Approximate Quadratic-Additive Mappings in Fuzzy Normed Spaces

Abstract

We examine the generalized Hyers-Ulam stability of the following functional equation: 2𝑓(𝑥+𝑦+𝑧+𝑤)+𝑓(−𝑥−𝑦+𝑧+𝑤)+𝑓(−𝑥+𝑦−𝑧+𝑤)+𝑓(−𝑥+𝑦+𝑧−𝑤)+𝑓(𝑥−𝑦−𝑧+𝑤)+𝑓(𝑥−𝑦+𝑧−𝑤)+𝑓(𝑥+𝑦−𝑧−𝑤)−5𝑓(𝑥)−3𝑓(−𝑥)−5𝑓(𝑦)−3𝑓(−𝑦)−5𝑓(𝑧)−3𝑓(−𝑧)−5𝑓(𝑤)−3𝑓(−𝑤)=0, in the fuzzy normed spaces with the fixed point method.

1. Introduction

The problem of stability for functional equations originated from questions of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] had answered affirmatively the question of Ulam for Banach spaces. The theorem of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. Thereafter, many interesting results of the generalized Hyers-Ulam stability to a number of functional equations and mappings have been investigated. Especially, Cădariu and Radu [5] observed that the existence of the solution for a functional equation and the estimation of the difference with the given mapping can be obtained from the fixed point alternative. This method is called a fixed point method. Also, they [6, 7] applied this method to prove the stability theorems of the additive functional equation.

Katsaras [8] 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, following Cheng and Mordeson, gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type [9–11]. In 2008, Mirmostafaee and Moslehian [12, 13] obtained a fuzzy stability for the additive functional equation and for the quadratic functional equation.

On the other hand, there are some papers where several results of stability for different functional equations are proved in probabilistic metric and random normed spaces (see, e.g., [14–17]); after that the results were established in fuzzy normed spaces or in non-Archimedean fuzzy normed spaces [18–21]. In these papers except [20], the fixed point method is used. Moreover, in some of them another type of metric is used (see, e.g., [17]).

In this paper, we take into account the generalized Hyers-Ulam stability of the following quadratic-additive type functional equation: (𝑥)()𝑦(𝑧)()(𝑤)()2𝑓𝑥+𝑦+𝑧+𝑤+𝑓−𝑥−𝑦+𝑧+𝑤+𝑓−𝑥+𝑦−𝑧+𝑤+𝑓−𝑥+𝑦+𝑧−𝑤+𝑓𝑥−𝑦−𝑧+𝑤+𝑓𝑥−𝑦+𝑧−𝑤+𝑓𝑥+𝑦−𝑧−𝑤−5𝑓−3𝑓−𝑥−5𝑓−3𝑓−𝑦−5𝑓−3𝑓−𝑧−5𝑓−3𝑓−𝑤=0, (1) in the fuzzy normed spaces via the fixed point method. First of all, it is known that if a mapping 𝑓 satisfies the functional equation (1), then 𝑓 is quadratic-additive mapping in [22]. Thus the functional equation (1) may be called the quadratic-additive type functional equation and the general solution of functional equation (1) may be called the quadratic-additive mapping. However, the stability problem for the functional equation (1.1) in [22] is not investigated and so in this paper we deal with the stability of this equation.

2. Preliminaries

We first introduce one of the fundamental results of the fixed point theory. For the proof, we refer to [23] or [24].

Theorem 1 (the fixed point alternative). Assume that (𝑋,𝑑) is a complete generalized metric space and Λ∶𝑋→𝑋 is a strict contraction with the Lipschitz constant 𝐿<1. If there exists a nonnegative integer 𝑛₀ such that 𝑑(Λ^𝑛0+1𝑥,Λ^𝑛0𝑥)<∞ for some 𝑥∈𝑋, then the following statements are true. (F₁)The sequence {Λ^𝑛𝑥} converges to a fixed point 𝑥^∗ of Λ.(F₂)𝑥^∗ is the unique fixed point of Λ in 𝑋^∗={𝑦∈𝑋∣𝑑(Λ^𝑛0𝑥,𝑦)<∞}.(F₃)If 𝑦∈𝑋^∗, then 𝑑𝑦,𝑥^∗≤1𝑑.1−𝐿Λ𝑦,𝑦 (2)

We now introduce the definition of fuzzy normed spaces to establish a reasonable fuzzy stability for the quadratic and additive functional equation (1) in the fuzzy normed spaces (cf. [9]).

Definition 2. 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: (N₁)𝑁(𝑥,𝑡)=0 for all 𝑥∈𝑋 and 𝑡≤0;(N₂)𝑥=0 if and only if 𝑁(𝑥,𝑡)=1 for all 𝑡>0;(N₃)𝑁(𝑐𝑥,𝑡)=𝑁(𝑥,𝑡/|𝑐|) for all 𝑥∈𝑋 and 𝑐,𝑡∈ℝ with 𝑐≠0;(N₄)𝑁(𝑥+𝑦,𝑠+𝑡)≥min{𝑁(𝑥,𝑠),𝑁(𝑦,𝑡)} for all 𝑥,𝑦∈𝑋 and 𝑠,𝑡∈ℝ;(N₅)𝑁(𝑥,⋅) is a nondecreasing function on ℝ and lim_𝑡→∞𝑁(𝑥,𝑡)=1 for all 𝑥∈𝑋.

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 this paper, we note that the triangular norm 𝑇=min is used (see the definition of the fuzzy norm, in the axiom (𝑁₄)), while in some recent papers properties of generalized Hyers-Ulam stability by taking other triangular norms have been discussed (e.g., of Hadžić type, in [25] ).

3. Generalized Hyers-Ulam Stability of (1)

Let (𝑋,𝑁) and (𝑌,𝑁^′) be a fuzzy normed space and a fuzzy Banach space, respectively. For a given mapping 𝑓∶𝑋→𝑌, we use the abbreviation (𝑥)()𝑦(𝑧)()(𝑤)()𝐷𝑓𝑥,𝑦,𝑧,𝑤∶=2𝑓𝑥+𝑦+𝑧+𝑤+𝑓−𝑥−𝑦+𝑧+𝑤+𝑓−𝑥+𝑦−𝑧+𝑤+𝑓−𝑥+𝑦+𝑧−𝑤+𝑓𝑥−𝑦−𝑧+𝑤+𝑓𝑥−𝑦+𝑧−𝑤+𝑓𝑥+𝑦−𝑧−𝑤−5𝑓−3𝑓−𝑥−5𝑓−3𝑓−𝑦−5𝑓−3𝑓−𝑧−5𝑓−3𝑓−𝑤 (3) for all 𝑥,𝑦,𝑧,𝑤∈𝑋.

In the following theorem, we investigate the stability problems of the functional equation (1) between fuzzy normed spaces.

Theorem 3. Let (𝑋,𝑁) and (𝑍,𝑁^′′) be fuzzy normed spaces and let (𝑌,𝑁^′) be a fuzzy Banach space. Assume that a mapping 𝜑∶𝑋⁴→𝑍 satisfies one of the following conditions: (i)𝑁^′′(𝛼𝜑(𝑥,𝑦,𝑧,𝑤),𝑡)≤𝑁^′′(𝜑(2𝑥,2𝑦,2𝑧,2𝑤),𝑡)≤𝑁^′′(𝛼^′𝜑(𝑥,𝑦,𝑧,𝑤),𝑡) for some 1≤𝛼^′≤𝛼<2;(ii)𝑁^′′(𝛼𝜑(𝑥,𝑦,𝑧,𝑤),𝑡)≤𝑁^′′(𝜑(2𝑥,2𝑦,2𝑧,2𝑤),𝑡) for some 0<𝛼<1;(iii)𝑁^′′(𝜑(2𝑥,2𝑦,2𝑧,2𝑤),𝑡)≤𝑁^′′(𝛼𝜑(𝑥,𝑦,𝑧,𝑤),𝑡) for some 𝛼>4 for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝑡>0. If a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies 𝑁^′𝐷𝑓𝑥,𝑦,𝑧,𝑤,𝑡≥𝑁^′′𝜑𝑥,𝑦,𝑧,𝑤,𝑡 (4) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝑡>0, then there exists a unique quadratic-additive mapping 𝐹∶𝑋→𝑌 such that 𝑁^′𝐹(𝑥)(𝑥)≥𝑀(()𝑡)(𝑖)(),𝑀(()𝑡)()−𝑓,𝑡𝑥,22−𝛼𝑖𝑓𝜑𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠or𝑖𝑖𝑥,2𝛼−4𝑖𝑓𝜑𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠𝑖𝑖𝑖 (5) for all 𝑥∈𝑋 and 𝑡>0, where 𝑀()𝑁𝑥,𝑡∶=min^′′𝜑(𝑥,𝑥,0,0),𝑡,𝑁^′′𝜑(−𝑥,−𝑥,0,0),𝑡. (6) Moreover, if 𝑁^′′(𝜑(𝑥,𝑦,0,0),𝑡) is continuous in 𝑥,𝑦 under the condition (𝑖𝑖), then the mapping 𝑓 is a quadratic-additive mapping.

Proof. We will take into account three different cases for the assumption of 𝜑.
Case 1. Assume that 𝜑 satisfies the condition (i). We consider the set of functions 𝑆∶=𝑔∶𝑋⟶𝑌∣𝑔(0)=0 (7) and introduce a generalized metric on 𝑆 by 𝑑𝑔,ℎ∶=inf𝑢∈ℝ⁺∣𝑁^′𝑔(𝑥)(𝑥)().−ℎ,𝑢𝑡≥𝑀𝑥,𝑡∀𝑥∈𝑋 (8)
We first prove that 𝑑 is a generalized metric on 𝑆. If 𝑑(𝑔,ℎ)=0, that is, inf𝑢∈ℝ⁺∣𝑁^′𝑔(𝑥)−ℎ(𝑥),𝑢𝑡≥𝑀(𝑥,𝑡)∀𝑥∈𝑋=0, (9) then we see that 𝑁^′𝑔(𝑥)−ℎ(𝑥),𝑢𝑡≥𝑀(𝑥,𝑡) (10) for all 𝑥∈𝑋 and all 𝑢>0, which means that 𝑁^′𝑔(𝑥)(𝑥)𝑡−ℎ,𝑡≥𝑀𝑥,𝑢 (11) for all 𝑥∈𝑋 and all 𝑢>0. It follows that 𝑁^′𝑔(𝑥)−ℎ(𝑥),𝑡≥lim_𝑢→0𝑀𝑡𝑥,𝑢=1. (12) So we get 𝑔(𝑥)=ℎ(𝑥) for all 𝑥∈𝑋.
Conversely, if 𝑔(𝑥)=ℎ(𝑥) for all 𝑥∈𝑋, then we have 𝑁^′𝑔(𝑥)−ℎ(𝑥),𝑢𝑡=𝑁^′(0,𝑢𝑡)=1≥𝑀(𝑥,𝑡) (13) for all 𝑢 and 𝑡. So we know that 𝑑𝑔,ℎ=inf𝑢∈ℝ⁺∣𝑁^′𝑔(𝑥)(𝑥)()−ℎ,𝑢𝑡≥𝑀𝑥,𝑡∀𝑥∈𝑋=0. (14)
Of course, it is easily checked that 𝑑(𝑔,ℎ)=𝑑(ℎ,𝑔) for all 𝑔,ℎ∈𝑆.
Let 𝑢,𝑣>0 such that 𝑑(𝑓,𝑔)<𝑢 and 𝑑(𝑔,ℎ)<𝑣. Then 𝑁^′𝑓(𝑥)(𝑥)(),𝑁−𝑔,𝑢𝑡≥𝑀𝑥,𝑡_′𝑔(𝑥)−ℎ(𝑥),𝑣𝑡≥𝑀(𝑥,𝑡) (15) for all 𝑥∈𝑋 and all 𝑡>0. Thus we find that 𝑁^′𝑓(𝑥)(𝑥),()𝑡𝑁−ℎ𝑢+𝑣≥min^′𝑓(𝑥)−𝑔(𝑥),𝑢𝑡,𝑁^′𝑔(𝑥)(𝑥)().−ℎ,𝑣𝑡≥𝑀𝑥,𝑡 (16) This implies that 𝑢+𝑣≥𝑑(𝑓,ℎ). Hence we yield that 𝑑(𝑓,ℎ)≤𝑑(𝑓,𝑔)+𝑑(𝑔,ℎ). Therefore 𝑑 is a generalized metric on 𝑆.
Now if we define a function 𝐽∶𝑆→𝑆 by (𝑥)1𝐽𝑓∶=8𝑓()(+12𝑥+𝑓−2𝑥)4𝑓(2𝑥)−𝑓(−2𝑥) (17) for all 𝑥∈𝑋, then we have 𝐽^𝑛𝑓(𝑥)=2^−2𝑛−1𝑓2^𝑛𝑥+𝑓−2^𝑛𝑥+2^−𝑛−1𝑓2^𝑛𝑥−𝑓−2^𝑛𝑥 (18) for all 𝑥∈𝑋 and all 𝑛∈ℕ∪{0}.
For any 𝑓,𝑔∈𝑆, let 𝑢∈[0,∞] be an arbitrary constant with 𝑑(𝑔,𝑓)≤𝑢. The definition of 𝑑 provides that, for 0<𝛼<2, 𝑁^′𝐽𝑔(𝑥)−𝐽𝑓(𝑥),𝛼𝑢𝑡2=𝑁^′38𝑔()()−12𝑥−𝑓2𝑥8𝑔(−2𝑥)−𝑓(−2𝑥),𝛼𝑢𝑡2≥min𝑁^′38𝑔(2𝑥)−𝑓(2𝑥),3𝛼𝑢𝑡8,𝑁^′18𝑔(−2𝑥)−𝑓(−2𝑥),𝛼𝑢𝑡8=min𝑁^′𝑔()(),𝑁2𝑥−𝑓2𝑥,𝛼𝑢𝑡_′𝑔()(){𝑀()(()−2𝑥−𝑓−2𝑥,𝛼𝑢𝑡≥min2𝑥,𝛼𝑡,𝑀−2𝑥,𝛼𝑡)}≥𝑀𝑥,𝑡 (19) for all 𝑥∈𝑋, which implies that 𝑑(𝐽𝑓,𝐽𝑔)≤(𝛼/2)𝑑(𝑓,𝑔). Thus 𝐽 is a strictly contractive self-mapping of 𝑆 with the Lipschitz constant 𝛼/2.
Moreover, by (4), we see that 𝑁^′𝑓(𝑥)(𝑥),𝑡−𝐽𝑓4=𝑁^′−5𝐷𝑓(𝑥,𝑥,0,0)+3𝐷𝑓(−𝑥,−𝑥,0,0),𝑡324≥min𝑁^′−5𝐷𝑓(𝑥,𝑥,0,0),325𝑡,𝑁32_′3𝐷𝑓(−𝑥,−𝑥,0,0),323𝑡𝑁32≥min^′(),𝑁𝐷𝑓𝑥,𝑥,0,0,𝑡_′()𝑁𝐷𝑓−𝑥,−𝑥,0,0,𝑡≥min^′′𝜑(),𝑁𝑥,𝑥,0,0,𝑡_′′𝜑()()−𝑥,−𝑥,0,0,𝑡=𝑀𝑥,𝑡 (20) for all 𝑥∈𝑋. The above inequality and the definition of 𝑑 show that 𝑑(𝑓,𝐽𝑓)≤1/4.
According to Theorem 1, the sequence {𝐽^𝑛𝑓} converges to a unique fixed point 𝐹∶𝑋→𝑌 of 𝐽 in the set 𝑇={𝑔∈𝑆∣𝑑(𝑓,𝑔)<∞}, which is represented by 𝐹(𝑥)∶=𝑁^′−lim_𝑛→∞2^−2𝑛−1𝑓2^𝑛𝑥+𝑓−2^𝑛𝑥+2^−𝑛−1𝑓2^𝑛𝑥−𝑓−2^𝑛𝑥 (21) for all 𝑥∈𝑋. We observe that 𝑑≤1𝑓,𝐹𝑑≤11−𝛼/2𝑓,𝐽𝑓2().2−𝛼 (22) This guarantees that inequality (5) holds.
Next, we are in the position to prove that 𝐹 is quadratic-additive mapping. Now, we figure out the relation 𝑁^′𝑁𝐷𝐹𝑥,𝑦,𝑧,𝑤,𝑡≥min^′2𝐹−𝐽^𝑛𝑓,𝑡𝑥+𝑦+𝑧+𝑤,𝑁30_′𝐹−𝐽^𝑛𝑓,𝑡𝑥−𝑦+𝑧+𝑤,𝑁30_′𝐹−𝐽^𝑛𝑓,𝑡𝑥+𝑦−𝑧+𝑤,𝑁30_′𝐹−𝐽^𝑛𝑓,𝑡𝑥+𝑦+𝑧−𝑤,𝑁30_′𝐹−𝐽^𝑛𝑓,𝑡𝑥−𝑦−𝑧+𝑤,𝑁30_′𝐹−𝐽^𝑛𝑓,𝑡𝑥−𝑦+𝑧−𝑤,𝑁30_′𝐹−𝐽^𝑛𝑓,𝑡𝑥+𝑦−𝑧−𝑤,𝑁30_′5𝐽^𝑛(𝑥),𝑡𝑓−𝐹,𝑁30_′3𝐽^𝑛(),𝑡𝑓−𝐹−𝑥,𝑁30_′5𝐽^𝑛𝑦,𝑡𝑓−𝐹,𝑁30_′3𝐽^𝑛,𝑡𝑓−𝐹−𝑦,𝑁30_′5𝐽^𝑛𝑧),𝑡𝑓−𝐹(,𝑁30_′3𝐽^𝑛(),𝑡𝑓−𝐹−𝑧,𝑁30_′5𝐽^𝑛(𝑤),𝑡𝑓−𝐹,𝑁30_′3𝐽^𝑛(),𝑡𝑓−𝐹−𝑤,𝑁30_′𝐷𝐽^𝑛𝑓,𝑡𝑥,𝑦,𝑧,𝑤2 (23) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and all 𝑛∈ℕ. The first fifteen terms on the right-hand side of the above inequality tend to 1 as 𝑛→∞ by the definition of 𝐹. Moreover, we find that 𝑁^′𝐷𝐽^𝑛𝑓,𝑡𝑥,𝑦,𝑧,𝑤2≥min𝑁^′2^−2𝑛−1𝐷𝑓2^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧,2^𝑛𝑤,𝑡8,𝑁^′2^−2𝑛−1𝐷𝑓−2^𝑛𝑥,−2^𝑛𝑦,−2^𝑛𝑧,−2^𝑛𝑤,𝑡8,𝑁^′2^−𝑛−1𝐷𝑓2^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧,2^𝑛𝑤,𝑡8,𝑁^′2^−𝑛−1𝐷𝑓−2^𝑛𝑥,−2^𝑛𝑦,−2^𝑛𝑧,−2^𝑛𝑤,𝑡8≥min𝑁^′𝐷𝑓2^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧,2^𝑛𝑤,2^2𝑛−2𝑡,𝑁^′𝐷𝑓−2^𝑛𝑥,−2^𝑛𝑦,−2^𝑛𝑧,−2^𝑛𝑤,2^2𝑛−2𝑡,𝑁^′𝐷𝑓2^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧,2^𝑛𝑤,2^𝑛−2𝑡,𝑁^′𝐷𝑓−2^𝑛𝑥,−2^𝑛𝑦,−2^𝑛𝑧,−2^𝑛𝑤,2^𝑛−2𝑡≥min𝑁^′′𝜑𝑥,𝑦,𝑧,𝑤,2^2𝑛−2𝛼^−𝑛𝑡,𝑁^′′𝜑−𝑥,−𝑦,−𝑧,−𝑤,2^2𝑛−2𝛼^−𝑛𝑡,𝑁^′′𝜑𝑥,𝑦,𝑧,𝑤,2^𝑛−2𝛼^−𝑛𝑡,𝑁^′′𝜑−𝑥,−𝑦,−𝑧,−𝑤,2^𝑛−2𝛼^−𝑛𝑡, (24) which tends to 1 as 𝑛→∞, since 2/𝛼>1. Therefore inequality (23) gives that 𝑁^′𝐷𝐹𝑥,𝑦,𝑧,𝑤,𝑡=1 (25) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝑡>0. So we deduce that 𝐷𝐹(𝑥,𝑦,𝑧,𝑤)=0 for all 𝑥,𝑦,𝑧,𝑤∈𝑋.
In order to show the uniqueness of 𝐹, we assume that 𝐹^′∶𝑋→𝑌 is another quadratic-additive mapping satisfying (5), and then we yield that 𝐽𝐹^′(𝑥)=5𝐷𝐹^′(𝑥,𝑥,0,0)−3𝐷𝐹^′(−𝑥,−𝑥,0,0)32+𝐹^′(𝑥)=𝐹^′(𝑥) (26) for all 𝑥∈𝑋. That is, 𝐹^′ is another fixed point of 𝐽. Since 𝐹 is a unique fixed point of 𝐽 in the set 𝑇, we conclude that 𝐹=𝐹^′.
Case 2. Assume that 𝜑 satisfies the condition (ii). The proof of this case can be carried out similarly as the proof of Case  1. In particular, assume that 𝑁^′′(𝜑(𝑥,𝑦,0,0),𝑡) is continuous in 𝑥,𝑦. If 𝑀, 𝑎,𝑏,𝑐,𝑑 are any fixed nonzero integers, then we have lim_𝑛→∞𝑁^′′𝜑2^𝑛𝑎+𝑏𝑥,2^𝑛𝑐+𝑑𝑦,0,0,𝑡≥lim_𝑛→∞𝑁^′′𝜑𝑎+2^−𝑛𝑏𝑥,𝑐+2^−𝑛𝑑𝑦,0,0,𝛼^−𝑛𝑡≥lim_𝑛→∞𝑁^′′𝜑𝑎+2^−𝑛𝑏𝑥,𝑐+2^−𝑛𝑑𝑦,0,0,𝑀𝑡=𝑁^′′𝜑𝑎𝑥,𝑐𝑦,0,0,𝑀𝑡 (27) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Since 𝑛 is arbitrary, we have lim_𝑛→∞𝑁^′′𝜑2^𝑛𝑎+𝑏𝑥,2^𝑛𝑐+𝑑𝑦,0,0,𝑡≥lim_𝑀→∞𝑁^′′𝜑𝑎𝑥,𝑐𝑦,0,0,𝑀𝑡=1 (28) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝑡>0. From these and the following equality: (𝑥)=3𝐹−𝑓8𝐷𝐹−𝐷𝑓2^𝑛+1𝑥,−2^𝑛−1𝑥,0,08𝐷𝐹−𝐷𝑓−2^𝑛+1𝑥,2^𝑛−1𝑥,0,02𝐹−𝑓2^𝑛+1𝑥−1+12𝐹−𝑓−2^𝑛+1𝑥+3+12𝐹−𝑓2^𝑛𝑥+1+12𝐹−𝑓−2^𝑛𝑥+3+12𝐹−𝑓2^𝑛𝑥+12𝐹−𝑓2^𝑛𝑥, (29) we get the inequality 𝑁^′𝐹−𝑓(𝑥),11𝑡2≥lim_𝑛→∞min𝑁^′′38𝜑2^𝑛+1𝑥,−2^𝑛𝑥,0,0,3𝑡8,𝑁^′′18𝜑−2^𝑛+1𝑥,2^𝑛,𝑡𝑥,0,08,𝑀2^𝑛+1()𝑡,𝑀2+1𝑥,22−𝛼_𝑛()𝑡,𝑀2+1𝑥,22−𝛼_𝑛𝑥,2(2−𝛼)𝑡=1 (30) for all 𝑥∈𝑋. Due to the previous inequality and the fact that 𝑓(0)=0=𝐹(0), we obtain that 𝑓≡𝐹.
Case 3. Assume that 𝜑 satisfies the condition (iii). Let the set (𝑆,𝑑) be as in the proof of Case  1. Now we take into account the function 𝐽∶𝑆→𝑆 defined by (𝑥)𝑥𝐽𝑔∶=𝑔2−𝑥−𝑔2𝑔𝑥+22−𝑥+𝑔2 (31) for all 𝑔∈𝑆 and 𝑥∈𝑋. Note that 𝐽^𝑛𝑔(𝑥)=2^𝑛−1𝑔2^−𝑛𝑥−𝑔−2^−𝑛𝑥+2^2𝑛−1𝑔2^−𝑛𝑥+𝑔−2^−𝑛𝑥 (32) and 𝐽⁰𝑔(𝑥)=𝑔(𝑥) for all 𝑥∈𝑋. Let 𝑓,𝑔∈𝑆 and let 𝑢∈[0,∞] be an arbitrary constant with 𝑑(𝑔,𝑓)≤𝑢. From the definition of 𝑑, we have 𝑁^′𝐽𝑔(𝑥)−𝐽𝑓(𝑥),4𝑢𝑡𝛼=𝑁^′3𝑔𝑥2𝑥−𝑓2+𝑔−𝑥2−𝑥−𝑓2,4𝑢𝑡𝛼≥min𝑁^′3𝑔𝑥2𝑥−𝑓2,3𝑢𝑡𝛼,𝑁^′𝑔−𝑥2−𝑥−𝑓2,𝑢𝑡𝛼≥min𝑁^′𝑔𝑥2𝑥−𝑓2,𝑢𝑡𝛼,𝑁^′𝑔−𝑥2−𝑥−𝑓2,𝑢𝑡𝛼𝑀𝑥≥min2,𝑡𝛼−𝑥,𝑀2,𝑡𝛼=𝑀(𝑥,𝑡) (33) for all 𝑥∈𝑋, which means that 𝑑(𝐽𝑓,𝐽𝑔)≤(4/𝛼)𝑑(𝑓,𝑔). Hence 𝐽 is a strictly contractive self-mapping of 𝑆 with the Lipschitz constant 0<4/𝛼<1.
Moreover, by (4), we see that 𝑁^′𝑓(𝑥)(𝑥),𝑡−𝐽𝑓2𝛼=𝑁^′3𝐷𝑓(𝑥/2,𝑥/2,0,0)8−𝐷𝑓(−𝑥/2,−𝑥/2,0,0)8,𝑡𝑁2𝛼≥min^′′𝑥3𝜑2,𝑥,2,0,03𝑡𝛼,𝑁^′′𝜑−𝑥2𝑥,−,𝑡2,0,0𝛼≥𝑀(𝑥,𝑡) (34) for all 𝑥∈𝑋. It implies that 𝑑(𝑓,𝐽𝑓)≤1/2𝛼 by the definition of 𝑑. Therefore, according to Theorem 1, the sequence {𝐽^𝑛𝑓} converges to a unique fixed point 𝐹∶𝑋→𝑌 of 𝐽 in the set 𝑇={𝑔∈𝑆∣𝑑(𝑓,𝑔)<∞}, which is represented by 𝐹(𝑥)=𝑁^′−lim_𝑛→∞2^𝑛−1𝑓2^−𝑛𝑥−𝑓−2^−𝑛𝑥+2^2𝑛−1𝑓2^−𝑛𝑥+𝑓−2^−𝑛𝑥 (35) for all 𝑥∈𝑋. Since 𝑑≤1𝑓,𝐹𝑑≤11−4/𝛼𝑓,𝐽𝑓2(),𝛼−4 (36) inequality (5) holds.
Next, we will show that 𝐹 is quadratic-additive mapping. As in the previous case, we have inequality (23) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and all 𝑛∈ℕ. The first terms on the right-hand side of inequality (23) tend to 1 as 𝑛→∞ by the definition of 𝐹. Now consider that 𝑁^′𝐷𝐽^𝑛𝑓,𝑡𝑥,𝑦,𝑧,𝑤2≥min𝑁^′2^2𝑛−1𝐷𝑓2^−𝑛𝑥,2^−𝑛𝑦,2^−𝑛𝑧,2^−𝑛𝑤,𝑡8,𝑁^′2^2𝑛−1𝐷𝑓−2^−𝑛𝑥,−2^−𝑛𝑦,−2^−𝑛𝑧,−2^−𝑛𝑤𝑡8,𝑁^′2^𝑛−1𝐷𝑓2^−𝑛𝑥,2^−𝑛𝑦,2^−𝑛𝑧,2^−𝑛𝑤,𝑡8,𝑁^′−2^𝑛−1𝐷𝑓−2^−𝑛𝑥,−2^−𝑛𝑦,−2^−𝑛𝑧,−2^−𝑛𝑤,𝑡8≥min𝑁^′′𝜑𝑥,𝑦,𝑧,𝑤,2^−2𝑛−2𝛼^𝑛𝑡,𝑁^′′𝜑−𝑥,−𝑦,−𝑧,−𝑤,2^−2𝑛−2𝛼^𝑛𝑡,𝑁^′′𝜑𝑥,𝑦,𝑧,𝑤,2^−𝑛−2𝛼^𝑛𝑡,𝑁^′′𝜑−𝑥,−𝑦,−𝑧,−𝑤,2^−𝑛−2𝛼^𝑛𝑡, (37) which tends to 1 as 𝑛→∞ for all 𝑥,𝑦,𝑧,𝑤∈𝑋. Therefore it follows from (23) that 𝑁^′𝐷𝐹𝑥,𝑦,𝑧,𝑤,𝑡=1 (38) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝑡>0. That is, 𝐷𝐹(𝑥,𝑦,𝑧,𝑤)=0 for all 𝑥,𝑦,𝑧,𝑤∈𝑋.

In particular, instead of the assumption of Theorem 3 that (𝑋,𝑁) is a fuzzy normed space, it is enough to consider that 𝑋 is a linear space. Moreover, we can use Theorem 3 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 following 𝑁_𝑋()=‖𝑥‖,‖𝑥‖,𝑥,𝑡0,𝑡≤1,𝑡> (39) where 𝑥∈𝑋 and 𝑡∈ℝ, and see [12].

Theorem 4. Let 𝑋 and 𝑌 be a normed space and a complete normed space, respectively. If a mapping 𝑓∶𝑋→𝑌 satisfies 𝐷𝑓𝑥,𝑦,𝑧,𝑤≤‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝+‖𝑤‖^𝑝 (40) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and a fixed 𝑝∈(0,1)∪(2,∞), then there exists a unique quadratic-additive mapping 𝐹∶𝑋→𝑌 such that 𝑓(𝑥)(𝑥≤‖𝑥‖−𝐹)2−2^𝑝𝑝‖𝑥‖𝑖𝑓0<𝑝<1,_𝑝2^𝑝−4𝑖𝑓𝑝>2 (41) for all 𝑥∈𝑋.

Proof. Let 𝑁_𝑌 be a fuzzy norm on 𝑌. Then we get 𝑁_𝑌=,𝐷𝑓𝑥,𝑦,𝑧,𝑤,𝑡0if𝑡≤𝐷𝑓𝑥,𝑦,𝑧,𝑤1if𝑡>𝐷𝑓𝑥,𝑦,𝑧,𝑤 (42) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and all 𝑡∈ℝ and 𝑁_ℝ‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝+‖𝑤‖^𝑝=‖𝑥‖,𝑡0if𝑡≤^𝑝+𝑦^𝑝+‖𝑧‖^𝑝+‖𝑤‖^𝑝,1if𝑡>‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝+‖𝑤‖^𝑝 (43) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and all 𝑡∈ℝ. Therefore, it follows from (40) that 𝑁_𝑌𝐷𝑓𝑥,𝑦,𝑧,𝑤,𝑡≥𝑁_ℝ‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝+‖𝑤‖^𝑝,𝑡. (44) If we define a mapping 𝜑∶𝑋⁴→ℝ by 𝜑𝑥,𝑦,𝑧,𝑤∶=‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝+‖𝑤‖^𝑝, (45) then we see that 𝑓 and 𝜑 satisfy the conditions of Theorem 3 with 𝛼=𝛼^′=2^𝑝 for 0<𝑝<1 and 𝛼=2^𝑝 for 𝑝>2. So we feel that (41) holds for all 𝑥∈𝑋.

Theorem 5. Let 𝑋 and 𝑌 be a normed space and a complete normed space, respectively. Assume that 𝜙∶𝑋→ℝ is a mapping defined by 𝜙(𝑥)=‖𝑥‖^𝑝𝑖𝑓𝑥≠0,0𝑖𝑓𝑥=0 (46) for all 𝑥∈𝑋 and a fixed 𝑝<0. Suppose that 𝑓∶𝑋→𝑌 is a mapping such that 𝐷𝑓𝑥,𝑦,𝑧,𝑤≤𝜙(𝑥)+𝜙𝑦+𝜙(𝑧)+𝜙(𝑤) (47) for all 𝑥,𝑦,𝑧,𝑤∈𝑋. Then the mapping 𝑓 is a quadratic-additive mapping.

Proof. If we define a mapping 𝜑∶𝑋⁴→ℝ by 𝜑𝑥,𝑦,𝑧,𝑤∶=𝜙(𝑥)+𝜙𝑦+𝜙(𝑧)+𝜙(𝑤), (48) then 𝑓 and 𝜑 are fulfilled in the conditions of Theorem 3 with 𝛼<2^𝑝<1. Based on the fact that 𝑁_ℝ(𝜑(𝑥,𝑦,0,0),𝑡) is continuous in 𝑥,𝑦 under the condition (ii), we arrive at the desired conclusion.

Conflict of Interests

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

Acknowledgments

The authors would like to thank the referees for giving useful suggestions and for the improvement of this paper. The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2A10004419).