On the Intuitionistic Fuzzy Stability of Ring Homomorphism and Ring Derivation

Abstract

We take into account the stability of ring homomorphism and ring derivation in intuitionistic fuzzy Banach algebra associated with the Jensen functional equation. In addition, we deal with the superstability of functional equation 𝑓(𝑥𝑦)=𝑥𝑓(𝑦)+𝑓(𝑥)𝑦 in an intuitionistic fuzzy normed algebra with unit.

1. Introduction and Preliminaries

The study of stability problems has originally been formulated by Ulam [1]: under what condition does there exists a homomorphism near an approximate homomorphism? 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. The paper work of Rassias [4] has had a lot of influence in the development of what is called the generalized Hyers-Ulam stability of functional equations. Thereafter, many interesting results of the generalized Hyers-Ulam stability to a number of functional equations have been investigated. In particular, Badora [5] gave a generalization of Bourgin's result [6], and he also dealt with the stability and the Bourgin-type superstability of derivations in [7]. Recently, fuzzy version is discussed in [8, 9]. Quite recently, the stability results in the setting of intuitionistic fuzzy normed space were studied in [10–13]; respectively, while the idea of intuitionistic fuzzy normed space was introduced in [14].

We now demonstrate some notations and basic definitions used in this work.

Definition 1. A binary operation ∗∶[0,1]×[0,1]→[0,1] is said to be a continuous t-norm if it satisfies the following conditions:(1)∗ is associative and commutative;(2)∗ is continuous;(3)𝑎∗1=𝑎 for all 𝑎∈[0,1];(4)𝑎∗𝑏≤𝑐∗𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑 for each 𝑎,𝑏,𝑐,𝑑∈[0,1].

Definition 2. A binary operation ⋄∶[0,1]×[0,1]→[0,1] is said to be a continuous t-conorm if it satisfies the following conditions:(1)⋄ is associative and commutative;(2)⋄ is continuous;(3)𝑎⋄0=𝑎 for all 𝑎∈[0,1];(4)𝑎⋄𝑏≤𝑐⋄𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑 for each 𝑎,𝑏,𝑐,𝑑∈[0,1].
Using the notions of continuous t-norm and t-conorm, Saadati and Park [14] have recently introduced the concept of intuitionistic fuzzy normed space as follows.

Definition 3. The five-tuple (𝒳,𝜇,𝜈,∗,⋄) is said to be an intuitionistic fuzzy normed space if 𝒳 is a vector space, ∗ is a continuous t-norm, ⋄ is a continuous t-conorm, and 𝜇,𝜈 are fuzzy sets on 𝒳×(0,∞) satisfying the following conditions. For every 𝑥,𝑦∈𝒳 and 𝑠,𝑡>0,(1)𝜇(𝑥,𝑡)+𝜈(𝑥,𝑡)≤1; (2)𝜇(𝑥,𝑡)>0; (3)𝜇(𝑥,𝑡)=1 if and only if 𝑥=0;(4)𝜇(𝛼𝑥,𝑡)=𝜇(𝛼𝑥,𝑡/|𝛼|) for each 𝛼≠0;(5)𝜇(𝑥,𝑡)∗𝜇(𝑦,𝑠)≤𝜇(𝑥+𝑦,𝑡+𝑠); (6)𝜇(𝑥,⋅)∶(0,∞)→[0,1] is continuous;(7)lim_𝑡→∞𝜇(𝑥,𝑡)=1 and lim_𝑡→0𝜇(𝑥,𝑡)=0;(8)𝜈(𝑥,𝑡)<1; (9)𝜈(𝑥,𝑡)=0 if and only if 𝑥=0;(10)𝜈(𝛼𝑥,𝑡)=𝜈(𝛼𝑥,𝑡/|𝛼|) for each 𝛼≠0;(11)𝜈(𝑥,𝑡)⋄𝜇(𝑦,𝑠)≥𝜈(𝑥+𝑦,𝑡+𝑠); (12)𝜈(𝑥,⋅)∶(0,∞)→[0,1] is continuous;(13)lim_𝑡→∞𝜈(𝑥,𝑡)=0 and lim_𝑡→0𝜈(𝑥,𝑡)=1.In this case, (𝜇,𝜈) is called an intuitionistic fuzzy norm.

Example 4. Let (𝒳,‖⋅‖) be a normed space, let 𝑎∗𝑏=𝑎𝑏, and let 𝑎⋄𝑏=min{𝑎+𝑏,1} for all 𝑎,𝑏∈[0,1]. For all 𝑥∈𝒳 and every 𝑡>0, consider 𝜇()=𝑡𝑥,𝑡‖𝑥‖𝜈()=‖𝑥‖𝑡+,if𝑡>0,0,if𝑡≤0,𝑥,𝑡𝑡+‖𝑥‖,if𝑡>0,0,if𝑡≤0. (1) Then (𝒳,𝜇,𝜈,∗,⋄) is an intuitionistic fuzzy normed space.

Definition 5 (see [15]). The five-tuple (𝒳,𝜇,𝜈,∗,⋄) is said to be an intuitionistic fuzzy normed algebra if 𝒳 is an algebra, ∗ is a continuous t-norm, ⋄ is a continuous t-conorm, and 𝜇,𝜈 are fuzzy sets on 𝒳×(0,∞) satisfying the conditions (1)–(13) of Definition 3. Furthermore, for every 𝑥,𝑦∈𝒳 and 𝑠,𝑡>0, (14) max{𝜇(𝑥,𝑡),𝜇(𝑦,𝑠)}≤𝜇(𝑥𝑦,𝑡+𝑠), (15) min{𝜈(𝑥,𝑡),𝜈(𝑦,𝑠)}≥𝜈(𝑥𝑦,𝑡+𝑠).
For an intuitionistic fuzzy normed algebra (𝒳,𝜇,𝜈,∗,⋄), we further assume that (16) 𝑎∗𝑎=𝑎 and 𝑎⋄𝑎=𝑎 for all 𝑎∈[0,1].
The concepts of convergence and Cauchy sequences in an intuitionistic fuzzy normed space are studied in [14]. Let (𝒳,𝜇,𝜈,∗,⋄) be an intuitionistic fuzzy normed space or intuitionistic fuzzy normed algebra. A sequence 𝑥={𝑥_𝑘} is said to be intuitionistic fuzzy convergent to 𝐿∈𝒳 if lim_𝑘→∞𝜇(𝑥_𝑘−𝐿,𝑡)=1 and lim_𝑘→∞𝜈(𝑥_𝑘−𝐿,𝑡)=0 for all 𝑡>0. In this case, we write (𝜇,𝜈)−lim_𝑛→∞𝑥_𝑘=𝐿 or 𝑥−−→_𝑘^𝐼𝐹𝐿 as 𝑘→∞. A sequence 𝑥={𝑥_𝑘} in (𝒳,𝜇,𝜈,∗,⋄) is said to be intuitionistic fuzzy Cauchy sequence if lim_𝑘→∞𝜇(𝑥_𝑘+𝑝−𝑥_𝑘,𝑡)=1 and lim_𝑘→∞𝜈(𝑥_𝑘+𝑝−𝑥_𝑘,𝑡)=0 for all 𝑡>0 and 𝑝=1,2,…. An intuitionistic fuzzy normed space or intuitionistic fuzzy normed algebra (𝒳,𝜇,𝜈,∗,⋄) is said to be complete if every intuitionistic fuzzy Cauchy sequence in (𝒳,𝜇,𝜈,∗,⋄) is intuitionistic fuzzy convergent in (𝒳,𝜇,𝜈,∗,⋄). A complete intuitionistic fuzzy normed space (resp., intuitionistic fuzzy normed algebra) is also called an intuitionistic fuzzy Banach space (resp., intuitionistic fuzzy Banach algebra).
In this work, we establish the stability of ring homomorphism and ring derivation in intuitionistic fuzzy Banach algebra associated to the Jensen functional equation 2𝑓(𝑥+𝑦/2)=𝑓(𝑥)+𝑓(𝑦). Moreover, we consider the superstability of functional equation 𝑓(𝑥𝑦)=𝑥𝑓(𝑦)+𝑓(𝑥)𝑦 in intuitionistic fuzzy normed algebra with unit.

2. The Main Results

Remark 6. The following theorem introduced in [10] is the generalized Hyers-Ulam theorem in intuitionistic fuzzy normed space for the Jensen functional equation. However, in order to arrive at the conclusion of the theorem, the assumptions 𝑎∗𝑎≥𝑎,𝑎⋄𝑎≤𝑎∀𝑎∈[0,1] (2) should be added.

Theorem 7. Let 𝒜 be a vector space and 𝑓 a mapping from 𝒜 to an intuitionistic fuzzy Banach space (ℬ,𝜇,𝜈,∗,⋄) with 𝑓(0)=0. Suppose that 𝜑 is a function from 𝒜 to an intuitionistic fuzzy normed space (𝐶,𝜇^′,𝜈^′,∗,⋄) such that 𝜇2𝑓𝑥+𝑦2−𝑓(𝑥)−𝑓𝑦,𝑡+𝑠≥𝜇^′𝜑(𝑥),𝑡∗𝜇^′𝜑𝑦,𝜈,𝑠2𝑓𝑥+𝑦2−𝑓(𝑥)−𝑓𝑦,𝑡+𝑠≤𝜈^′𝜑(𝑥),𝑡⋄𝜈^′𝜑𝑦,𝑠 (3) for all 𝑥,𝑦∈𝒜⧵{0}, 𝑡>0, and 𝑠>0. If 𝜑(3𝑥)=𝛼𝜑(𝑥) for some real number 𝛼 with 0<|𝛼|<3, then there exists a unique additive mapping ℒ∶𝒜→ℬ such that ℒ(𝑥)∶=(𝜇,𝜈)−lim_𝑛→∞(𝑓(3^𝑛𝑥)/3^𝑛), 𝜇ℒ(𝑥)(𝑥)()𝑡−𝑓,𝑡≥𝑀𝑥,3−𝛼6,𝜈ℒ(𝑥)(𝑥)()𝑡−𝑓,𝑡≤𝑁𝑥,3−𝛼6, (4) where 𝑀(𝑥,𝑡)∶=𝜇^′𝜑(𝑥),34𝑡∗𝜇^′𝜑(),3−𝑥4𝑡∗𝜇^′𝜑(),33𝑥4𝑡,𝑁(𝑥,𝑡)∶=𝜈^′𝜑(𝑥),34𝑡⋄𝜈^′𝜑(),3−𝑥4𝑡⋄𝜈^′𝜑(),33𝑥4𝑡. (5)

We begin with a generalized Hyers-Ulam theorem in intuitionistic fuzzy Banach algebra for the ring homomorphism.

Theorem 8. Let 𝒜 be an algebra and 𝑓 a mapping from 𝒜 to an intuitionistic fuzzy Banach algebra (ℬ,𝜇,𝜈,∗,⋄) with 𝑓(0)=0. Suppose that 𝜑 is a function from 𝒜 to an intuitionistic fuzzy normed algebra (𝐶,𝜇^′,𝜈^′,∗,⋄) satisfying (3) and that Φ is a function from 𝒜 to an intuitionistic fuzzy normed space (𝐷,𝜇^′′,𝜈^′′,∗,⋄) such that 𝜇𝑓(𝑥)𝑓𝑦𝜇𝑥𝑦−𝑓,𝑡+𝑠≥max^′′(Φ(𝑥),𝑡),𝜇^′′Φ𝑦,𝜈𝑓(𝑥)𝑓𝑦𝜈,𝑠𝑥𝑦−𝑓,𝑡+𝑠≤min^′′(Φ(𝑥),𝑡),𝜈^′′Φ𝑦,𝑠 (6) for all 𝑥,𝑦∈𝒜, 𝑡>0, and 𝑠>0. If 𝜑(3𝑥)=𝛼𝜑(𝑥) for some real number 𝛼 with 0<|𝛼|<3 and Φ(3𝑥)=𝛽Φ(𝑥) for some real number 𝛽 with 0<|𝛽|<3, then there exists a unique ring homomorphism ℒ∶𝒜→ℬ satisfying (4).

Proof. It follows by Theorem 7 that there exists a unique additive mapping ℒ∶𝒜→ℬ satisfying (4), where ℒ(𝑥)∶=(𝜇,𝜈)−lim_𝑛→∞(𝑓(3^𝑛𝑥)/3^𝑛).
Without loss of generality, we suppose that 0<𝛽<3. We prove that ℒ is a ring homomorphism. Note that 𝜇ℒ(𝑥)𝑓𝑦ℒ−𝑓3𝑥𝑦−ℒ,𝑡≥𝜇𝑥𝑦^𝑛𝑥𝑦3^𝑛,𝑡3𝑓3∗𝜇^𝑛𝑥𝑦3^𝑛−𝑓3^𝑛𝑥3^𝑛𝑓𝑦,𝑡3𝑓3∗𝜇^𝑛𝑥3^𝑛𝑓𝑦(𝑥)𝑓𝑦,𝑡−ℒ3,𝜈ℒ(𝑥)𝑓𝑦ℒ−𝑓3𝑥𝑦−ℒ,𝑡≤𝜈𝑥𝑦^𝑛𝑥𝑦3^𝑛,𝑡3𝑓3⋄𝜈^𝑛𝑥𝑦3^𝑛−𝑓3^𝑛𝑥3^𝑛𝑓𝑦,𝑡3𝑓3⋄𝜇^𝑛𝑥3^𝑛𝑓𝑦(𝑥)𝑓𝑦,𝑡−ℒ3 (7) for all 𝑥,𝑦∈𝒜 and 𝑡>0. On the other hand, we see that 𝜇𝑓3^𝑛𝑥⋅𝑦3^𝑛−𝑓3^𝑛𝑥3^𝑛𝑓𝑦,𝑡3=𝜇𝑓3^𝑛𝑥𝑦−𝑓3^𝑛𝑥𝑓𝑦,3^𝑛−1𝑡≥max𝜇^′′Φ(𝑥),3𝛽^𝑛𝑡6,𝜇^′′Φ𝑦,3^𝑛−1𝑡2,𝜈𝑓3^𝑛𝑥⋅𝑦3^𝑛−𝑓3^𝑛𝑥3^𝑛𝑓𝑦,𝑡3=𝜈𝑓3^𝑛𝑥𝑦−𝑓3^𝑛𝑥𝑓𝑦,3^𝑛−1𝑡≤min𝜈^′′Φ(𝑥),3𝛽^𝑛𝑡6,𝜈^′′Φ𝑦,3^𝑛−1𝑡2 (8) for all 𝑥,𝑦∈𝒜 and 𝑡>0 and 𝜇𝑓3^𝑛𝑥3^𝑛𝑓𝑦(𝑥)𝑓𝑦,𝑡−ℒ3=𝜇𝑓3^𝑛𝑥𝑓𝑦−3^𝑛ℒ(𝑥)𝑓𝑦,3^𝑛−1𝑡𝜇𝑓3≥max^𝑛𝑥3^𝑛(𝑥),𝑡−ℒ6,𝜇𝑓𝑦,3^𝑛−1𝑡2,𝜈𝑓3^𝑛𝑥3^𝑛𝑓𝑦(𝑥)𝑓𝑦,𝑡−ℒ3=𝜈𝑓3^𝑛𝑥𝑓𝑦−3^𝑛ℒ(𝑥)𝑓𝑦,3^𝑛−1𝑡𝜈𝑓3≤min^𝑛𝑥3^𝑛(𝑥),𝑡−ℒ6𝑓𝑦,3,𝜈^𝑛−1𝑡2 (9) for all 𝑥,𝑦∈𝒜 and 𝑡>0. Letting 𝑛→∞ in (7), (8), and (9), we get 𝜇ℒ(𝑥)𝑓𝑦𝜈ℒ(𝑥)𝑓𝑦𝑥𝑦−ℒ,𝑡=1,𝑥𝑦−ℒ,𝑡=0. (10) This implies that ℒ𝑥𝑦=ℒ(𝑥)𝑓𝑦 (11) for all 𝑥,𝑦∈𝒜.
Using additivity of ℒ and (11), we find that 3^𝑛ℒ(𝑥)𝑓𝑦=ℒ3^𝑛𝑥⋅𝑦=ℒ𝑥⋅3^𝑛𝑦=ℒ(𝑥)𝑓3^𝑛𝑦. (12) So we obtain ℒ(𝑥)𝑓(𝑦)=ℒ(𝑥)(𝑓(3^𝑛𝑦)/3^𝑛); that is, 𝜇ℒ(𝑥)𝑓𝑦(𝑥)𝑓3−ℒ^𝑛𝑦3^𝑛𝜈ℒ(𝑥)𝑓𝑦(𝑥)𝑓3,𝑡=1,−ℒ^𝑛𝑦3^𝑛,𝑡=0 (13) for all 𝑥,𝑦∈𝒜 and 𝑡>0. This relation yields that 𝜇ℒ(𝑥)𝑓𝑦(𝑥)ℒ𝑦ℒ(𝑥)𝑓𝑦(𝑥)𝑓3−ℒ,𝑡≥𝜇−ℒ^𝑛𝑦3^𝑛,𝑡2ℒ(𝑥)𝑓3∗𝜇^𝑛𝑦3^𝑛(𝑥)ℒ𝑦,𝑡−ℒ2ℒ(𝑥)𝑓3=𝜇^𝑛𝑦3^𝑛(𝑥)ℒ𝑦,𝑡−ℒ2,𝜈ℒ(𝑥)𝑓𝑦(𝑥)ℒ𝑦ℒ(𝑥)𝑓𝑦(𝑥)𝑓3−ℒ,𝑡≤𝜈−ℒ^𝑛𝑦3^𝑛,𝑡2ℒ(𝑥)𝑓3⋄𝜈^𝑛𝑦3^𝑛(𝑥)ℒ𝑦,𝑡−ℒ2ℒ(𝑥)𝑓3=𝜈^𝑛𝑦3^𝑛(𝑥)ℒ𝑦,𝑡−ℒ2 (14) for all 𝑥,𝑦∈𝒜 and 𝑡>0. On the other hand, we see that 𝜇ℒ(𝑥)𝑓3^𝑛𝑦3^𝑛(𝑥)ℒ𝑦,𝑡−ℒ2=𝜇ℒ(𝑥)𝑓3^𝑛𝑦−3^𝑛ℒ(𝑥)ℒ𝑦,3^𝑛𝑡2𝜇ℒ(𝑥),3≥max^𝑛𝑡4𝑓3,𝜇^𝑛𝑦3^𝑛𝑦,𝑡−ℒ4,𝜈ℒ(𝑥)𝑓3^𝑛𝑦3^𝑛(𝑥)ℒ𝑦,𝑡−ℒ2=𝜈ℒ(𝑥)𝑓3^𝑛𝑦−3^𝑛ℒ(𝑥)ℒ𝑦,3^𝑛𝑡2𝜈ℒ(𝑥),3≤min^𝑛𝑡4𝑓3,𝜈^𝑛𝑦3^𝑛𝑦,𝑡−ℒ4 (15) for all 𝑥,𝑦∈𝒜 and 𝑡>0. Sending 𝑛→∞ in (14) and (15), we have 𝜇ℒ(𝑥)𝑓𝑦(𝑥)ℒ𝑦𝜈ℒ(𝑥)𝑓𝑦(𝑥)ℒ𝑦−ℒ,𝑡=1,−ℒ,𝑡=0. (16) Thus, we conclude that ℒ(𝑥)𝑓𝑦=ℒ(𝑥)ℒ𝑦 (17) for all 𝑥,𝑦∈𝒜.
Therefore, by combining (11) and (17), we get ℒ(𝑥𝑦)=ℒ(𝑥)ℒ(𝑦), which completes the proof.

Now we recall that an additive mapping 𝑑 on an algebra 𝒜 is said to be a ring derivation if the functional equation 𝑑(𝑥𝑦)=𝑥𝑑(𝑦)+𝑑(𝑥)𝑦 holds for all 𝑥,𝑦∈𝒜.

Theorem 9. Let (𝒜,𝜇,𝜈,∗,⋄) be an intuitionistic fuzzy Banach algebra and 𝑓∶𝒜→𝒜 a mapping with 𝑓(0)=0. Assume that 𝜑 is a function from 𝒜 to an intuitionistic fuzzy normed space (𝐶,𝜇^′,𝜈^′,∗,⋄) satisfying (3) and that Φ is a function from 𝒜 to an intuitionistic fuzzy normed space (𝐷,𝜇^′′,𝜈^′′,∗,⋄) such that 𝜇𝑓𝑦(𝑥)𝜇𝑥𝑦−𝑥𝑓−𝑓𝑦,𝑡+𝑠≥max^′′(Φ(𝑥),𝑡),𝜇^′′Φ𝑦,𝜈𝑓𝑦(𝑥)𝜈,𝑠𝑥𝑦−𝑥𝑓−𝑓𝑦,𝑡+𝑠≤min^′′(Φ(𝑥),𝑡),𝜈^′′Φ𝑦,𝑠 (18) for all 𝑥,𝑦∈𝒜, 𝑡>0, and 𝑠>0. If 𝜑(3𝑥)=𝛼𝜑(𝑥) for some real number 𝛼 with 0<|𝛼|<3 and Φ(3𝑥)=𝛽Φ(𝑥) for some real number 𝛽 with 0<|𝛽|<3, then there exists a unique ring derivation ℒ∶𝒜→𝒜 satisfying (4). Moreover, 𝑥𝑓𝑦−ℒ𝑦=0 (19) is fulfilled for all 𝑥,𝑦∈𝒜.

Proof. By Theorem 7, there exists a unique additive mapping ℒ∶𝒜→ℬ satisfying (4), where ℒ(𝑥)∶=(𝜇,𝜈)−lim_𝑛→∞(𝑓(3^𝑛𝑥)/3^𝑛).
As in the proof of Theorem 8, we consider 0<|𝛽|<3. We show that ℒ is a ring derivation. Observe that 𝜇ℒ𝑦(𝑥)ℒ−𝑓3𝑥𝑦−𝑥𝑓−ℒ𝑦,𝑡≥𝜇𝑥𝑦^𝑛𝑥𝑦3^𝑛,𝑡3𝑓3∗𝜇^𝑛𝑥𝑦3^𝑛𝑦−𝑓3−𝑥𝑓^𝑛𝑥3^𝑛𝑡𝑦,3𝑓3∗𝜇^𝑛𝑥3^𝑛(𝑥)𝑡𝑦−ℒ𝑦,3,𝜈ℒ𝑦(𝑥)ℒ−𝑓3𝑥𝑦−𝑥𝑓−ℒ𝑦,𝑡≤𝜈𝑥𝑦^𝑛𝑥𝑦3^𝑛,𝑡3𝑓3⋄𝜈^𝑛𝑥𝑦3^𝑛𝑦−𝑓3−𝑥𝑓^𝑛𝑥3^𝑛𝑡𝑦,3𝑓3⋄𝜈^𝑛𝑥3^𝑛(𝑥)𝑡𝑦−ℒ𝑦,3 (20) for all 𝑥,𝑦∈𝒜 and 𝑡>0. On the other hand, we yield that 𝜇𝑓3^𝑛𝑥𝑦3^𝑛𝑦−𝑓3−𝑥𝑓^𝑛𝑥3^𝑛𝑡𝑦,3=𝜇𝑓3^𝑛𝑥⋅𝑦−3^𝑛𝑥⋅𝑓𝑦−𝑓3^𝑛𝑥𝑦,3^𝑛−1𝑡≥max𝜇^′′Φ(𝑥),3𝛽^𝑛𝑡6,𝜇^′′Φ𝑦,3^𝑛−1𝑡2,𝜈𝑓3^𝑛𝑥𝑦3^𝑛𝑦−𝑓3−𝑥𝑓^𝑛𝑥3^𝑛𝑡𝑦,3=𝜈𝑓3^𝑛𝑥⋅𝑦−3^𝑛𝑥⋅𝑓𝑦−𝑓3^𝑛𝑥𝑦,3^𝑛−1𝑡≤min𝜈^′′Φ(𝑥),3𝛽^𝑛𝑡6,𝜈^′′Φ𝑦,3^𝑛−1𝑡2 (21) for all 𝑥,𝑦∈𝒜 and 𝑡>0 and 𝜇𝑓3^𝑛𝑥3^𝑛(𝑥)𝑡𝑦−ℒ𝑦,3=𝜇𝑓3^𝑛𝑥𝑦−3^𝑛ℒ(𝑥)𝑦,3^𝑛−1𝑡𝜇𝑓3≥max^𝑛𝑥3^𝑛(𝑥),𝑡−ℒ63,𝜇𝑦,^𝑛−1𝑡2,𝜈𝑓3^𝑛𝑥3^𝑛(𝑥)𝑡𝑦−ℒ𝑦,3=𝜈𝑓3^𝑛𝑥𝑦−3^𝑛ℒ(𝑥)𝑦,3^𝑛−1𝑡𝜈𝑓3≤min^𝑛𝑥3^𝑛(𝑥),𝑡−ℒ63,𝜈𝑦,^𝑛−1𝑡2 (22) for all 𝑥,𝑦∈𝒜 and 𝑡>0. Letting 𝑛→∞ in (20), (21), and (22), we have 𝜇ℒ𝑦(𝑥)𝜈ℒ𝑦(𝑥)𝑥𝑦−𝑥𝑓−ℒ𝑦,𝑡=1,𝑥𝑦−𝑥𝑓−ℒ𝑦,𝑡=0. (23) So we get ℒ𝑥𝑦=𝑥𝑓𝑦+ℒ(𝑥)𝑦 (24) for all 𝑥,𝑦∈𝒜.
Due to additivity of ℒ and (24), we see that 3^𝑛𝑥𝑓𝑦+3^𝑛ℒ(𝑥)𝑦=ℒ3^𝑛𝑥⋅𝑦=ℒ𝑥⋅3^𝑛𝑦=𝑥𝑓3^𝑛𝑦+3^𝑛ℒ(𝑥)𝑦. (25) So we obtain 𝑥𝑓(𝑦)=𝑥(𝑓(3^𝑛𝑦)/3^𝑛); that is, 𝜇𝑦𝑓3𝑥𝑓−𝑥^𝑛𝑦3^𝑛𝜈𝑦𝑓3,𝑡=1,𝑥𝑓−𝑥^𝑛𝑦3^𝑛,𝑡=0 (26) for all 𝑥,𝑦∈𝒜 and 𝑡>0. From this, it follows that 𝜇𝑦𝑦𝑦𝑓3𝑥𝑓−𝑥ℒ,𝑡≥𝜇𝑥𝑓−𝑥^𝑛𝑦3^𝑛,𝑡2𝑥𝑓3∗𝜇^𝑛𝑦3^𝑛𝑦,𝑡−𝑥ℒ2𝑥𝑓3=𝜇^𝑛𝑦3^𝑛𝑦,𝑡−𝑥ℒ2,𝜈𝑦𝑦𝑦𝑓3𝑥𝑓−𝑥ℒ,𝑡≤𝜈𝑥𝑓−𝑥^𝑛𝑦3^𝑛,𝑡2𝑥𝑓3⋄𝜈^𝑛𝑦3^𝑛𝑦,𝑡−𝑥ℒ2𝑥𝑓3=𝜈^𝑛𝑦3^𝑛𝑦,𝑡−𝑥ℒ2 (27) for all 𝑥,𝑦∈𝒜 and 𝑡>0. On the other hand, we find that 𝜇𝑥𝑓3^𝑛𝑦3^𝑛𝑦,𝑡−𝑥ℒ2=𝜇𝑥𝑓3^𝑛𝑦−3^𝑛𝑦,3𝑥ℒ^𝑛𝑡2𝜇3≥max𝑥,^𝑛𝑡4𝑓3,𝜇^𝑛𝑦3^𝑛𝑦,𝑡−ℒ4,𝜈𝑥𝑓3^𝑛𝑦3^𝑛𝑦,𝑡−𝑥ℒ2=𝜈𝑥𝑓3^𝑛𝑦−3^𝑛𝑦,3𝑥ℒ^𝑛𝑡2𝜈3≤min𝑥,^𝑛𝑡4𝑓3,𝜈^𝑛𝑦3^𝑛𝑦,𝑡−ℒ4 (28) for all 𝑥,𝑦∈𝒜 and 𝑡>0. Taking 𝑛→∞ in (27) and (28), we obtain that 𝜇𝑥𝑓𝑦−𝑥ℒ𝑦,𝑡=1,𝜈𝑥𝑓𝑦−𝑥ℒ𝑦,𝑡=0. (29) Therefore, 𝑥𝑓𝑦=𝑥ℒ𝑦 (30) for all 𝑥,𝑦∈𝒜, which implies that condition (19) holds.
Comparing (24) and (30), we have ℒ(𝑥𝑦)=𝑥ℒ(𝑦)+ℒ(𝑥)𝑦. This completes the proof.

By considering the unit, the following result can be obtained easily from Theorem 9.

Corollary 10. Let (𝒜,𝜇,𝜈,∗,⋄) be an intuitionistic fuzzy Banach algebra with unit, and let 𝑓∶𝒜→𝒜 be a mapping with 𝑓(0)=0. Assume that 𝜑 is a function from 𝒜 to an intuitionistic fuzzy normed space (𝐶,𝜇^′,𝜈^′,∗,⋄) satisfying (3), and that Φ is a function from 𝒜 to an intuitionistic fuzzy normed space (𝐷,𝜇^′′,𝜈^′′,∗,⋄) satisfying (18). If 𝜑(3𝑥)=𝛼𝜑(𝑥) for some real number 𝛼 with 0<|𝛼|<3 and Φ(3𝑥)=𝛽Φ(𝑥) for some real number 𝛽 with 0<|𝛽|<3, then 𝑓∶𝒜→ℬ is a ring derivation.

We can also prove the preceding results for the case when 𝛼>3 and 𝛽>3. In this case, the mapping ℒ(𝑥)∶=lim_𝑛→∞3^𝑛𝑓(3^−𝑛𝑥). In the case of intuitionistic fuzzy normed algebras with unit, we can also prove the superstability of functional equation 𝑓(𝑥𝑦)=𝑥𝑓(𝑦)+𝑓(𝑥)𝑦 as follows.

Theorem 11. Let (𝒜,𝜇,𝜈,∗,⋄) be an intuitionistic fuzzy normed algebra with unit. Suppose that 𝑓∶𝒜→𝒜 is a mapping for which there exists a function Φ from 𝒜 to an intuitionistic fuzzy normed space (𝐷,𝜇^′′,𝜈^′′,∗,⋄) satisfying (18). If 𝑙>1 is a fixed integer and Φ(𝑙𝑥)=𝛽Φ(𝑥) for some real number 𝛽 with 0<|𝛽|<𝑙, then 𝑓 satisfies the functional equation 𝑓(𝑥𝑦)=𝑥𝑓(𝑦)+𝑓(𝑥)𝑦 for all 𝑥,𝑦∈𝒜.

Proof. Without loss of generality, we assume that 0<𝛽<𝑙. Here, we will denote the unit by 𝑒. We first note that 𝜇𝑓𝑙^𝑛𝑥⋅𝑦𝑙^𝑛𝑦−𝑓𝑙−𝑥𝑓^𝑛𝑥𝑙^𝑛𝑓𝑙𝑦,𝑡=𝜇^𝑛𝑥⋅𝑦−𝑙^𝑛𝑥⋅𝑓𝑦−𝑓𝑙^𝑛𝑥𝑦,𝑙^𝑛𝑡≥max𝜇^′′Φ(𝑥),𝑙𝛽^𝑛𝑡2,𝜇^′′Φ𝑦,𝑙^𝑛𝑡2,𝜈𝑓𝑙^𝑛𝑥⋅𝑦𝑙^𝑛𝑦−𝑓𝑙−𝑥𝑓^𝑛𝑥𝑙^𝑛𝑓𝑙𝑦,𝑡=𝜈^𝑛𝑥⋅𝑦−𝑙^𝑛𝑥⋅𝑓𝑦−𝑓𝑙^𝑛𝑥𝑦,𝑙^𝑛𝑡≤min𝜈^′′Φ(𝑥),𝑙𝛽^𝑛𝑡2,𝜈^′′Φ𝑦,𝑙^𝑛𝑡2 (31) for all 𝑥,𝑦∈𝒜 and 𝑡>0, which mean that 𝜇,𝜈−lim_𝑛⟶∞𝑓𝑙^𝑛𝑥⋅𝑦𝑙^𝑛−𝑓𝑙^𝑛𝑥𝑙^𝑛𝑦=𝑥𝑓𝑦 (32) for all 𝑥,𝑦∈𝒜. Secondly, we find that 𝜇𝑓𝑥⋅𝑙^𝑛𝑦𝑙^𝑛𝑓𝑙−𝑥^𝑛𝑦𝑙^𝑛(𝑥)𝑓−𝑓𝑦,𝑡=𝜇𝑥⋅𝑙^𝑛𝑦−𝑥𝑓𝑙^𝑛𝑦−𝑓(𝑥)⋅𝑙^𝑛𝑦,𝑙^𝑛𝑡≥max𝜇^′′Φ(𝑥),𝑙^𝑛𝑡2,𝜇^′′Φ𝑦,𝑙𝛽^𝑛𝑡2,𝜈𝑓𝑥⋅𝑙^𝑛𝑦𝑙^𝑛𝑓𝑙−𝑥^𝑛𝑦𝑙^𝑛(𝑥)𝑓−𝑓𝑦,𝑡=𝜈𝑥⋅𝑙^𝑛𝑦−𝑥𝑓𝑙^𝑛𝑦−𝑓(𝑥)⋅𝑙^𝑛𝑦,𝑙^𝑛𝑡≤min𝜈^′′Φ(𝑥),𝑙^𝑛𝑡2,𝜈^′′Φ𝑦,𝑙𝛽^𝑛𝑡2 (33) for all 𝑥,𝑦∈𝒜 and 𝑡>0. This implies that 𝜇,𝜈−lim_𝑛⟶∞𝑓𝑥⋅𝑙^𝑛𝑦𝑙^𝑛𝑓𝑙−𝑥^𝑛𝑦𝑙^𝑛=𝑓(𝑥)𝑦 (34) for all 𝑥,𝑦∈𝒜.
We also know that 𝜇𝑓𝑦(𝑥)𝑓−𝑓𝑙𝑥𝑦−𝑥𝑓−𝑓𝑦,𝑡≥𝜇𝑥𝑦^𝑛𝑒⋅𝑥⋅𝑦𝑙^𝑛+𝑓𝑙^𝑛𝑒𝑙^𝑛𝑦+𝑓𝑙𝑥𝑦−𝑥𝑓^𝑛𝑥⋅𝑦𝑙^𝑛−𝑓𝑙^𝑛𝑥𝑙^𝑛(𝑥)𝑦+𝑓𝑙𝑦−𝑓_𝑛𝑥⋅𝑦𝑙^𝑛𝑓𝑙−𝑥^𝑛𝑦𝑙^𝑛,𝑡2𝑓𝑙∗𝜇^𝑛𝑒⋅𝑥⋅𝑦𝑙^𝑛−𝑓𝑙^𝑛𝑒𝑙^𝑛𝑓𝑙𝑥𝑦−^𝑛𝑥⋅𝑦𝑙^𝑛+𝑓𝑙^𝑛𝑥𝑙^𝑛𝑓𝑙𝑦−^𝑛𝑥⋅𝑦𝑙^𝑛𝑓𝑙+𝑥^𝑛𝑦𝑙^𝑛,𝑡2,𝜈𝑓𝑦(𝑥)𝑓−𝑓𝑙𝑥𝑦−𝑥𝑓−𝑓𝑦,𝑡≥𝜈𝑥𝑦^𝑛𝑒⋅𝑥⋅𝑦𝑙^𝑛+𝑓𝑙^𝑛𝑒𝑙^𝑛𝑦+𝑓𝑙𝑥𝑦−𝑥𝑓^𝑛𝑥⋅𝑦𝑙^𝑛−𝑓𝑙^𝑛𝑥𝑙^𝑛(𝑥)𝑦+𝑓𝑙𝑦−𝑓_𝑛𝑥⋅𝑦𝑙^𝑛𝑓𝑙−𝑥^𝑛𝑦𝑙^𝑛,𝑡2𝑓𝑙⋄𝜈^𝑛𝑒⋅𝑥⋅𝑦𝑙^𝑛−𝑓𝑙^𝑛𝑒𝑙^𝑛𝑓𝑙𝑥𝑦−^𝑛𝑥⋅𝑦𝑙^𝑛+𝑓𝑙^𝑛𝑥𝑙^𝑛𝑓𝑙𝑦−^𝑛𝑥⋅𝑦𝑙^𝑛𝑓𝑙+𝑥^𝑛𝑦𝑙^𝑛,𝑡2, (35) for all 𝑥,𝑦∈𝒜 and 𝑡>0. On the other hand, we see that 𝜇𝑓−𝑓𝑙𝑥𝑦^𝑛𝑒⋅𝑥⋅𝑦𝑙^𝑛+𝑓𝑙^𝑛𝑒𝑙^𝑛𝑦+𝑓𝑙𝑥𝑦−𝑥𝑓_𝑛𝑥⋅𝑦𝑙^𝑛−𝑓𝑙^𝑛𝑥𝑙^𝑛(𝑥)𝑦+𝑓𝑙𝑦−𝑓_𝑛𝑥⋅𝑦𝑙^𝑛𝑓𝑙−𝑥^𝑛𝑦𝑙^𝑛,𝑡2𝑓−𝑓𝑙≥𝜇𝑥𝑦^𝑛𝑒⋅𝑥⋅𝑦𝑙^𝑛−𝑓𝑙^𝑛𝑒𝑙^𝑛,𝑡𝑥𝑦6𝑓𝑙∗𝜇^𝑛𝑥⋅𝑦𝑙^𝑛−𝑓𝑙^𝑛𝑥𝑙^𝑛𝑦𝑦,𝑡−𝑥𝑓6𝑓𝑙∗𝜇^𝑛𝑥⋅𝑦𝑙^𝑛𝑓𝑙−𝑥^𝑛𝑦𝑙^𝑛(𝑥)𝑡−𝑓𝑦,6,𝜈𝑓−𝑓𝑙𝑥𝑦^𝑛𝑒⋅𝑥⋅𝑦𝑙^𝑛+𝑓𝑙^𝑛𝑒𝑙^𝑛𝑦+𝑓𝑙𝑥𝑦−𝑥𝑓^𝑛𝑥⋅𝑦𝑙^𝑛−𝑓𝑙^𝑛𝑥𝑙^𝑛𝑦(𝑥)𝑓𝑙−𝑓𝑦+^𝑛𝑥⋅𝑦𝑙^𝑛𝑓𝑙−𝑥^𝑛𝑦𝑙^𝑛,𝑡2𝑓−𝑓𝑙≤𝜈𝑥𝑦^𝑛𝑒⋅𝑥⋅𝑦𝑙^𝑛−𝑓𝑙^𝑛𝑒𝑙^𝑛,𝑡𝑥𝑦6𝑓𝑙⋄𝜈^𝑛𝑥⋅𝑦𝑙^𝑛−𝑓𝑙^𝑛𝑥𝑙^𝑛𝑦𝑦,𝑡−𝑥𝑓6𝑓𝑙⋄𝜈^𝑛𝑥⋅𝑦𝑙^𝑛𝑓𝑙−𝑥^𝑛𝑦𝑙^𝑛(𝑥)𝑡−𝑓𝑦,6 (36) for all 𝑥,𝑦∈𝒜 and 𝑡>0 and 𝜇𝑓𝑙^𝑛𝑒⋅𝑥⋅𝑦𝑙^𝑛−𝑓𝑙^𝑛𝑒𝑙^𝑛𝑓𝑙𝑥𝑦−^𝑛𝑥⋅𝑦𝑙^𝑛+𝑓𝑙^𝑛𝑥𝑙^𝑛𝑓𝑙𝑦−^𝑛𝑥⋅𝑦𝑙^𝑛𝑓𝑙+𝑥^𝑛𝑦𝑙^𝑛,𝑡2=𝜇𝑓𝑙^𝑛𝑒⋅𝑥−𝑙^𝑛𝑒⋅𝑓(𝑥)−𝑓𝑙^𝑛𝑒𝑥𝑦+𝑙^𝑛𝑓(𝑥)𝑦+𝑥𝑓𝑙^𝑛𝑦−𝑓𝑙^𝑛,𝑙𝑥⋅𝑦^𝑛𝑡2≥max𝜇𝑓𝑙^𝑛𝑒⋅𝑥−𝑙^𝑛𝑒⋅𝑓(𝑥)−𝑓𝑙^𝑛𝑒𝑥𝑙𝑦,^𝑛𝑡4∗𝜇𝑙^𝑛𝑓(𝑥)𝑦+𝑥𝑓𝑙^𝑛𝑦−𝑓𝑙^𝑛,𝑙𝑥⋅𝑦^𝑛𝑡4,𝜈𝑓𝑙^𝑛𝑒⋅𝑥⋅𝑦𝑙^𝑛−𝑓𝑙^𝑛𝑒𝑙^𝑛𝑓𝑙𝑥𝑦−^𝑛𝑥⋅𝑦𝑙^𝑛+𝑓𝑙^𝑛𝑥𝑙^𝑛𝑓𝑙𝑦−^𝑛𝑥⋅𝑦𝑙^𝑛𝑓𝑙+𝑥^𝑛𝑦𝑙^𝑛,𝑡2=𝜈𝑓𝑙^𝑛𝑒⋅𝑥−𝑙^𝑛𝑒⋅𝑓(𝑥)−𝑓𝑙^𝑛𝑒𝑥𝑦+𝑙^𝑛𝑓(𝑥)𝑦+𝑥𝑓𝑙^𝑛𝑦−𝑓𝑙^𝑛,𝑙𝑥⋅𝑦^𝑛𝑡2≤min𝜈𝑓𝑙^𝑛𝑒⋅𝑥−𝑙^𝑛(𝑥)𝑙𝑒⋅𝑓−𝑓^𝑛𝑒𝑥𝑙𝑦,^𝑛𝑡4⋄𝜈𝑙^𝑛𝑓(𝑥)𝑦+𝑥𝑓𝑙^𝑛𝑦−𝑓𝑙^𝑛,𝑙𝑥⋅𝑦^𝑛𝑡4 (37) for all 𝑥,𝑦∈𝒜 and 𝑡>0. In particular, we obtain the following: 𝜇𝑓𝑙^𝑛𝑒⋅𝑥−𝑙^𝑛𝑒⋅𝑓(𝑥)−𝑓𝑙^𝑛𝑒𝑥𝑙𝑦,^𝑛𝑡4≥max𝜇𝑓𝑙^𝑛𝑒⋅𝑥−𝑙^𝑛(𝑥)𝑙𝑒⋅𝑓−𝑓^𝑛𝑒𝑙𝑥,^𝑛𝑡8𝑙,𝜇𝑦,^𝑛𝑡8,𝜈𝑓𝑙^𝑛𝑒⋅𝑥−𝑙^𝑛𝑒⋅𝑓(𝑥)−𝑓𝑙^𝑛𝑒𝑥𝑙𝑦,^𝑛𝑡4≤min𝜈𝑓𝑙^𝑛𝑒⋅𝑥−𝑙^𝑛(𝑥)𝑙𝑒⋅𝑓−𝑓^𝑛𝑒𝑙𝑥,^𝑛𝑡8𝑙,𝜈𝑦,^𝑛𝑡8,𝜇𝑓𝑙^𝑛𝑒⋅𝑥−𝑙^𝑛𝑒⋅𝑓(𝑥)−𝑓𝑙^𝑛𝑒𝑙𝑥,^𝑛𝑡8≥max𝜇^′′Φ(𝑒),𝑙𝛽^𝑛𝑡16,𝜇^′′Φ(𝑥),𝑙^𝑛𝑡,𝜈𝑓𝑙16_𝑛𝑒⋅𝑥−𝑙^𝑛𝑒⋅𝑓(𝑥)−𝑓𝑙^𝑛𝑒𝑙𝑥,^𝑛𝑡8≤min𝜈^′′Φ(𝑒),𝑙𝛽^𝑛𝑡16,𝜈^′′Φ(𝑥),𝑙^𝑛𝑡16 (38) for all 𝑥,𝑦∈𝒜 and 𝑡>0. Also, we arrive at 𝜇𝑙^𝑛𝑓(𝑥)𝑦+𝑥𝑓𝑙^𝑛𝑦−𝑓𝑙^𝑛,𝑙𝑥⋅𝑦^𝑛𝑡4=𝜇𝑓𝑥⋅𝑙^𝑛𝑦−𝑥𝑓𝑙^𝑛𝑦−𝑙^𝑛𝑓(𝑥)𝑙𝑦,^𝑛𝑡4≥max𝜇^′′Φ(𝑥),𝑙^𝑛𝑡8,𝜇^′′Φ𝑦,𝑙𝛽^𝑛𝑡8,𝜈𝑙^𝑛𝑓(𝑥)𝑦+𝑥𝑓𝑙^𝑛𝑦−𝑓𝑙^𝑛,𝑙𝑥⋅𝑦^𝑛𝑡4=𝜈𝑓𝑥⋅𝑙^𝑛𝑦−𝑥𝑓𝑙^𝑛𝑦−𝑙^𝑛𝑓(𝑥)𝑙𝑦,^𝑛𝑡4≤min𝜈^′′Φ(𝑥),𝑙^𝑛𝑡8,𝜈^′′Φ𝑦,𝑙𝛽^𝑛𝑡8. (39)
Letting 𝑛→∞ in (35)–(39) with (32) and (34), we get the desired result. This completes the proof.

We remark that we can also verify Theorem 11 for the case when 𝛽>𝑙.

Acknowledgments

The authors would like to thank the referees for giving useful suggestions and for the improvement of this paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2010-002338 and no. 2013R1A1A2A10004419). This research was also supported by the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (no. 2012R1A1A2021721).