥,^𝑞𝑡^𝑞∑^𝑚′−1_𝑖=0(𝑛^𝑝/𝑛)^𝑖𝑞 (15) for all 𝑥∈𝑋 and for any 𝑡>0.
Next, we will show that 𝐹 is a quadratic-additive function. Using (𝐹₄) and (4), we have 𝑁₂𝐷𝐹𝑥₁,…,𝑥_𝑛,𝑁,𝑡≥min₂2𝐹−𝐽^𝑚𝑓^𝑛_𝑖=1𝑥_𝑖,𝑡5,min_{1≤𝑖,𝑗≤𝑛,𝑖≠𝑗}𝑁₂𝐹−𝐽^𝑚𝑓𝑥_𝑖−𝑥_𝑗,𝑡(),5𝑛𝑛−1min_{1≤𝑖≤𝑛}𝑁₂(𝑛+1)𝐽^𝑚𝑓−𝐹𝑥_𝑖,𝑡,5𝑛min_{1≤𝑖≤𝑛}𝑁₂(𝑛−1)𝐽^𝑚𝑓−𝐹−𝑥_𝑖,𝑡,𝑁5𝑛₂𝐷𝐽^𝑚𝑓𝑥₁,𝑥₂,…,𝑥_𝑛,𝑡5 (16) for all 𝑥₁,…,𝑥_𝑛∈𝑋, 𝑡>0, and 𝑚∈ℕ. In view of the definition of 𝐹 and (𝐹₂), the first four terms on the right hand side of (16) tend to 1 as 𝑚→∞. It follows from (4) and (8) that 𝐷𝐽^𝑚𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=12𝑛^2𝑚𝐷𝑓𝑛^𝑚𝑥₁,…,𝑛^𝑚𝑥_𝑛+12𝑛^2𝑚𝐷𝑓−𝑛^𝑚𝑥₁,…,−𝑛^𝑚𝑥_𝑛+12𝑛^𝑚𝐷𝑓𝑛^𝑚𝑥₁,…,𝑛^𝑚𝑥_𝑛−12𝑛^𝑚𝐷𝑓−𝑛^𝑚𝑥₁,…,−𝑛^𝑚𝑥_𝑛 (17) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and 𝑚∈ℕ₀. Hence, by (𝐹₃) and (𝐹₄), we have 𝑁₂𝐷𝐽^𝑚𝑓𝑥₁,𝑥₂,…,𝑥_𝑛,𝑡5≥min𝑁₂𝐷𝑓𝑛^𝑚𝑥₁,…,𝑛^𝑚𝑥_𝑛2𝑛^2𝑚,𝑡,𝑁20₂𝐷𝑓−𝑛^𝑚𝑥₁,…,−𝑛^𝑚𝑥_𝑛2𝑛^2𝑚,𝑡,𝑁20₂𝐷𝑓𝑛^𝑚𝑥₁,…,𝑛^𝑚𝑥_𝑛2𝑛^𝑚,𝑡,𝑁20₂𝐷𝑓−𝑛^𝑚𝑥₁,…,−𝑛^𝑚𝑥_𝑛2𝑛^𝑚,𝑡20 (18) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and 𝑡>0.
By (𝐹₃) and (5), we obtain 𝑁₂𝐷𝑓±𝑛^𝑚𝑥₁,…,±𝑛^𝑚𝑥_𝑛2𝑛^2𝑚,𝑡𝑁20≥min₁𝑥₁,10^−𝑞𝑛^{(2𝑞−1)𝑚−𝑞}𝑡^𝑞𝑁,…,₁𝑥_𝑛,10^−𝑞𝑛^{(2𝑞−1)𝑚−𝑞}𝑡^𝑞,𝑁₂𝐷𝑓±𝑛^𝑚𝑥₁,…,±𝑛^𝑚𝑥_𝑛2𝑛^𝑚,𝑡𝑁20≥min₁𝑥₁,10^−𝑞𝑛^{(𝑞−1)𝑚−𝑞}𝑡^𝑞𝑁,…,₁𝑥_𝑛,10^−𝑞𝑛^{(𝑞−1)𝑚−𝑞}𝑡^𝑞 (19)for all 𝑥₁,…,𝑥_𝑛∈𝑋, 𝑡>0, and 𝑚∈ℕ. Since 𝑞>1, it follows from (𝐹₅) that the last term in (16) also goes to 1 as 𝑚→∞. In view of (16), we conclude that 𝑁₂𝐷𝐹𝑥₁,𝑥₂,…,𝑥_𝑛,𝑡=1 (20) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and 𝑡>0. By (𝐹₂), this implies that 𝐷𝐹𝑥₁,𝑥₂,…,𝑥_𝑛=0 (21) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋.
We will now prove the first inequality in (6). Let 𝑥∈𝑋, let 𝑡>𝑡′>0, and let 0<𝜀<1 be given. Since 𝐹 is the limit of the sequence {𝐽^𝑚𝑓(𝑥)}, there exists an 𝑚∈ℕ such that 𝑁₂𝐹(𝑥)−𝐽^𝑚𝑓(𝑥),𝑡−𝑡^′≥1−𝜀. (22)By (𝐹₄) and (15), we have 𝑁₂𝐹(𝑥)(𝑥)𝑁−𝑓,𝑡≥min₂𝐹(𝑥)−𝐽^𝑚𝑓(𝑥),𝑡−𝑡^′,𝑁₂𝐽^𝑚𝑓(𝑥)−𝑓(𝑥),𝑡^′≥min1−𝜀,𝑁₁2𝑥,^𝑞𝑡′^𝑞∑^𝑚−1_𝑖=0(𝑛^𝑝/𝑛)^𝑖𝑞≥min1−𝜀,𝑁₁𝑥,2^𝑞𝑛^−𝑞𝑛−𝑛^𝑝𝑞𝑡^′𝑞. (23)Because 0<𝜀<1 is arbitrary, we get the first inequality in (6) for the case of 𝑞>1.
Finally, it remains to prove the uniqueness of 𝐹. Let 𝐺∶𝑋→𝑌 be another quadratic-additive function satisfying the first inequality in (6). Then, by (9) and (21), we get 𝐹(𝑥)−𝐽^𝑚𝐹(𝑥)=^𝑚−1_𝑖=0𝐽^𝑖𝐹(𝑥)−𝐽^𝑖+1𝐹(𝑥𝐺(𝑥))=0,−𝐽^𝑚𝐺(𝑥)=^𝑚−1_𝑖=0𝐽^𝑖𝐺(𝑥)−𝐽^𝑖+1𝐺(𝑥)=0 (24) for all 𝑥∈𝑋 and 𝑚∈ℕ. Together with (𝐹₃), (𝐹₄), (𝐹₅), (6), and (8), this implies that 𝑁₂(𝐹(𝑥)−𝐺(𝑥),𝑡)=𝑁₂𝐽^𝑚𝐹(𝑥)−𝐽^𝑚𝐺(𝑥)𝑁,𝑡≥min₂𝐽^𝑚𝐹−𝐽^𝑚𝑓(𝑥),𝑡2,𝑁₂𝐽^𝑚𝑓−𝐽^𝑚𝐺(𝑥),𝑡2≥min𝑁₂𝐹−𝑓𝑛^𝑚𝑥2𝑛^2𝑚,𝑡8,𝑁₂𝑓−𝐺𝑛^𝑚𝑥2𝑛^2𝑚,𝑡8,𝑁₂𝐹−𝑓−𝑛^𝑚𝑥2𝑛^2𝑚,𝑡8,𝑁₂𝑓−𝐺−𝑛^𝑚𝑥2𝑛^2𝑚,𝑡8,𝑁₂𝐹−𝑓𝑛^𝑚𝑥2𝑛^𝑚,𝑡8,𝑁₂𝑓−𝐺𝑛^𝑚𝑥2𝑛^𝑚,𝑡8,𝑁₂𝐹−𝑓−𝑛^𝑚𝑥2𝑛^𝑚,𝑡8,𝑁₂𝑓−𝐺−𝑛^𝑚𝑥2𝑛^𝑚,𝑡8≥sup_{0<𝑡′<𝑡}𝑁₁𝑥,𝑛^{(𝑞−1)𝑚−𝑞}2^−𝑞𝑛−𝑛^𝑝𝑞𝑡′^𝑞⟶1as𝑚⟶∞ (25)for all 𝑥∈𝑋 and 𝑡>0. This implies that 𝑁₂(𝐹(𝑥)−𝐺(𝑥),𝑡)=1 for all 𝑥∈𝑋 and 𝑡>0. Thus, it follows from (𝐹₂) that 𝐹(𝑥)=𝐺(𝑥) (26) for all 𝑥∈𝑋.Case 2. Assume that 1/2<𝑞<1. Let us define a function 𝐽^𝑚𝑓∶𝑋→𝑌 by 𝐽^𝑚𝑓(𝑥)1∶=2𝑛^2𝑚𝑓𝑛^𝑚𝑥+𝑓−𝑛^𝑚𝑥+𝑛^𝑚2𝑓𝑥𝑛^𝑚−𝑥−𝑓𝑛^𝑚 (27) for all 𝑥∈𝑋 and 𝑚∈ℕ₀. Then, we have 𝐽⁰𝑓(𝑥)=𝑓(𝑥), and it follows from (4) and (27) that 𝐽^𝑖𝑓(𝑥)−𝐽^𝑖+1𝑓(𝑥)𝑛=−𝐷𝑓^𝑖𝑥,…,𝑛^𝑖𝑥4𝑛^2𝑖+2−𝐷𝑓−𝑛^𝑖𝑥,…,−𝑛^𝑖𝑥4𝑛^2𝑖+2+𝑛^𝑖4𝑥𝐷𝑓𝑛^𝑖+1𝑥,…,𝑛^𝑖+1−𝑛^𝑖4−𝑥𝐷𝑓𝑛^𝑖+1𝑥,…,−𝑛^𝑖+1 (28) for all 𝑥∈𝑋 and 𝑖∈ℕ₀. If 𝑚′+𝑚>𝑚≥0, then it follows from (𝐹₃), (𝐹₄), and (5) that 𝑁₂𝐽^𝑚𝑓(𝑥)−𝐽^𝑚′+𝑚𝑓(𝑥),^{𝑚′+𝑚−1}_𝑖=𝑚1𝑛2𝑛𝑛^𝑝2𝑖+𝑛2𝑛^𝑝𝑛𝑛^𝑝𝑖𝑡^𝑝=𝑁₂^{𝑚′+𝑚−1}_𝑖=𝑚𝐽^𝑖𝑓(𝑥)−𝐽^𝑖+1𝑓(𝑥)_{𝑚′+𝑚−1𝑖=𝑚},1𝑛2𝑛𝑛^𝑝2𝑖+𝑛2𝑛^𝑝𝑛𝑛^𝑝𝑖𝑡^𝑝≥min_{𝑚≤𝑖<𝑚′+𝑚}min𝑁₂−𝐷𝑓𝑛^𝑖𝑥,…,𝑛^𝑖𝑥4𝑛^2𝑖+2,𝑛^𝑖𝑝𝑡^𝑝4𝑛^2𝑖+1,𝑁₂−𝐷𝑓−𝑛^𝑖𝑥,…,−𝑛^𝑖𝑥4𝑛^2𝑖+2,𝑛^𝑖𝑝𝑡^𝑝4𝑛^2𝑖+1,𝑁₂𝑛^𝑖4𝑥𝐷𝑓𝑛^𝑖+1𝑥,…,𝑛^𝑖+1,𝑛^𝑖+1𝑡^𝑝4𝑛^(𝑖+1)𝑝,𝑁₂−𝑛^𝑖4−𝑥𝐷𝑓𝑛^𝑖+1𝑥,…,−𝑛^𝑖+1,𝑛^𝑖+1𝑡^𝑝4𝑛^(𝑖+1)𝑝≥min_{𝑚≤𝑖<𝑚′+𝑚}min𝑁₁𝑛^𝑖𝑥,𝑛^𝑖𝑡,𝑁₁𝑥𝑛^𝑖+1,𝑡𝑛^𝑖+1=𝑁₁(𝑥,𝑡) (29) for all 𝑥∈𝑋 and 𝑡>0.
By a similar argument of the preceding part, we can define the limit 𝐹(𝑥)∶=𝑁₂-lim_𝑚→∞𝐽^𝑚𝑓(𝑥) of the Cauchy sequence {𝐽^𝑚𝑓(𝑥)} in the fuzzy Banach space 𝑌. Moreover, putting 𝑚=0 and replacing 𝑡 with 𝑡^𝑞/(∑^𝑚′−1_𝑖=0((1/2𝑛)(𝑛^𝑝/𝑛²)^𝑖+(𝑛/2𝑛^𝑝)(𝑛/𝑛^𝑝)^𝑖))^𝑞 in the last inequality yield 𝑁₂𝑓(𝑥)−𝐽^𝑚′𝑓(𝑥),𝑡≥𝑁₁𝑡𝑥,^𝑞∑^𝑚′−1_𝑖=0(1/2𝑛)𝑛^𝑝/𝑛^2𝑖+(𝑛/2𝑛^𝑝)(𝑛/𝑛^𝑝)^𝑖𝑞 (30) for each 𝑥∈𝑋 and for each 𝑚′∈ℕ.
To prove that 𝐹 is a quadratic-additive function, it suffices to show that the last term of (16) tends to 1 as 𝑚→∞. By (4) and (27), we obtain 𝐷𝐽^𝑚𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=12𝑛^2𝑚𝐷𝑓𝑛^𝑚𝑥₁,…,𝑛^𝑚𝑥_𝑛+12𝑛^2𝑚𝐷𝑓−𝑛^𝑚𝑥₁,…,−𝑛^𝑚𝑥_𝑛+𝑛^𝑚2𝑥𝐷𝑓¹𝑛^𝑚𝑥,…,^𝑛𝑛^𝑚−𝑛^𝑚2−𝑥𝐷𝑓¹𝑛^𝑚𝑥,…,−^𝑛𝑛^𝑚 (31) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and 𝑚∈ℕ₀. Hence, it follows from (𝐹₃), (𝐹₄), (𝐹₅), and (5) that 𝑁₂𝐷𝐽^𝑚𝑓𝑥₁,𝑥₂,…,𝑥_𝑛,𝑡5≥min𝑁₂12𝑛^2𝑚𝐷𝑓𝑛^𝑚𝑥₁,…,𝑛^𝑚𝑥_𝑛,𝑡,𝑁20₂12𝑛^2𝑚𝐷𝑓−𝑛^𝑚𝑥₁,…,−𝑛^𝑚𝑥_𝑛,𝑡,𝑁20₂𝑛^𝑚2𝑥𝐷𝑓¹𝑛^𝑚𝑥,…,^𝑛𝑛^𝑚,𝑡,𝑁20₂−𝑛^𝑚2−𝑥𝐷𝑓¹𝑛^𝑚𝑥,…,−^𝑛𝑛^𝑚,𝑡𝑁20≥min₁𝑥₁,10^−𝑞𝑛^{(2𝑞−1)𝑚−𝑞}𝑡^𝑞𝑁,…,₁𝑥_𝑛,10^−𝑞𝑛^{(2𝑞−1)𝑚−𝑞}𝑡^𝑞,𝑁₁𝑥₁,10^−𝑞𝑛^{(1−𝑞)𝑚−𝑞}𝑡^𝑞𝑁,…,₁𝑥_𝑛,10^−𝑞𝑛^{(1−𝑞)𝑚−𝑞}𝑡^𝑞⟶1as𝑚⟶∞ (32) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and 𝑡>0, since 1/2<𝑞<1. Hence, as we did for the preceding case of 𝑞>1, it follows from (16) that 𝐷𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛)=0 for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋.
By a similar way as the first inequality in (6) follows from (15), we see that the second inequality of (6) follows from (30).
We will prove the uniqueness of 𝐹. Let 𝐺 be another quadratic-additive function satisfying the second inequality in (6). Notice that (24) also holds true for the case of 1/2<𝑞<1. Then, by (𝐹₃), (𝐹₄), (𝐹₅), (6), (24), and (27), we have 𝑁₂(𝐹(𝑥)−𝐺(𝑥),𝑡)=𝑁₂𝐽^𝑚𝐹(𝑥)−𝐽^𝑚𝐺(𝑥)𝑁,𝑡≥min₂𝐽^𝑚𝐹−𝐽^𝑚𝑓(𝑥),𝑡2,𝑁₂𝐽^𝑚𝑓−𝐽^𝑚𝐺(𝑥),𝑡2≥min𝑁₂𝐹−𝑓𝑛^𝑚𝑥2𝑛^2𝑚,𝑡8,𝑁₂𝑓−𝐺𝑛^𝑚𝑥2𝑛^2𝑚,𝑡8,𝑁₂𝐹−𝑓−𝑛^𝑚𝑥2𝑛^2𝑚,𝑡8,𝑁₂𝑓−𝐺−𝑛^𝑚𝑥2𝑛^2𝑚,𝑡8,𝑁₂𝑛^𝑚2𝑥𝐹−𝑓𝑛^𝑚,𝑡8,𝑁₂𝑛^𝑚2𝑥𝑓−𝐺𝑛^𝑚,𝑡8,𝑁₂−𝑛^𝑚2−𝑥𝐹−𝑓𝑛^𝑚,𝑡8,𝑁₂−𝑛^𝑚2−𝑥𝑓−𝐺𝑛^𝑚,𝑡8≥minsup_{0<𝑡′<𝑡}𝑁₁2𝑥,^−𝑞𝑛^{(2𝑞−1)𝑚}𝑡^′𝑞𝑛/𝑛²−𝑛^𝑝+𝑛/(𝑛^𝑝−𝑛)^𝑞,sup_{0<𝑡′<𝑡}𝑁₁2𝑥,^−𝑞𝑛^{(1−𝑞)𝑚}𝑡^′𝑞𝑛/𝑛²−𝑛^𝑝+𝑛/(𝑛^𝑝−𝑛)^𝑞⟶1as𝑚⟶∞ (33)for all 𝑥∈𝑋 and 𝑡>0, since lim_𝑚→∞𝑛^{(2𝑞−1)𝑚}=lim_𝑚→∞𝑛^{(1−𝑞)𝑚}=∞ in this case. This implies that 𝑁₂(𝐹(𝑥)−𝐺(𝑥),𝑡)=1 for all 𝑥∈𝑋 and 𝑡>0, and hence 𝐹(𝑥)=𝐺(𝑥) for all 𝑥∈𝑋 by (𝐹₂).Case 3. Finally, we consider the case of 0<𝑞<1/2. Let us define a function 𝐽^𝑚𝑓∶𝑋→𝑌 by 𝐽^𝑚𝑓(𝑥)𝑛∶=^2𝑚2𝑓𝑥𝑛^𝑚−𝑥+𝑓𝑛^𝑚+𝑛^𝑚2𝑓𝑥𝑛^𝑚−𝑥−𝑓𝑛^𝑚 (34) for all 𝑥∈𝑋 and 𝑚∈ℕ₀. Then, we have 𝐽⁰𝑓(𝑥)=𝑓(𝑥), and a somewhat tedious manipulation yields 𝐽^𝑖𝑓(𝑥)−𝐽^𝑖+1𝑓(𝑥)=𝑛^𝑖𝑛^𝑖+14𝑥𝐷𝑓𝑛^𝑖+1𝑥,…,𝑛^𝑖+1+𝑛^𝑖𝑛^𝑖−14−𝑥𝐷𝑓𝑛^𝑖+1𝑥,…,−𝑛^𝑖+1 (35) for all 𝑥∈𝑋 and 𝑖∈ℕ₀.
If 𝑚 and 𝑚′ are nonnegative integers with 𝑚′+𝑚>𝑚, then it follows from (𝐹₃), (𝐹₄), and (5) that 𝑁₂𝐽^𝑚𝑓(𝑥)−𝐽^𝑚′+𝑚𝑓(𝑥),^{𝑚′+𝑚−1}_𝑖=𝑚𝑛²𝑛^𝑝𝑖+1𝑡^𝑝2𝑛=𝑁₂^{𝑚′+𝑚−1}_𝑖=𝑚𝐽^𝑖𝑓(𝑥)−𝐽^𝑖+1𝑓(𝑥),^{𝑚′+𝑚−1}_𝑖=𝑚𝑛²𝑛^𝑝𝑖+1𝑡^𝑝2𝑛≥min_{𝑚≤𝑖<𝑚′+𝑚}min𝑁₂𝑛^𝑖𝑛^𝑖+14𝑥𝐷𝑓𝑛^𝑖+1𝑥,…,𝑛^𝑖+1,𝑛^𝑖+1𝑛^𝑖+1𝑡^𝑝4𝑛^(𝑖+1)𝑝,𝑁₂𝑛^𝑖𝑛^𝑖−14−𝑥𝐷𝑓𝑛^𝑖+1𝑥,…,−𝑛^𝑖+1,𝑛^𝑖+1𝑛^𝑖−1𝑡^𝑝4𝑛^(𝑖+1)𝑝≥𝑁₁𝑥𝑛^𝑖+1,𝑡𝑛^𝑖+1=𝑁₁(𝑥,𝑡) (36)for all 𝑥∈𝑋 and 𝑡>0.
Similarly to the preceding cases, let us define a function 𝐹∶𝑋→𝑌 by 𝐹(𝑥)∶=𝑁₂-lim_𝑚→∞𝐽^𝑚𝑓(𝑥). Putting 𝑚=0 and replacing 𝑡 with 2^𝑞𝑛^𝑞𝑡^𝑞/(∑^𝑚′−1_𝑖=0(𝑛²/𝑛^𝑝)^𝑖+1)^𝑞 in the last inequality yield 𝑁₂𝑓(𝑥)−𝐽^𝑚′𝑓(𝑥),𝑡≥𝑁₁2𝑥,^𝑞𝑛^𝑞𝑡^𝑞∑^𝑚′−1_𝑖=0(𝑛²/𝑛^𝑝)^𝑖+1𝑞 (37) for all 𝑥∈𝑋 and 𝑡>0. Notice that 𝐷𝐽^𝑚𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=𝑛^2𝑚2𝑥𝐷𝑓¹𝑛^𝑚𝑥,…,^𝑛𝑛^𝑚+𝑛^2𝑚2−𝑥𝐷𝑓¹𝑛^𝑚𝑥,…,−^𝑛𝑛^𝑚+𝑛^𝑚2𝑥𝐷𝑓¹𝑛^𝑚𝑥,…,^𝑛𝑛^𝑚−𝑛^𝑚2−𝑥𝐷𝑓¹𝑛^𝑚𝑥,…,−^𝑛𝑛^𝑚 (38) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋. Hence, it follows from (𝐹₃), (𝐹₄) and (5) that 𝑁₂𝐷𝐽^𝑚𝑓𝑥₁,𝑥₂,…,𝑥_𝑛,𝑡5≥min𝑁₂𝑛^2𝑚2𝑥𝐷𝑓¹𝑛^𝑚𝑥,…,^𝑛𝑛^𝑚,𝑡,𝑁20₂𝑛^2𝑚2−𝑥𝐷𝑓¹𝑛^𝑚𝑥,…,−^𝑛𝑛^𝑚,𝑡,𝑁20₂𝑛^𝑚2𝑥𝐷𝑓¹𝑛^𝑚𝑥,…,^𝑛𝑛^𝑚,𝑡,𝑁20₂−𝑛^𝑚2−𝑥𝐷𝑓¹𝑛^𝑚𝑥,…,−^𝑛𝑛^𝑚,𝑡𝑁20≥min₁𝑥₁,10^−𝑞𝑛^{(1−2𝑞)𝑚−𝑞}𝑡^𝑞𝑁,…,₁𝑥_𝑛,10^−𝑞𝑛^{(1−2𝑞)𝑚−𝑞}𝑡^𝑞,𝑁₁𝑥₁,10^−𝑞𝑛^{(1−𝑞)𝑚−𝑞}𝑡^𝑞𝑁,…,₁𝑥_𝑛,10^−𝑞𝑛^{(1−𝑞)𝑚−𝑞}𝑡^𝑞⟶1as𝑚⟶∞ (39)for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and 𝑡>0, since 0<𝑞<1/2. Thus, all terms on the right hand side of (16) tend to 1 as 𝑚→∞. Therefore, it follows from (𝐹₂) and (16) that 𝐷𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛)=0 for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋. Moreover, using the same argument as in the first case, the third inequality in (6) follows from (37) for the case of 0<𝑞<1/2.
We will prove the uniqueness of 𝐹. Let 𝐺∶𝑋→𝑌 be another quadratic-additive function satisfying the third inequality in (6). Then, (𝐹₃), (𝐹₄), (𝐹₅), (6), (24), and by (34), we get 𝑁₂(𝐹(𝑥)(𝑥))𝑁−𝐺,𝑡≥min₂𝐽^𝑚𝐹−𝐽^𝑚𝑓(𝑥),𝑡2,𝑁₂𝐽^𝑚𝑓−𝐽^𝑚𝐺𝑥),𝑡(2≥min𝑁₂𝑛^2𝑚2𝑥𝐹−𝑓𝑛^𝑚,𝑡8,𝑁₂𝑛^2𝑚2𝑥𝑓−𝐺𝑛^𝑚,𝑡8,𝑁₂𝑛^2𝑚2−𝑥𝐹−𝑓𝑛^𝑚,𝑡8,𝑁₂𝑛^2𝑚2−𝑥𝑓−𝐺𝑛^𝑚,𝑡8,𝑁₂𝑛^𝑚2𝑥𝐹−𝑓𝑛^𝑚,𝑡8,𝑁₂𝑛^𝑚2𝑥𝑓−𝐺𝑛^𝑚,𝑡8,𝑁₂−𝑛^𝑚2−𝑥𝐹−𝑓𝑛^𝑚,𝑡8,𝑁₂−𝑛^𝑚2−𝑥𝑓−𝐺𝑛^𝑚,𝑡8≥sup_{0<𝑡′<𝑡}𝑁₁𝑥,2^−𝑞𝑛^𝑝−𝑛^2𝑞𝑛^{(1−2𝑞)𝑚−𝑞}𝑡′^𝑞⟶1as𝑚⟶∞ (40) for all 𝑥∈𝑋 and 𝑡>0. This implies that 𝑁₂(𝐹(𝑥)−𝐺(𝑥),𝑡)=1 for all 𝑥∈𝑋 and 𝑡>0, and hence 𝐹(𝑥)=𝐺(𝑥) for all 𝑥∈𝑋 by (𝐹₂).

4. Applications

Corollary 3. If an even function 𝑓∶𝑋→𝑌 satisfies all of the conditions of Theorem 2, then there exists a unique quadratic function 𝐹∶𝑋→𝑌 such that 𝑁₂𝐹(𝑥)−𝑓(𝑥),𝑡≥sup_{0<𝑡′<𝑡}𝑁₁𝑥,2^𝑞𝑛^𝑝−𝑛^2𝑞𝑛^−𝑞𝑡′^𝑞 (41) for all 𝑥∈𝑋 and 𝑡>0, where 𝑞∉{1/2,1}.

Proof. Let 𝐽^𝑚𝑓∶𝑋→𝑌 be defined as in Theorem 2. Since 𝑓 is an even function, it follows from (8), (27), and (34) that 𝐽^𝑚𝑓(𝑥)=12𝑛^2𝑚𝑓𝑛^𝑚𝑥+𝑓−𝑛^𝑚𝑥if𝑛𝑞>1/2,𝑞≠1,_2𝑚2𝑓𝑥𝑛^𝑚−𝑥+𝑓𝑛^𝑚if0<𝑞<1/2 (42) for all 𝑥∈𝑋. Notice that 𝐽⁰𝑓(𝑥)=𝑓(𝑥) and 𝐽^𝑖𝑓(𝑥)−𝐽^𝑖+1𝑓(𝑥)=−𝐷𝑓𝑛^𝑖𝑥,…,𝑛^𝑖𝑥4𝑛^2𝑖+2−𝐷𝑓−𝑛^𝑖𝑥,…,−𝑛^𝑖𝑥4𝑛^2𝑖+2if𝑛𝑞>1/2,𝑞≠1,_2𝑖4𝑥𝐷𝑓𝑛^𝑖+1𝑥,…,𝑛^𝑖+1+𝑛^2𝑖4−𝑥𝐷𝑓𝑛^𝑖+1𝑥,…,−𝑛^𝑖+1if0<𝑞<1/2 (43)for all 𝑥∈𝑋 and 𝑖∈ℕ₀. From these equalities, using the similar method as presented in Theorem 2, we obtain the quadratic-additive function 𝐹 satisfying (41). Notice that 𝐹(𝑥)∶=𝑁₂-lim_𝑛→∞𝐽^𝑛𝑓(𝑥) for all 𝑥∈𝑋, 𝐹 is even, and 𝐷𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛)=0 for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋. Hence, we get 𝐹(𝑥)𝑦=1𝑥+𝑦+𝐹𝑥−𝑦−2𝐹−2𝐹2𝐷𝐹𝑥,𝑦,0,…,0=0 (44) for all 𝑥, 𝑦∈𝑋. This implies that 𝐹 is a quadratic function.

Corollary 4. If an odd function 𝑓∶𝑋→𝑌 satisfies all of the conditions of Theorem 2, then there exists a unique additive function 𝐹∶𝑋→𝑌 such that 𝑁₂𝐹(𝑥)−𝑓(𝑥),𝑡≥sup_{0<𝑡′<𝑡}𝑁₁𝑥,2^𝑞𝑛^−𝑞𝑛−𝑛^𝑝𝑞𝑡^′𝑞 (45) for all 𝑥∈𝑋 and 𝑡>0, where 𝑝=1/𝑞 and 𝑞∉{1/2,1}.

Proof. Let 𝐽^𝑚𝑓∶𝑋→𝑌 be defined as in Theorem 2. Since 𝑓 is an odd function, we obtain 𝐽^𝑚𝑓(𝑥)=12𝑛^𝑚𝑓𝑛^𝑚𝑥−𝑓−𝑛^𝑚𝑥if𝑛𝑞>1,_𝑚2𝑓𝑥𝑛^𝑚−𝑥−𝑓𝑛^𝑚if0<𝑞<1𝑞≠1/2 (46) for all 𝑥∈𝑋. Notice that 𝐽⁰𝑓(𝑥)=𝑓(𝑥) and 𝐽^𝑖𝑓(𝑥)−𝐽^𝑖+1𝑓(𝑥)=−14𝑛^𝑖+1𝐷𝑓𝑛^𝑖𝑥,…,𝑛^𝑖𝑥+14𝑛^𝑖+1𝐷𝑓−𝑛^𝑖𝑥,…,−𝑛^𝑖𝑥if𝑛𝑞>1,_𝑖4𝑥𝐷𝑓𝑛^𝑖+1𝑥,…,𝑛^𝑖+1−𝑛^𝑖4−𝑥𝐷𝑓𝑛^𝑖+1𝑥,…,−𝑛^𝑖+1if0<𝑞<1,𝑞≠1/2 (47) for all 𝑥∈𝑋 and 𝑖∈ℕ₀. From these equalities, by using a similar method in Theorem 2, we obtain the quadratic-additive function 𝐹 satisfying (45). Notice that 𝐹(𝑥)∶=𝑁₂-lim_𝑛→∞𝐽^𝑛𝑓(𝑥) for all 𝑥∈𝑋, 𝐹 is odd, and 𝐷𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛)=0 for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋. Hence, we get 𝐹(𝑥)𝑦=1𝑥+𝑦−𝐹−𝐹2𝐷𝐹𝑥,−𝑦,0,…,0=0 (48) for all 𝑥, 𝑦∈𝑋. This implies that 𝐹 is an additive function.

We can use Theorem 2 to prove a classical result in the framework of normed spaces. Let (𝑋,‖⋅‖) be a normed space and let (𝑌,‖⋅‖) be a Banach space. Then we can define fuzzy norms by𝑁_𝑋()=0𝑥,𝑡if‖𝑥‖,1𝑡≤if‖𝑥‖,𝑁𝑡>_𝑌=0𝑦,𝑡if𝑦,1𝑡≤if𝑡>𝑦, (49) where 𝑥∈𝑋, 𝑦∈𝑌 and 𝑡∈ℝ (see [22]). Suppose a function 𝑓∶𝑋→𝑌 satisfies𝐷𝑓𝑥₁,𝑥₂,…,𝑥_𝑛≤𝑥₁^𝑝+𝑥₂^𝑝+⋯+𝑥_𝑛^𝑝 (50) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋, where 𝑝>0 and 𝑝∉{1,2}. Let 𝑁_𝑌 be a fuzzy norm on 𝑌. Then we get𝑁_𝑌𝐷𝑓𝑥₁,…,𝑥_𝑛,𝑡₁+⋯+𝑡_𝑛=0if𝑡₁+⋯+𝑡_𝑛≤𝐷𝑓𝑥₁,…,𝑥_𝑛,1if𝑡₁+⋯+𝑡_𝑛>𝐷𝑓𝑥₁,…,𝑥_𝑛 (51) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and 𝑡₁,…,𝑡_𝑛∈ℝ.

Consider the case𝑁_𝑌𝐷𝑓𝑥₁,…,𝑥_𝑛,𝑡₁+⋯+𝑡_𝑛=0. (52)

This implies that𝑥₁^𝑝+𝑥₂^𝑝+⋯+𝑥_𝑛^𝑝≥𝐷𝑓𝑥₁,𝑥₂,…,𝑥_𝑛≥𝑡₁+⋯+𝑡_𝑛. (53)

Hence, there exists an 𝑖∈{1,…,𝑛} such that ‖𝑥_𝑖‖^𝑝≥𝑡_𝑖. Thus, with 𝑞=1/𝑝, we havemin𝑁_𝑋𝑥₁,𝑡^𝑞₁,…,𝑁_𝑋𝑥_𝑛,𝑡^𝑞_𝑛=0 (54)

for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 and 𝑡₁,…,𝑡_𝑛>0. Therefore, the inequality𝑁_𝑌𝐷𝑓𝑥₁,…,𝑥_𝑛,𝑡₁+⋯+𝑡_𝑛≥min𝑁_𝑋𝑥₁,𝑡^𝑞₁,…,𝑁_𝑋𝑥_𝑛,𝑡^𝑞_𝑛 (55)

holds true. That is, 𝑓 is a fuzzy 𝑞-almost quadratic-additive function, and by Theorem 2, we obtain the following stability result.

Corollary 5. Let (𝑋,‖⋅‖) be a normed linear space and let (𝑌,‖⋅‖) be a Banach space. Assume that 𝑛 is an integer greater than 1 and 𝑝 is a positive real number with 𝑝∉{1,2}. If a function 𝑓∶𝑋→𝑌 satisfies 𝐷𝑓𝑥₁,𝑥₂,…,𝑥_𝑛≤𝑥₁^𝑝+𝑥₂^𝑝+⋯+𝑥_𝑛^𝑝 (56) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋, then there exists a unique quadratic-additive function 𝐹∶𝑋→𝑌 such that (𝑥)(𝑥)‖=𝑛‖𝐹−𝑓2𝑛^𝑝−𝑛²‖𝑥‖^𝑝if𝑛𝑝>2,2|𝑛^𝑝|+𝑛−𝑛2𝑛²−𝑛^𝑝‖𝑥‖^𝑝if𝑛‖𝑥‖1<𝑝<2,_𝑝2(𝑛−𝑛^𝑝)if𝑝<1 (57) for all 𝑥∈𝑋.

Acknowledgment

The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2011-0004919).