Fuzzy Stability of Jensen-Type Quadratic Functional Equations

Abstract

We prove the generalized Hyers-Ulam stability of the following quadratic functional equations 2𝑓((𝑥+𝑦)/2)+2𝑓((𝑥−𝑦)/2)=𝑓(𝑥)+𝑓(𝑦) and 𝑓(𝑎𝑥+𝑎𝑦)+(𝑎𝑥−𝑎𝑦)=2𝑎²𝑓(𝑥)+2𝑎²𝑓(𝑦) in fuzzy Banach spaces for a nonzero real number 𝑎 with 𝑎≠±1/2.

1. Introduction and Preliminaries

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The work of Th. M. Rassias [4] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Găvruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach.

J. M. Rassias [6] proved a similar stability theorem in which he replaced the factor ‖𝑥‖^𝑝+‖𝑦‖^𝑝 by ‖𝑥‖^𝑝⋅‖𝑦‖^𝑞 for 𝑝,𝑞∈ℝ with 𝑝+𝑞≠1 (see also [7, 8] for a number of other new results). The papers of J. M. Rassias [6–8] introduced the Ulam- Găvruţa-Rassias stability of functional equations. See also [9–11].

The functional equation 𝑓(𝑥+𝑦)+𝑓(𝑥−𝑦)=2𝑓(𝑥)+2𝑓(𝑦) (1.1) is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [12] for mappings 𝑓∶𝑋→𝑌, where 𝑋 is a normed space and 𝑌 is a Banach space. Cholewa [13] noticed that the theorem of Skof is still true if the relevant domain 𝑋 is replaced by an Abelian group. In [14], Czerwik proved the generalized Hyers-Ulam stability of the quadratic functional equation.

J. M. Rassias [15] introduced and solved the stability problem of Ulam for the Euler-Lagrange-type quadratic functional equation

𝑓(𝑟𝑥+𝑠𝑦)+𝑓(𝑠𝑥−𝑟𝑦)=(𝑟²+𝑠²)[𝑓(𝑥)+𝑓(𝑦)], (1.2) motivated from the following pertinent algebraic equation

|𝑎𝑥+𝑏𝑦|²+|𝑏𝑥−𝑎𝑦|²=(𝑎²+𝑏²)(|𝑥|²+|𝑦|²). (1.3) The solution of the functional equation (1.2) is called a Euler-Lagrange-type quadratic mapping. J. M. Rassias [16, 17] introduced and investigated the relative functional equations. In addition, J. M. Rassias [18] generalized the algebraic equation (1.3) to the following equation 𝑚𝑛|𝑎𝑥+𝑏𝑦|²+|𝑛𝑏𝑥−𝑚𝑎𝑦|²=(𝑚𝑎²+𝑛𝑏²)(𝑛|𝑥|²+𝑚|𝑦|²), (1.4) and introduced and investigated the general pertinent Euler-Lagrange quadratic mappings. Analogous quadratic mappings were introduced and investigated in [19, 20].

These Euler-Lagrange mappings are named Euler-Lagrange-Rassias mappings and the corresponding Euler-Lagrange equations are called Euler-Lagrange-Rassias equations. Before 1992, these mappings and equations were not known at all in functional equations and inequalities. However, a completely different kind of Euler-Lagrange partial differential equations are known in calculus of variations. Therefore, we think that J. M. Rassias' introduction of Euler-Lagrange mappings and equations in functional equations and inequalities provides an interesting cornerstone in analysis. Already some mathematicians have employed these Euler-Lagrange mappings.

Recently, Jun and Kim [21] solved the stability problem of Ulam for another Euler-Lagrange-Rassias-type quadratic functional equation. Jun and Kim [22] introduced and investigated the following quadratic functional equation of Euler-Lagrange-Rassias type:

^𝑛_𝑖=1𝑟_𝑖𝑄(^𝑛_𝑗=1𝑟_𝑗(𝑥_𝑖−𝑥_𝑗))+(^𝑛_𝑖=1𝑟_𝑖)𝑄(^𝑛_𝑖=1𝑟_𝑖𝑥_𝑖)=(^𝑛_𝑖=1𝑟_𝑖)^2𝑛_𝑖=1𝑟_𝑖𝑄(𝑥_𝑖), (1.5) whose solution is said to be a generalized quadratic mapping of Euler-Lagrange-Rassias type.

During the last two decades a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings (see [9, 23–26]).

Katsaras [27] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [28–30]. In particular, Bag and Samanta [31], following Cheng and Mordeson [32], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type [33]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [34].

We use the definition of fuzzy normed spaces given in [31] and [35–38] to investigate a fuzzy version of the generalized Hyers-Ulam stability for the quadratic functional equations

2𝑓(𝑥+𝑦2)+2𝑓(𝑥−𝑦2)=𝑓(𝑥)+𝑓(𝑦), (1.6)𝑓(𝑎𝑥+𝑎𝑦)+𝑓(𝑎𝑥−𝑎𝑦)=2𝑎²𝑓(𝑥)+2𝑎²𝑓(𝑦) (1.7) in the fuzzy normed vector space setting.

Definition 1.1 (see [31, 35–38]). Let 𝑋 be a real vector space. A function 𝑁∶𝑋×ℝ→[0,1] is called a fuzzy norm on 𝑋 if for all 𝑥,𝑦∈𝑋 and all 𝑠,𝑡∈ℝ,
(𝑁₁) 𝑁(𝑥,𝑡)=0 for 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|) if 𝑐≠0;
(𝑁₄) 𝑁(𝑥+𝑦,𝑠+𝑡)≥min{𝑁(𝑥,𝑠),𝑁(𝑦,𝑡)};
(𝑁₅) 𝑁(𝑥,⋅) is a non-decreasing function of ℝ and lim_𝑡→∞𝑁(𝑥,𝑡)=1;
(𝑁₆) for 𝑥≠0, 𝑁(𝑥,⋅) is continuous on ℝ.
The pair (𝑋,𝑁) is called a fuzzy normed vector space.
The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [35–38].

Definition 1.2 (see [31, 35–38]). Let (𝑋,𝑁) be a fuzzy normed vector space. A sequence {𝑥_𝑛} in 𝑋 is said to be convergent or converge 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 denote it by N-lim_𝑛→∞𝑥_𝑛=𝑥.

Definition 1.3 (see [31, 35–38]). Let (𝑋,𝑁) be a fuzzy normed vector space. 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 for all 𝑛≥𝑛₀ and all 𝑝>0, we have 𝑁(𝑥_𝑛+𝑝−𝑥_𝑛,𝑡)>1−𝜀.
It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.
We say that a mapping 𝑓∶𝑋→𝑌 between fuzzy normed vector spaces 𝑋 and 𝑌 is continuous at a point 𝑥₀∈𝑋 if for each sequence {𝑥_𝑛} converging to 𝑥₀ in 𝑋, then the sequence {𝑓(𝑥_𝑛)} converges to 𝑓(𝑥₀). If 𝑓∶𝑋→𝑌 is continuous at each 𝑥∈𝑋, then 𝑓∶𝑋→𝑌 is said to be continuous on 𝑋 (see [34]).
In this paper, we prove the generalized Hyers-Ulam stability of the quadratic functional equations (1.6) and (1.7) in fuzzy Banach spaces.
Throughout this paper, assume that 𝑋 is a vector space and that (𝑌,𝑁) is a fuzzy Banach space. Let 𝑎 be a nonzero real number with 𝑎≠(±1/2).

2. Fuzzy Stability of Quadratic Functional Equations

We prove the fuzzy stability of the quadratic functional equation (1.6).

Theorem 2.1. Let𝑓∶𝑋→𝑌be an even mapping with𝑓(0)=0. Suppose that𝜑 is a mapping from 𝑋to a fuzzy normed space(𝑍,𝑁^′)such that𝑁(2𝑓(𝑥+𝑦2)+2𝑓(𝑥−𝑦2)−𝑓(𝑥)−𝑓(𝑦),𝑡+𝑠)≥min{𝑁^′(𝜑(𝑥),𝑡),𝑁^′(𝜑(𝑦),𝑠)} (2.1)for all𝑥,𝑦∈𝑋⧵{0}and all positive real numbers𝑡,𝑠. If𝜑(3𝑥)=𝛼𝜑(𝑥)for some positive real number𝛼with𝛼<9, then there is a unique quadratic mapping𝑄∶𝑋→𝑌such that𝑄(𝑥)=𝑁-lim_𝑛→∞𝑓(3^𝑛𝑥)/9^𝑛 and 𝑁(𝑄(𝑥)−𝑓(𝑥),𝑡)≥𝑀(𝑥,(9−𝛼)𝑡18), (2.2) where 𝑀(𝑥,𝑡)∶=min{𝑁^′3(𝜑(𝑥),2𝑡),𝑁^′3(𝜑(2𝑥),2𝑡),𝑁^′3(𝜑(3𝑥),2𝑡),𝑁^′3(𝜑(0),2𝑡)}. (2.3)

Proof. Putting 𝑦=3𝑥 and 𝑠=𝑡 in (2.1), we get 𝑁(2𝑓(2𝑥)+2𝑓(−𝑥)−𝑓(𝑥)−𝑓(3𝑥),2𝑡)≥min{𝑁^′(𝜑(𝑥),𝑡),𝑁^′(𝜑(3𝑥),𝑡)} (2.4) for all 𝑥∈𝑋 and all 𝑡>0. Replacing 𝑥 by 2𝑥,𝑦 by 0, and 𝑠 by 𝑡 in (2.1), we obtain 𝑁(4𝑓(𝑥)−𝑓(2𝑥),2𝑡)≥min{𝑁^′(𝜑(2𝑥),𝑡),𝑁^′(𝜑(0),𝑡)}. (2.5) Thus 𝑁(9𝑓(𝑥)−𝑓(3𝑥),6𝑡)≥min{𝑁^′(𝜑(𝑥),𝑡),𝑁^′(𝜑(2𝑥),𝑡),𝑁^′(𝜑(3𝑥),𝑡),𝑁^′(𝜑(0),𝑡)}, (2.6) and so 𝑁(𝑓(𝑥)−𝑓(3𝑥)9,𝑡)≥min{𝑁^′3(𝜑(𝑥),2𝑡),𝑁^′3(𝜑(2𝑥),2𝑡)𝑁^′3(𝜑(3𝑥),2𝑡),𝑁^′3(𝜑(0),2𝑡)}. (2.7) Then by the assumption, 𝑡𝑀(3𝑥,𝑡)=𝑀(𝑥,𝛼). (2.8) Replacing 𝑥 by 3^𝑛𝑥 in (2.7) and applying (2.8), we get 𝑁(𝑓(3^𝑛𝑥)9^𝑛−𝑓(3^𝑛+1𝑥)9^𝑛+1,𝛼^𝑛𝑡9^𝑛)=𝑁(𝑓(3^𝑛𝑥)−𝑓(3^𝑛+1𝑥)9,𝛼^𝑛𝑡)≥𝑀(3^𝑛𝑥,𝛼^𝑛𝑡)=𝑀(𝑥,𝑡). (2.9) Thus for each 𝑛>𝑚 we have 𝑁(𝑓(3^𝑚𝑥)9^𝑚−𝑓(3^𝑛𝑥)9^𝑛,^𝑛−1_𝑘=𝑚𝛼^𝑘𝑡9^𝑘)=𝑁(^𝑛−1_𝑘=𝑚(𝑓(3^𝑘𝑥)9^𝑘−𝑓(3^𝑘+1𝑥)9^𝑘+1),^𝑛−1_𝑘=𝑚𝛼^𝑘𝑡9^𝑘)≥min{^𝑛−1_𝑘=𝑚{𝑁(𝑓(3^𝑘𝑥)9^𝑘−𝑓(3^𝑘+1𝑥)9^𝑘+1,𝛼^𝑘𝑡9^𝑘)}}≥𝑀(𝑥,𝑡). (2.10) Let 𝜀>0 and 𝛿>0 be given. Since lim_𝑡→∞𝑀(𝑥,𝑡)=1, there is some 𝑡₀>0 such that 𝑀(𝑥,𝑡₀)>1−𝜀. Since ∑^∞_𝑘=0𝛼^𝑘𝑡₀/9^𝑘<∞, there is some 𝑛₀∈ℕ such that ∑^𝑛−1_𝑘=𝑚𝛼^𝑘𝑡₀/9^𝑘<𝛿 for 𝑛>𝑚≥𝑛₀. It follows that 𝑁(𝑓(3^𝑚𝑥)9^𝑚−𝑓(3^𝑛𝑥)9^𝑛,𝛿)≥𝑁(𝑓(3^𝑚𝑥)9^𝑚−𝑓(3^𝑛𝑥)9^𝑛,^𝑛−1_𝑘=𝑚𝛼^𝑘𝑡9^𝑘)≥𝑀(𝑥,𝑡₀)≥1−𝜀 (2.11) for all 𝑡≥𝑡₀. This shows that the sequence {𝑓(3^𝑛𝑥)/9^𝑛} is Cauchy in (𝑌,𝑁). Since (𝑌,𝑁) is complete, {𝑓(3^𝑛𝑥)/9^𝑛} converges to some 𝑄(𝑥)∈𝑌. Thus we can define a mapping 𝑄∶𝑋→𝑌 by 𝑄(𝑥)∶=𝑁−lim_𝑡→∞𝑓(3^𝑛𝑥)/9^𝑛. Moreover, if we put 𝑚=0 in (2.10), then we observe that 𝑁(𝑓(3^𝑛𝑥)9^𝑛−𝑓(𝑥),^𝑛−1_𝑘=0𝛼^𝑘𝑡9^𝑘)≥𝑀(𝑥,𝑡). (2.12) Thus 𝑁(𝑓(3^𝑛𝑥)9^𝑛𝑡−𝑓(𝑥),𝑡)≥𝑀(𝑥,∑^𝑛−1_𝑘=0(𝛼/9)^𝑘). (2.13) Next we show that 𝑄 is quadratic. Let 𝑥,𝑦∈𝑋. Then we have 𝑁(2𝑄(𝑥+𝑦2)+2𝑄(𝑥−𝑦2)−𝑄(𝑥)−𝑄(𝑦),𝑡)≥min{𝑁(2𝑄(𝑥+𝑦2)−2𝑓(3^𝑛(𝑥+𝑦)/2)9^𝑛,𝑡5),𝑁(2𝑄(𝑥−𝑦2)−2𝑓(3^𝑛(𝑥−𝑦)/2)9^𝑛,𝑡5),𝑁(𝑓(3^𝑛𝑥)9^𝑛𝑡−𝑄(𝑥),5),𝑁(𝑓(3^𝑛𝑦)9^𝑛𝑡−𝑄(𝑦),5),𝑁(2𝑓(3^𝑛(𝑥+𝑦)/2)9^𝑛+2𝑓(3^𝑛(𝑥−𝑦)/2)9^𝑛−𝑓(3^𝑛𝑥)9^𝑛−𝑓(3^𝑛𝑦)9^𝑛,𝑡5)}. (2.14) The first four terms on the right-hand side of the above inequality tend to 1 as 𝑛→∞ and the fifth term, by (2.1), is greater than or equal to min{𝑁^′(𝜑(3^𝑛9𝑥),^𝑛𝑡10),𝑁^′(𝜑(3^𝑛9𝑦),^𝑛𝑡10)}=min{𝑁^′9(𝜑(𝑥),(𝛼)^𝑛𝑡910),(𝜑(𝑦),(𝛼)^𝑛𝑡10)}, (2.15) which tends to 1 as 𝑛→∞. Hence 𝑁(2𝑄(𝑥+𝑦2)+2𝑄(𝑥−𝑦2)−𝑄(𝑥)−𝑄(𝑦),𝑡)=1 (2.16) for all 𝑥,𝑦∈𝑋 and all 𝑡>0. This means that 𝑄 satisfies the Jensen quadratic functional equation and so it is quadratic.
Next, we approximate the difference between 𝑓 and 𝑄 in a fuzzy sense. For every 𝑥∈𝑋 and 𝑡>0, by (2.13), for large enough 𝑛, we have 𝑁(𝑄(𝑥)−𝑓(𝑥),𝑡)≥min{𝑁(𝑄(𝑥)−𝑓(3^𝑛𝑦)9^𝑛,𝑡2),𝑁(𝑓(3^𝑛𝑦)9^𝑛𝑡−𝑓(𝑥),2𝑡)}≥𝑀(𝑥,2∑^∞_𝑘=0(𝛼/9)^𝑘)=𝑀(𝑥,(9−𝛼)𝑡18). (2.17) The uniqueness assertion can be proved by a standard fashion; cf. [36]: Let 𝑄^′ be another quadratic mapping from 𝑋 into 𝑌, which satisfies the required inequality. Then for each 𝑥∈𝑋 and 𝑡>0, 𝑁(𝑄(𝑥)−𝑄^′𝑡(𝑥),𝑡)≥min{𝑁(𝑄(𝑥)−𝑓(𝑥),2),𝑁(𝑄^′𝑡(𝑥)−𝑓(𝑥),2)}≥𝑀(𝑥,(9−𝛼)𝑡36). (2.18) Since 𝑄 and 𝑄^′ are quadratic, 𝑁(𝑄(𝑥)−𝑄^′(𝑥),𝑡)=𝑁(𝑄(3^𝑛𝑥)−𝑄^′(3^𝑛𝑥),9^𝑛𝑡)≥𝑀(𝑥,(9/𝛼)^𝑛(9−𝛼)𝑡36). (2.19) for all 𝑥∈𝑋, all 𝑡>0 and all 𝑛∈ℕ.
Since 0<𝛼<9, lim_𝑛→∞(9/𝛼)^𝑛=∞. Hence the right-hand side of the above inequality tends to 1 as 𝑛→∞. It follows that 𝑄(𝑥)=𝑄^′(𝑥) for all 𝑥∈𝑋.

Theorem 2.2. Let𝑓∶𝑋→𝑌be an even mapping with𝑓(0)=0. Suppose that𝜑is a mapping from𝑋to a fuzzy normed space (𝑍,𝑁^′)satisfying (2.1). If𝜑(3𝑥)=𝛼𝜑(𝑥)for some real number𝛼with 𝛼>9, then there is a unique quadratic mapping𝑄∶𝑋→𝑌such that𝑄(𝑥)=𝑁-lim_𝑛→∞9^𝑛𝑓(𝑥/3^𝑛) and𝑁(𝑄(𝑥)−𝑓(𝑥),𝑡)≥𝑀(𝑥,(𝛼−9)𝑡2𝛼), (2.20) where 𝑀(𝑥,𝑡)∶=min{𝑁^′𝑥(𝜑(3𝑡),6),𝑁^′(𝜑(2𝑥3𝑡),6),𝑁^′𝑡(𝜑(𝑥),6),𝑁^′𝑡(𝜑(0),6)}. (2.21)

Proof. It follows from (2.7) that 𝑥𝑁(𝑓(𝑥)−𝑓(3),𝑡)≥min{𝑁^′𝑥(𝜑(3𝑡),6),𝑁^′(𝜑(2𝑥3𝑡),6),𝑁^′𝑡(𝜑(𝑥),6),𝑁^′𝑡(𝜑(0),6)}. (2.22) Then by the assumption, 𝑥𝑀(3,𝑡)=𝑀(𝑥,𝛼𝑡). (2.23) Replacing 𝑥 by 𝑥/3^𝑛 in (2.22) and applying (2.23), we get 𝑁(9^𝑛𝑥𝑓(3^𝑛)−9^𝑛+1𝑥𝑓(3^𝑛+19),^𝑛𝑡𝛼^𝑛𝑥)=𝑁(𝑓(3^𝑛𝑥)−9𝑓(3^𝑛+1𝑡),𝛼^𝑛)𝑥≥𝑀(3^𝑛,𝑡𝛼^𝑛)=𝑀(𝑥,𝑡). (2.24) Thus for each 𝑛>𝑚 we have 𝑁(9^𝑚𝑥𝑓(3^𝑚)−9^𝑛𝑥𝑓(3^𝑛),^𝑛−1_𝑘=𝑚9^𝑘𝑡𝛼^𝑘)=𝑁(^𝑛−1_𝑘=𝑚(9^𝑘𝑥𝑓(3^𝑘)−9^𝑘+1𝑥𝑓(3^𝑘+1)),^𝑛−1_𝑘=𝑚9^𝑘𝑡𝛼^𝑘)≥min{^𝑛−1_𝑘=𝑚{𝑁(9^𝑘𝑥𝑓(3^𝑘)−9^𝑘+1𝑥𝑓(3^𝑘+19),^𝑘𝑡𝛼^𝑘)}}≥𝑀(𝑥,𝑡). (2.25)
Let 𝜀>0 and 𝛿>0 be given. Since lim_𝑡→∞𝑀(𝑥,𝑡)=1, there is some 𝑡₀>0 such that 𝑀(𝑥,𝑡₀)>1−𝜀. Since ∑^∞_𝑘=09^𝑘𝑡₀/𝛼^𝑘<∞, there is some 𝑛₀∈ℕ such that ∑^𝑛−1_𝑘=𝑚9^𝑘𝑡₀/𝛼^𝑘<𝛿 for 𝑛>𝑚≥𝑛₀. It follows that 𝑁(9^𝑚𝑥𝑓(3^𝑚)−9^𝑛𝑥𝑓(3^𝑛),𝛿)≥𝑁(9^𝑚𝑥𝑓(3^𝑚)−9^𝑛𝑥𝑓(3^𝑛),^𝑛−1_𝑘=𝑚9^𝑘𝑡𝛼^𝑘)≥𝑀(𝑥,𝑡₀)≥1−𝜀 (2.26) for all 𝑡≥𝑡₀. This shows that the sequence {9^𝑛𝑓(𝑥/3^𝑛)} is Cauchy in (𝑌,𝑁). Since (𝑌,𝑁) is complete, {9^𝑛𝑓(𝑥/3^𝑛)} converges to some 𝑄(𝑥)∈𝑌. Thus we can define a mapping 𝑄∶𝑋→𝑌 by 𝑄(𝑥)∶=𝑁-lim_𝑡→∞9^𝑛𝑓(𝑥/3^𝑛). Moreover, if we put 𝑚=0 in (2.8), then we observe that 𝑁(9^𝑛𝑥𝑓(3^𝑛)−𝑓(𝑥),^𝑛−1_𝑘=09^𝑘𝑡𝛼^𝑘)≥𝑀(𝑥,𝑡). (2.27) Thus 𝑁(9^𝑛𝑥𝑓(3^𝑛𝑡)−𝑓(𝑥),𝑡)≥𝑀(𝑥,∑^𝑛−1_𝑘=0(9/𝛼)^𝑘). (2.28) The rest of the proof is similar to the proof of Theorem 2.1.

Theorem 2.3. Let𝑓∶𝑋→𝑌be a mapping with𝑓(0)=0. Suppose that𝜑is a mapping from𝑋to a fuzzy normed space (𝑍,𝑁^′)satisfying (2.1). If𝜑(2𝑥)=𝛼𝜑(𝑥)for some positive real number 𝛼 with𝛼<4, then there is a unique quadratic mapping𝑄∶𝑋→𝑌such that𝑄(𝑥)=𝑁-lim_𝑛→∞𝑓(2^𝑛𝑥)/4^𝑛and𝑁(𝑄(𝑥)−𝑓(𝑥),𝑡)≥𝑀(𝑥,(4−𝛼)𝑡8) (2.29) where 𝑀(𝑥,𝑡)∶=min{𝑁^′(𝜑(2𝑥),2𝑡),𝑁^′(𝜑(0),2𝑡)}.

Proof. Letting 𝑦=0 and replacing 𝑥 by 2𝑥 and 𝑠 by 𝑡 in (2.1), we obtain 𝑁(4𝑓(𝑥)−𝑓(2𝑥),2𝑡)≥min{𝑁^′(𝜑(2𝑥),𝑡),𝑁^′(𝜑(0),𝑡)}. (2.30) Thus 𝑁(𝑓(𝑥)−𝑓(2𝑥)4,𝑡)≥min{𝑁^′(𝜑(2𝑥),2𝑡),𝑁^′(𝜑(0),2𝑡)}. (2.31) Then by the assumption, 𝑡𝑀(2𝑥,𝑡)=𝑀(𝑥,𝛼). (2.32) Replacing 𝑥 by 2^𝑛𝑥 in (2.31) and applying (2.32), we get 𝑁(𝑓(2^𝑛𝑥)4^𝑛−𝑓(2^𝑛+1𝑥)4^𝑛+1,𝛼^𝑛𝑡4^𝑛)=𝑁(𝑓(2^𝑛𝑥)−𝑓(4^𝑛+1𝑥)4,𝛼^𝑛𝑡)≥𝑀(2^𝑛𝑥,𝛼^𝑛𝑡)=𝑀(𝑥,𝑡). (2.33) Thus for each 𝑛>𝑚 we have 𝑁(𝑓(2^𝑚𝑥)4^𝑚−𝑓(2^𝑛𝑥)4^𝑛,^𝑛−1_𝑘=𝑚𝛼^𝑘𝑡4^𝑘)=𝑁(^𝑛−1_𝑘=𝑚(𝑓(2^𝑘𝑥)4^𝑘−𝑓(2^𝑘+1𝑥)4^𝑘+1),^𝑛−1_𝑘=𝑚𝛼^𝑘𝑡4^𝑘)≥min{^𝑛−1_𝑘=𝑚{𝑁(𝑓(2^𝑘𝑥)4^𝑘−𝑓(2^𝑘+1𝑥)4^𝑘+1,𝛼^𝑘𝑡4^𝑘)}}≥𝑀(𝑥,𝑡). (2.34)
Let 𝜀>0 and 𝛿>0 be given. Since lim_𝑡→∞𝑀(𝑥,𝑡)=1, there is some 𝑡₀>0 such that 𝑀(𝑥,𝑡₀)>1−𝜀. Since ∑^∞_𝑘=0𝛼^𝑘𝑡₀/4^𝑘<∞, there is some 𝑛₀∈ℕ such that ∑^𝑛−1_𝑘=𝑚𝛼^𝑘𝑡₀/4^𝑘<𝛿 for 𝑛>𝑚≥𝑛₀. It follows that 𝑁(𝑓(2^𝑚𝑥)4^𝑚−𝑓(2^𝑛𝑥)4^𝑛,𝛿)≥𝑁(𝑓(2^𝑚𝑥)4^𝑚−𝑓(2^𝑛𝑥)4^𝑛,^𝑛−1_𝑘=𝑚𝛼^𝑘𝑡4^𝑘)≥𝑀(𝑥,𝑡₀)≥1−𝜀 (2.35) for all 𝑡≥𝑡₀. This shows that the sequence {𝑓(2^𝑛𝑥)/4^𝑛} is Cauchy in (𝑌,𝑁). Since (𝑌,𝑁) is complete, {𝑓(2^𝑛𝑥)/4^𝑛} converges to some 𝑄(𝑥)∈𝑌. Thus we can define a mapping 𝑄∶𝑋→𝑌 by 𝑄(𝑥)∶=𝑁-lim_𝑡→∞𝑓(2^𝑛𝑥)/4^𝑛. Moreover, if we put 𝑚=0 in (2.34), then we observe that 𝑁(𝑓(2^𝑛𝑥)4^𝑛−𝑓(𝑥),^𝑛−1_𝑘=0𝛼^𝑘𝑡4^𝑘)≥𝑀(𝑥,𝑡). (2.36) Thus 𝑁(𝑓(2^𝑛𝑥)4^𝑛𝑡−𝑓(𝑥),𝑡)≥𝑀(𝑥,∑^𝑛−1_𝑘=0(𝛼/4)^𝑘). (2.37) The rest of the proof is similar to the proof of Theorem 2.1.

Theorem 2.4. Let 𝑓∶𝑋→𝑌be a mapping with𝑓(0)=0. Suppose that𝜑is a mapping from𝑋to a fuzzy normed space(𝑍,𝑁^′) satisfying (2.1). If𝜑(2𝑥)=𝛼𝜑(𝑥)for some real number𝛼with𝛼>4, then there is a unique quadratic mapping𝑄∶𝑋→𝑌such that𝑄(𝑥)=𝑁-lim_𝑛→∞4^𝑛𝑓(𝑥/2^𝑛)and𝑁(𝑄(𝑥)−𝑓(𝑥),𝑡)≥𝑀(𝑥,(𝛼−4)𝑡2𝛼), (2.38) where 𝑀(𝑥,𝑡)∶=min{𝑁^′(𝜑(𝑥),𝑡/2),𝑁^′(𝜑(0),𝑡/2)}.

Proof. It follows from (2.31) that 𝑥𝑁(𝑓(𝑥)−4𝑓(2),𝑡)≥min{𝑁^′𝑡(𝜑(𝑥),2),𝑁^′𝑡(𝜑(0),2)}. (2.39) Then by the assumption, 𝑥𝑀(2,𝑡)=𝑀(𝑥,𝛼𝑡). (2.40) Replacing 𝑥 by 𝑥/2^𝑛 in (2.39) and applying (2.40), we get 𝑁(4^𝑛𝑥𝑓(2^𝑛)−4^𝑛+1𝑥𝑓(2^𝑛+14),^𝑛𝑡𝛼^𝑛𝑥)=𝑁(𝑓(2^𝑛𝑥)−4𝑓(2^𝑛+1𝑡),𝛼^𝑛)𝑥≥𝑀(2^𝑛,𝑡𝛼^𝑛)=𝑀(𝑥,𝑡). (2.41) Thus for each 𝑛>𝑚 we have 𝑁(4^𝑚𝑥𝑓(2^𝑚)−4^𝑛𝑥𝑓(2^𝑛),^𝑛−1_𝑘=𝑚4^𝑘𝑡𝛼^𝑘)=𝑁(^𝑛−1_𝑘=𝑚(4^𝑘𝑥𝑓(2^𝑘)−4^𝑘+1𝑥𝑓(2^𝑘+1)),^𝑛−1_𝑘=𝑚4^𝑘𝑡𝛼^𝑘)≥min{^𝑛−1_𝑘=𝑚{𝑁(4^𝑘𝑥𝑓(2^𝑘)−4^𝑘+1𝑥𝑓(2^𝑘+14),^𝑘𝑡𝛼^𝑘)}}≥𝑀(𝑥,𝑡). (2.42)
Let 𝜀>0 and 𝛿>0 be given. Since lim_𝑡→∞𝑀(𝑥,𝑡)=1, there is some 𝑡₀>0 such that 𝑀(𝑥,𝑡₀)>1−𝜀. Since ∑^∞_𝑘=04^𝑘𝑡₀/𝛼^𝑘<∞, there is some 𝑛₀∈ℕ such that ∑^𝑛−1_𝑘=𝑚4^𝑘𝑡₀/𝛼^𝑘<𝛿 for 𝑛>𝑚≥𝑛₀. It follows that 𝑁(4^𝑚𝑥𝑓(2^𝑚)−4^𝑛𝑥𝑓(2^𝑛),𝛿)≥𝑁(4^𝑚𝑥𝑓(2^𝑚)−4^𝑛𝑥𝑓(2^𝑛),^𝑛−1_𝑘=𝑚4^𝑘𝑡𝛼^𝑘)≥𝑀(𝑥,𝑡₀)≥1−𝜀 (2.43) for all 𝑡≥𝑡₀. This shows that the sequence {4^𝑛𝑓(𝑥/2^𝑛)} is Cauchy in (𝑌,𝑁). Since (𝑌,𝑁) is complete, {4^𝑛𝑓(𝑥/2^𝑛)} converges to some 𝑄(𝑥)∈𝑌. Thus we can define a mapping 𝑄∶𝑋→𝑌 by 𝑄(𝑥)∶=𝑁-lim_𝑡→∞4^𝑛𝑓(𝑥/2^𝑛). Moreover, if we put 𝑚=0 in (2.42), then we observe that 𝑁(4^𝑛𝑥𝑓(2^𝑛)−𝑓(𝑥),^𝑛−1_𝑘=04^𝑘𝑡𝛼^𝑘)≥𝑀(𝑥,𝑡). (2.44) Thus 𝑁(4^𝑛𝑥𝑓(2^𝑛𝑡)−𝑓(𝑥),𝑡)≥𝑀(𝑥,∑^𝑛−1_𝑘=0(4/𝛼)^𝑘). (2.45) The rest of the proof is similar to the proof of Theorem 2.1.

Now we prove the fuzzy stability of the quadratic functional equation (1.7) for the case 𝑎≠(±1/2).

Theorem 2.5. Let|2𝑎|>1and𝑓∶𝑋→𝑌a mapping with𝑓(0)=0. Suppose that𝜑is a mapping from𝑋to a fuzzy normed space(𝑍,𝑁^′)such that𝑁(𝑓(𝑎𝑥+𝑎𝑦)+𝑓(𝑎𝑥−𝑎𝑦)−2𝑎²𝑓(𝑥)−2𝑎²𝑓(𝑦),𝑡+𝑠)≥min{𝑁^′(𝜑(𝑥),𝑡),𝑁^′(𝜑(𝑦),𝑠)} (2.46)for all𝑥,𝑦∈𝑋⧵{0}and all positive real numbers𝑡,𝑠. If𝜑(2𝑎𝑥)=𝛼𝜑(𝑥)for some positive real number𝛼with0<𝛼<4𝑎², then there is a unique quadratic mapping𝑄∶𝑋→𝑌such that𝑄(𝑥)=𝑁-lim_𝑛→∞𝑓((2𝑎)^𝑛𝑥)/(2𝑎)^2𝑛and𝑁(𝑄(𝑥)−𝑓(𝑥),𝑡)≥𝑁^′(𝜑(𝑥),(4𝑎²−𝛼)𝑡4) (2.47) for all 𝑥∈𝑋and all𝑡>0.

Proof. Putting 𝑦=𝑥 and 𝑠=𝑡 in (2.46), we get 𝑁(𝑓(2𝑎𝑥)−4𝑎²𝑓(𝑥),2𝑡)≥𝑁^′(𝜑(𝑥),𝑡) (2.48) for all 𝑥∈𝑋 and all 𝑡>0. Thus 𝑁(𝑓(𝑥)−𝑓(2𝑎𝑥)4𝑎²,𝑡2𝑎²)≥𝑁^′(𝜑(𝑥),𝑡) (2.49) and so 𝑁(𝑓(𝑥)−𝑓(2𝑎𝑥)4𝑎²,𝑡)≥𝑁^′(𝜑(𝑥),2𝑎²𝑡). (2.50) Replacing 𝑥 by (2𝑎)^𝑛𝑥 in (2.50), we get 𝑁(𝑓((2𝑎)^𝑛𝑥)(2𝑎)^2𝑛−𝑓((2𝑎)^𝑛+1𝑥)(2𝑎)^2𝑛+2,𝛼^𝑛𝑡(2𝑎)^2𝑛)=𝑁(𝑓((2𝑎)^𝑛𝑥)−𝑓((2𝑎)^𝑛+1𝑥)4𝑎²,𝛼^𝑛𝑡)≥𝑁^′(𝜑(𝑥),2𝑎²𝑡). (2.51) Thus for each 𝑛>𝑚 we have 𝑁(𝑓((2𝑎)^𝑚𝑥)(2𝑎)^2𝑚−𝑓((2𝑎)^𝑛𝑥)(2𝑎)^2𝑛,^𝑛−1_𝑘=𝑚𝛼^𝑘𝑡(2𝑎)^2𝑘)=𝑁(^𝑛−1_𝑘=𝑚(𝑓((2𝑎)^𝑘𝑥)(2𝑎)^2𝑘−𝑓((2𝑎)^𝑘+1𝑥)(2𝑎)^2𝑘+2),^𝑛−1_𝑘=𝑚𝛼^𝑘𝑡(2𝑎)^2𝑘)≥min{^𝑛−1_𝑘=𝑚{𝑁(𝑓((2𝑎)^𝑘𝑥)(2𝑎)^2𝑘−𝑓((2𝑎)^𝑘+1𝑥)(2𝑎)^2𝑘+2,𝛼^𝑘𝑡(2𝑎)^2𝑘)}}≥𝑁^′(𝜑(𝑥),2𝑎²𝑡). (2.52)
Let 𝜀>0 and 𝛿>0 be given. Since lim_𝑡→∞𝑁^′(𝜑(𝑥),2𝑎²𝑡)=1, there is some 𝑡₀>0 such that 𝑁^′(𝜑(𝑥),2𝑎²𝑡₀)>1−𝜀. Since ∑^∞_𝑘=0𝛼^𝑘𝑡₀/(2𝑎)^2𝑘<∞, there is some 𝑛₀∈ℕ such that ∑^𝑛−1_𝑘=𝑚𝛼^𝑘𝑡₀/(2𝑎)^2𝑘<𝛿 for 𝑛>𝑚≥𝑛₀. It follows that 𝑁(𝑓((2𝑎)^𝑚𝑥)(2𝑎)^2𝑚−𝑓((2𝑎)^𝑛𝑥)(2𝑎)^2𝑛,𝛿)≥𝑁(𝑓((2𝑎)^𝑚𝑥)(2𝑎)^2𝑚−𝑓((2𝑎)^𝑛𝑥)(2𝑎)^2𝑛,^𝑛−1_𝑘=𝑚𝛼^𝑘𝑡(2𝑎)^2𝑘)≥𝑁^′(𝜑(𝑥),2𝑎²𝑡₀)≥1−𝜀 (2.53) for all 𝑡≥𝑡₀. This shows that the sequence {𝑓((2𝑎)^𝑛𝑥)/(2𝑎)^2𝑛} is Cauchy in (𝑌,𝑁). Since (𝑌,𝑁) is complete, {𝑓((2𝑎)^𝑛𝑥)/(2𝑎)^2𝑛} converges to some 𝑄(𝑥)∈𝑌. Thus we can define a mapping 𝑄∶𝑋→𝑌 by 𝑄(𝑥)∶=𝑁-lim_𝑡→∞𝑓((2𝑎)^𝑛𝑥)/(2𝑎)^2𝑛. Moreover, if we put 𝑚=0 in (2.52), then we observe that 𝑁(𝑓((2𝑎)^𝑛𝑥)(2𝑎)^2𝑛−𝑓(𝑥),^𝑛−1_𝑘=0𝛼^𝑘𝑡(2𝑎)^2𝑘)≥𝑁^′(𝜑(𝑥),2𝑎²𝑡). (2.54) Thus 𝑁(𝑓((2𝑎)^𝑛𝑥)(2𝑎)^2𝑛−𝑓(𝑥),𝑡)≥𝑁^′(𝜑(𝑥),2𝑎²𝑡∑^𝑛−1_𝑘=0(𝛼/(2𝑎)²)^𝑘). (2.55) The rest of the proof is similar to the proof of Theorem 2.1.

Theorem 2.6. Let|2𝑎|<1and𝑓∶𝑋→𝑌a mapping with𝑓(0)=0. Suppose that𝜑is a mapping from𝑋to a fuzzy normed space(𝑍,𝑁^′)satisfying (2.46). If𝜑(2𝑎𝑥)=𝛼𝜑(𝑥)for some real number𝛼with𝛼>4𝑎², then there is a unique quadratic mapping𝑄∶𝑋→𝑌such that𝑄(𝑥)=𝑁-lim_𝑛→∞(2𝑎)^2𝑛𝑓(𝑥/(2𝑎)^𝑛)and𝑁(𝑄(𝑥)−𝑓(𝑥),𝑡)≥𝑀(𝑥,(𝛼−4𝑎²)𝑡4) (2.56)for all𝑥∈𝑋and all𝑡>0.

Proof. It follows from (2.50) that 𝑁(𝑓(𝑥)−(2𝑎)²𝑥𝑓(2𝑎),2𝑡)≥𝑁^′𝑥(𝜑(2𝑎),𝑡) (2.57) for all 𝑥∈𝑋 and all 𝑡>0. Thus 𝑁(𝑓(𝑥)−4𝑎²𝑥𝑓(2𝑎),𝑡)≥𝑁^′𝑥(𝜑(𝑡2𝑎),2)=𝑁^′𝛼(𝜑(𝑥),2𝑡). (2.58) Replacing 𝑥 by 𝑥/(2𝑎)^𝑛 in (2.58), we get 𝑁((2𝑎)^2𝑛𝑥𝑓((2𝑎)^𝑛)−(2𝑎)^2𝑛+2𝑥𝑓((2𝑎)^𝑛+1),(2𝑎)^2𝑛𝑡𝛼^𝑛)𝑥=𝑁(𝑓((2𝑎)^𝑛)−4𝑎²𝑥𝑓((2𝑎)^𝑛+1),𝛼^𝑛𝑡)≥𝑁^′𝛼(𝜑(𝑥),2𝑡). (2.59) Thus for each 𝑛>𝑚 we have 𝑁((2𝑎)^2𝑚𝑥𝑓((2𝑎)^𝑚)−(2𝑎)^2𝑛𝑥𝑓((2𝑎)^𝑛),^𝑛−1_𝑘=𝑚(2𝑎)^2𝑘𝑡𝛼^𝑘)=𝑁(^𝑛−1_𝑘=𝑚((2𝑎)^2𝑘𝑥𝑓((2𝑎)^𝑘)−(2𝑎)^2𝑘+2𝑥𝑓((2𝑎)^𝑘+1)),^𝑛−1_𝑘=𝑚(2𝑎)^2𝑘𝑡𝛼^𝑘)≥min{^𝑛−1_𝑘=𝑚{𝑁((2𝑎)^2𝑘𝑥𝑓((2𝑎)^𝑘)−(2𝑎)^2𝑘+2𝑥𝑓((2𝑎)^𝑘+1),(2𝑎)^2𝑘𝑡𝛼^𝑘)}}≥𝑁^′𝛼(𝜑(𝑥),2𝑡). (2.60) Let 𝜀>0 and 𝛿>0 be given. Since lim_𝑡→∞𝑁^′(𝜑(𝑥),(𝛼/2)𝑡)=1, there is some 𝑡₀>0 such that 𝑁^′(𝜑(𝑥),(𝛼/2)𝑡₀)>1−𝜀. Since ∑^∞_𝑘=0(2𝑎)^2𝑘𝑡₀/𝛼^𝑘<∞, there is some 𝑛₀∈ℕ such that ∑^𝑛−1_𝑘=𝑚(2𝑎)^2𝑘𝑡₀/𝛼^𝑘<𝛿 for 𝑛>𝑚≥𝑛₀. It follows that 𝑁((2𝑎)^2𝑚𝑥𝑓((2𝑎)^𝑚)−(2𝑎)^2𝑛𝑥𝑓((2𝑎)^𝑛),𝛿)≥𝑁((2𝑎)^2𝑚𝑥𝑓((2𝑎)^𝑚)−(2𝑎)^2𝑛𝑥𝑓((2𝑎)^𝑛),^𝑛−1_𝑘=𝑚(2𝑎)^2𝑘𝑡𝛼^𝑘)≥𝑁^′𝛼(𝜑(𝑥),2𝑡₀)≥1−𝜀 (2.61) for all 𝑡≥𝑡₀. This shows that the sequence {(2𝑎)^2𝑛𝑓(𝑥/(2𝑎)^𝑛)} is Cauchy in (𝑌,𝑁). Since (𝑌,𝑁) is complete, {(2𝑎)^2𝑛𝑓(𝑥/(2𝑎)^𝑛)} converges to some 𝑄(𝑥)∈𝑌. Thus we can define a mapping 𝑄∶𝑋→𝑌 by 𝑄(𝑥)∶=𝑁-lim_𝑡→∞(2𝑎)^2𝑛𝑓(𝑥/(2𝑎)^𝑛). Moreover, if we put 𝑚=0 in (2.60), then we observe that 𝑁((2𝑎)^2𝑛𝑥𝑓((2𝑎)^𝑛)−𝑓(𝑥),^𝑛−1_𝑘=0(2𝑎)^2𝑘𝑡𝛼^𝑘)≥𝑁^′𝛼(𝜑(𝑥),2𝑡). (2.62) Thus 𝑁((2𝑎)^2𝑛𝑥𝑓((2𝑎)^𝑛)−𝑓(𝑥),𝑡)≥𝑁^′(𝜑(𝑥),𝛼𝑡2∑^𝑛−1_𝑘=0((2𝑎)²/𝛼)^𝑘). (2.63) The rest of the proof is similar to the proof of Theorem 2.1.

Acknowledgment

Dr. Sun-Young Jang was supported by the Research Fund of University ofUlsan in 2008, and Dr. Choonkil Park was supported by National ResearchFoundation of Korea (NRF-2009-0070788).