# Higher Ring Derivation and Intuitionistic Fuzzy Stability

Authors: Ick-Soon Chang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We take account of the stability of higher ring derivation in intuitionistic fuzzy Banach algebra associated to the Jensen type functional equation. In addition, we deal with the superstability of higher ring derivation in intuitionistic fuzzy Banach algebra with unit.


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## Body

## 1. Introduction and Preliminaries

The stability problem of functional equations has originally been formulated by Ulam [1]: under what condition does there exist a homomorphism near an approximate homomorphism? Hyers [2] answered the problem of Ulam under the assumption that the groups are Banach spaces. A generalized version of the theorem of Hyers for approximately additive mappings was given by Aoki [3] and for approximately linear mappings was presented by Rassias [4] 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. Since then, more generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings have been investigated (e.g., [5–7]). In particular, Badora [8] gave a generalization of the Bourgin's result [9], and he also dealt with the stability and the Bourgin-type superstability of derivations in [10]. Recently, fuzzy version is discussed in [11, 12]. Quite recently, the intuitionistic fuzzy stability problem for Jensen functional equation and cubic functional equation is considered in [13–15], respectively, while the idea of intuitionistic fuzzy normed space was introduced in [16], and there are some recent and important results which are directly related to the central theme of this paper, that is, intuitionistic fuzziness (see e.g., [17–20]).In this paper, we establish the stability of higher ring derivation in intuitionistic fuzzy Banach algebra associated to the Jensen type functional equationlf(x+y/l)=f(x)+f(y). Moreover, we consider the superstability of higher ring derivation in intuitionistic fuzzy Banach algebra with unit.We now recall some notations and basic definitions used in this paper.Definition 1.1 (see [5]).
Let𝒜 and ℬ be algebras over the real or complex field 𝔽. Let ℕ be the set of the natural numbers. From m∈ℕ∪{0}, a sequence H={h0,h1,…,hm} (resp., H={h0,h1,…,hk,…}) of additive operators from 𝒜 into ℬ is called a higher ring derivation of rank m (resp., infinite rank) if the functional equation hk(xy)=∑i=0k‍hi(x)hk-i(y) holds for each k=0,1,…,m (resp., k=0,1,…) and for all x,y∈𝒜. A higher ring derivation H of additive operators on 𝒜, particularly, is called strong if h0 is an identity operator.
Of course, a higher ring derivation of rank 0 from𝒜 into ℬ (resp., a strong higher ring derivation of rank 1 on 𝒜) is a ring homomorphism (resp., a ring derivation). Note that a higher ring derivation is a generalization of both a ring homomorphism and a ring derivation.Definition 1.2.
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],and(4)a*b≤c*d whenever a≤c and b≤d for each a,b,c,d∈[0,1].Definition 1.3.
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],and(4)a⋄b≤c⋄d whenever a≤c and b≤d for each a,b,c,d∈[0,1].Using the notions of continuoust-norm and t-conorm, Saadati and Park [16] have recently introduced the concept of intuitionistic fuzzy normed space as follows.Definition 1.4.
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(x,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) limt→∞μ(x,t)=1 and limt→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(x,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) limt→∞ν(x,t)=0 and limt→0ν(x,t)=1.
In this case,(μ,ν) is called an intuitionistic fuzzy norm.Example 1.5.
Let(𝒳,∥·∥) be a normed space, a*b=ab, and a⋄b=min{a+b,1} for all a,b∈[0,1]. For all x∈𝒳 and every t>0, consider
(1.1)μ(x,t)={1,ift>‖x‖,0,ift≤‖x‖,ν(x,t)={0,ift>‖x‖,1,ift≤‖x‖.
Then (𝒳,μ,ν,*,⋄) is an intuitionistic fuzzy normed space.Example 1.6.
Let(𝒳,∥·∥) be a normed space, a*b=ab, and a⋄b=min{a+b,1} for all a,b∈[0,1]. For all x∈𝒳 and every t>0 and k=1,2, consider
(1.2)μ(x,t)={tt+‖x‖,ift>0,0,ift≤0,ν(x,t)={k‖x‖t+k‖x‖,ift>0,0,ift≤0.
Then (𝒳,μ,ν,*,⋄) is an intuitionistic fuzzy normed space.Definition 1.7 (see [21]).
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 the Definition 1.4. 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 [16]. Let (𝒳,μ,ν,*,⋄) be an intuitionistic fuzzy normed space or intuitionistic fuzzy normed algebra. A sequence x={xk} is said to be intuitionistic fuzzy convergent to L∈𝒳 if limk→∞μ(xk-L,t)=1 and limk→∞ν(xk-L,t)=0 for all t>0. In this case, we write (μ,ν)-limk→∞xk=L or xk→IFL as k→∞. A sequence x={xk} in (𝒳,μ,ν,*,⋄) is said to be intuitionistic fuzzy Cauchy sequence if limk→∞μ(xk+p-xk,t)=1 and limk→∞ν(xk+p-xk,t)=0 for all t>0 and p=1,2,…. An intuitionistic fuzzy normed space (resp., 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).


## 2. Stability of Higher Ring Derivation in Intuitionistic FuzzyBanach Algebra

As a matter of convenience in this paper, we use the following abbreviation:(2.1)∏j=0n‍aj:=a1*a2*⋯*an,∏j=0∞‍aj:=a1*a2*⋯.
In addition,
(2.2)∐j=0n‍aj:=a1⋄a2⋄⋯⋄an,∐j=0∞‍aj:=a1⋄a2⋄⋯.We begin with a generalized Hyers-Ulam theorem in intuitionistic fuzzy Banach space for the Jensen type functional equation. The following result is also the generalization of the theorem introduced in [13].Theorem 2.1.
Let𝒜 be a vector space, and let f be 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
(2.3)μ(lf(x+yl)-f(x)-f(y),t+s)≥μ′(φ(x),t)*μ′(φ(y),s),(2.4)ν(lf(x+yl)-f(x)-f(y),t+s)≤ν′(φ(x),t)⋄ν′(φ(y),s)
for all x,y∈𝒜∖{0},t>0 and s>0. If l>1 is a fixed integer, and φ((l+1)x)=αφ(x) for some real number α with 0<|α|<l+1, then there exists a unique additive mapping ℒ:𝒜→ℬ such that ℒ(x):=(μ,ν)-limn→∞(f((l+1)nx)/(l+1)n),
(2.5)μ(L(x)-f(x),t)≥∏j=0∞‍M(x,((l+1)-α)t2(l+1)),ν(L(x)-f(x),t)≤∐j=0∞‍N(x,((l+1)-α)t2(l+1))
for all x∈𝒜 and t>0, where
(2.6)M(x,t):=μ′(φ(x),l+14t)*μ′(φ(-x),l+14t)*μ′(φ(-x),l+14t)*μ′(φ((l+1)x),l+14t),N(x,t):=ν′(φ(x),l+14t)⋄ν′(φ(-x),l+14t)⋄ν′(φ(-x),l+14t)⋄ν′(φ((l+1)x),l+14t).Proof.
Without loss of generality, we assume that0<α<l+1. From (2.3) and (2.4), we get
(2.7)μ(f(x)+f(-x),lt)≥μ′(φ(x),l2t)*μ′(φ(-x),l2t),ν(f(x)+f(-x),lt)≤ν′(φ(x),l2t)⋄(φ(-x),l2t)
for all x∈𝒜 and t>0. Again, by (2.3) and (2.4), we obtain
(2.8)μ(lf(x)-f(-x)-f((l+1)x),lt)≥μ′(φ(-x),l2t)*μ′(φ((l+1)x),l2t),ν(lf(x)-f(-x)-f((l+1)x),lt)≤ν′(φ(-x),l2t)⋄ν′(φ((l+1)x),l2t)
for all x∈𝒜 and t>0. Combining (2.7) and (2.8), we arrive at
(2.9)μ((l+1)f(x)-f((l+1)x),2lt)≥μ′(φ(x),l2t)*μ′(φ(-x),l2t)*μ′(φ(-x),l2t)*μ′(φ((l+1)x),l2t),ν((l+1)f(x)-f((l+1)x),2lt)≤ν′(φ(x),l2t)⋄ν′(φ(-x),l2t)⋄ν′(φ(-x),l2t)⋄ν′(φ((l+1)x),l2t),
for all x∈𝒜 and t>0. This implies that
(2.10)μ(f(x)-f((l+1)x)(l+1),t)≥μ′(φ(x),l+14t)*μ′(φ(-x),l+14t)*μ′(φ(-x),l+14t)*μ′(φ((l+1)x),l+14t),ν(f(x)-f((l+1)x)(l+1),t)≤ν′(φ(x),l+14t)⋄ν′(φ(-x),l+14t)⋄ν′(φ(-x),l+14t)⋄ν′(φ((l+1)x),l+14t),
for all x∈𝒜 and t>0. Now we define
(2.11)M(x,t):=μ′(φ(x),l+14t)*μ′(φ(-x),l+14t)*μ′(φ(-x),l+14t)*μ′(φ((l+1)x),l+14t),N(x,t):=ν′(φ(x),l+14t)⋄ν′(φ(-x),l+14t)⋄ν′(φ(-x),l+14t)⋄ν′(φ((l+1)x),l+14t),
for all x∈𝒜 and t>0. Then we have by assumption
(2.12)M((l+1)x,t)=M(x,tα),N((l+1)x,t)=N(x,tα),
for all x∈𝒜 and t>0. Using (2.10) and (2.12), we get
(2.13)μ(f((l+1)nx)(l+1)n-f((l+1)n+1x)(l+1)n+1,αnt(l+1)n)=μ(f((l+1)nx)-f((l+1)n+1x)l+1,αnt)≥M((l+1)nx,αnt)=M(x,t),ν(f((l+1)nx)(l+1)n-f((l+1)n+1x)(l+1)n+1,αnt(l+1)n)=ν(f((l+1)nx)-f((l+1)n+1x)l+1,αnt)≤N((l+1)nx,αnt)=N(x,t),
for all x∈𝒜 and t>0. Therefore, for all n>m, we have
(2.14)μ(f((l+1)mx)(l+1)m-f((l+1)nx)(l+1)n,∑j=mn-1‍αjt(l+1)j)=μ(∑j=mn-1‍[f((l+1)jx)(l+1)j-f((l+1)j+1x)(l+1)j+1],∑j=mn-1‍αjt(l+1)j)≥∏j=mn-1‍μ(f((l+1)jx)(l+1)j-f((l+1)j+1x)(l+1)j+1,αjt(l+1)j)≥∏j=mn-1‍M(x,t),ν(f((l+1)mx)(l+1)m-f((l+1)nx)(l+1)n,∑j=mn-1‍αjt(l+1)j)=ν(∑j=mn-1‍[f((l+1)jx)(l+1)j-f((l+1)j+1x)(l+1)j+1],∑j=mn-1‍αjt(l+1)j)≤∐j=mn-1‍ν(f((l+1)jx)(l+1)j-f((l+1)j+1x)(l+1)j+1,αjt(l+1)j)≤∐j=mn-1‍N(x,t),
for all x∈𝒜 and t>0. Let ɛ>0 and δ>0 be given. Since limt→∞∏j=mn-1‍M(x,t)=1 and limt→∞∐j=mn-1‍N(x,t)=0, there exists some t0 such that ∏j=mn-1‍M(x,t0)>1-ɛ,∐j=mn-1‍N(x,t0)<ɛ. Since ∑j=0∞‍(αjt/(l+1)j)<∞, there exists a positive integer n0 such that ∑j=mn-1‍(αjt/(l+1)j)<δ for all n>m≥n0.
Then(2.15)μ(f((l+1)mx)(l+1)m-f((l+1)nx)(l+1)n,δ)≥μ(f((l+1)mx)(l+1)m-f((l+1)nx)(l+1)n,∑j=mn-1‍αjt0(l+1)j)≥∏j=mn-1‍M(x,t0)>1-ɛ,ν(f((l+1)mx)(l+1)m-f((l+1)nx)(l+1)n,δ)≤ν(f((l+1)mx)(l+1)m-f((l+1)nx)(l+1)n,∑j=mn-1‍αjt0(l+1)j)≤∐j=mn-1‍N(x,t0)<ɛ.
This shows that {(f((l+1)nx))/((l+1)n)} is a Cauchy sequence in (ℬ,μ′,ν′,*,⋄). Since ℬ is complete, we can define a mapping ℒ by ℒ(x):=(μ,ν)-limn→∞(f((l+1)nx)/(l+1)n) for all x∈𝒜. Moreover, if we let m=0 in (2.14), then we get
(2.16)μ(f((l+1)nx)(l+1)n-f(x),∑j=0n-1‍αjt(l+1)j)≥∏j=0n-1‍M(x,t),ν(f((l+1)nx)(l+1)n-f(x),∑j=0n-1‍αjt(l+1)j)≤∐j=0n-1‍N(x,t),
for all x∈𝒜 and t>0. Therefore, we find that
(2.17)μ(f((l+1)nx)(l+1)n-f(x),t)≥∏j=0n-1‍M(x,t∑j=0n-1(αj/(l+1)j)),ν(f((l+1)nx)(l+1)n-f(x),t)≤∐j=0n-1‍N(x,t∑j=0n-1(αj/(l+1)j)).
Next, we will show thatℒ is additive mapping. Note that
(2.18)μ(lL(x+yl)-L(x)-L(y),t)≥μ(lL(x+yl)-lf(((l+1)n(x+y))/l)(l+1)n,t4)*μ(f((l+1)nx)(l+1)n-L(x),t4)*μ(f((l+1)ny)(l+1)n-L(y),t4)*μ(lf(((l+1)n(x+y))/(l))(l+1)n-f((l+1)nx)(l+1)n-f((l+1)ny)(l+1)n,t4),ν(lL(x+yl)-L(x)-L(y),t)≤ν(lL(x+yl)-lf(((l+1)n(x+y))/l)(l+1)n,t4)⋄ν(f((l+1)nx)(l+1)n-L(x),t4)⋄ν(f((l+1)ny)(l+1)n-L(y),t4)⋄ν(lf(((l+1)n(x+y))/(l))(l+1)n-f((l+1)nx)(l+1)n-f((l+1)ny)(l+1)n,t4).
On the other hand, (2.3) and (2.4) give the following:
(2.19)μ(lf(((l+1)n(x+y))/l)(l+1)n-f((l+1)nx)(l+1)n-f((l+1)ny)(l+1)n,t4)≥μ′(φ(x),(l+1α)nt8)*μ′(φ(y),(l+1α)nt8),ν(lf(((l+1)n(x+y))/l)(l+1)n-f((l+1)nx)(l+1)n-f((l+1)ny)(l+1)n,t4)≤ν′(φ(x),(l+1α)nt8)⋄ν′(φ(y),(l+1α)nt8).
Letting n→∞ in (2.18) and (2.19), we yield
(2.20)μ(lL(x+yl)-L(x)-L(y),t)=1,ν(lL(x+yl)-L(x)-L(y),t)=0.
So we see that ℒ is additive mapping.
Now, we approximate the difference betweenf and ℒ in an intuitionistic fuzzy sense. By (2.17), we get
(2.21)μ(L(x)-f(x),t)≥μ(L(x)-f((l+1)nx)(l+1)n,t2)*μ(f((l+1)nx)(l+1)n-f(x),t2)≥∏j=0∞‍M(x,((l+1)-α)t2(l+1)),ν(L(x)-f(x),t)≤ν(L(x)-f((l+1)nx)(l+1)n,t2)⋄ν(f((l+1)nx)(l+1)n-f(x),t2)≤∐j=0∞‍N(x,((l+1)-α)t2(l+1)),
for all x∈𝒜 and t>0 and sufficiently large n.
In order to prove the uniqueness ofℒ, we assume that T is another additive mapping from 𝒜 to ℬ, which satisfies the inequality (2.5). Then
(2.22)μ(L(x)-T(x),t)≥μ(L(x)-f(x),t2)*μ(T(x)-f(x),t2)≥∏j=0∞‍M(x,((l+1)-α)t4(l+1)),ν(L(x)-T(x),t)≤ν(L(x)-f(x),t2)⋄ν(T(x)-f(x),t2)≤∐j=0∞‍N(x,((l+1)-α)t4(l+1)),
for all x∈𝒜 and t>0. Therefore, due to the additivity of ℒ and T, we obtain that
(2.23)μ(L(x)-T(x),t)=μ(L((l+1)nx)-T((l+1)nx),(l+1)nt)≥∏j=0∞‍M(x,(l+1α)n((l+1)-α)t4(l+1)),ν(L(x)-T(x),t)=ν(L((l+1)nx)-T((l+1)nx),(l+1)nt)≤∐j=0∞‍M(x,(l+1α)n((l+1)-α)t4(l+1)).
Since 0<α<l+1,limn→∞((l+1)/α)n=∞, and we get
(2.24)lim⁡n→∞M(x,(l+1α)n((l+1)-α)t4(l+1))=1,lim⁡n→∞N(x,(l+1α)n((l+1)-α)t4(l+1))=0,
that is, μ(ℒ(x)-T(x),t)=1 and ν(ℒ(x)-T(x),t)=0 for all x∈𝒜,t>0. So ℒ=T, which completes the proof.In particular, we can prove the preceding result for the case whenα>l+1. In this case, the mapping ℒ(x):=(μ,ν)-limn→∞(l+1)nf((l+1)-nx). We now establish a generalized Hyers-Ulam stability in intuitionistic fuzzy Banach algebra for the higher ring derivation.Theorem 2.2.
Let𝒜 be an algebra, and let F={f0,f1,…,fk,…} be a sequence of mappings from 𝒜 to an intuitionistic fuzzy Banach algebra (ℬ,μ,ν,*,⋄) with fk(0)=0 for each k=0,1,…. Suppose that φ is a function from 𝒜 to an intuitionistic fuzzy normed algebra (C,μ′,ν′,*,⋄) such that for each k=0,1,…,
(2.25)μ(lfk(x+yl)-fk(x)-fk(y),t+s)≥μ′(φ(x),t)*μ′(φ(y),s),ν(lfk(x+yl)-fk(x)-fk(y),t+s)≤ν′(φ(x),t)⋄ν′(φ(y),s)
for all x,y∈𝒜∖{0},t>0 and s>0, and that Φ is a function from 𝒜 to an intuitionistic fuzzy normed space (D,μ′′,ν′′,*,⋄) such that for each k=0,1,…,
(2.26)μ(fk(xy)-∑i=0k‍fi(x)fk-i(y),t+s)≥max⁡{μ′′(Φ(x),t),μ′′(Φ(y),s)},ν(fk(xy)-∑i=0k‍fi(x)fk-i(y),t+s)≤min⁡{ν′′(Φ(x),t),ν′′(Φ(y),s)}
for all x,y∈𝒜,t>0, and s>0. If l>1 is a fixed integer, φ((l+1)x)=αφ(x), and Φ((l+1)x)=βΦ(x) for some real numbers α and β with 0<|α|<l+1 and 0<|β|<l+1, then there exists a unique higher ring derivation H={ℒ0,ℒ1,…,ℒk,…} of any rank such that for each k=0,1,…,
(2.27)μ(Lk(x)-fk(x),t)≥M(x,((l+1)-α)t2(l+1)),ν(Lk(x)-fk(x),t)≤N(x,((l+1)-α)t2(l+1)),
for all x∈𝒜 and t>0. In this case,
(2.28)M(x,t):=μ′(φ(x),l+14t)*μ′(φ(-x),l+14t)*μ′(φ((l+1)x),l+14t),N(x,t):=ν′(φ(x),l+14t)⋄ν′(φ(-x),l+14t)⋄ν′(φ((l+1)x),l+14t).
Moreover, the identity
(2.29)∑i=0k‍Li(y){Lk-i(y)-fk-i(y)}=0
holds for each k=0,1,… and all x,y∈𝒜.Proof.
It follows by Theorem2.1 that for each k=0,1,… and all x∈𝒜, there exists a unique additive mapping ℒk:𝒜→ℬ given by
(2.30)Lk(x):=(μ,ν)-lim⁡n→∞fk((l+1)nx)(l+1)n,
satisfying (2.27) since (C,μ′,ν′,*,⋄) is an intuitionistic fuzzy normed algebra.
Without loss of generality, we suppose that0<β<l+1. Now, we need to prove that the sequence H={ℒ0,ℒ1,…,ℒk,…} satisfies the identity ℒk(xy)=∑i=0k‍ℒi(x)ℒk-i(y) for each k=0,1,… and all x∈𝒜. It is observed that for each k=0,1,…,
(2.31)μ(Lk(xy)-∑i=0k‍Li(x)fk-i(y),t)≥μ(Lk(xy)-fk((l+1)nxy)(l+1)n,t3)*μ(fk((l+1)nxy)(l+1)n-∑i=0k‍fi((l+1)nx)(l+1)nfk-i(y),t3)*μ(∑i=0k‍fi((l+1)nx)(l+1)nfk-i(y)-∑i=0k‍Li(x)fk-i(y),t3),ν(Lk(xy)-∑i=0k‍Li(x)fk-i(y),t)≤ν(Lk(xy)-fk((l+1)nxy)(l+1)n,t3)⋄ν(fk((l+1)nxy)(l+1)n-∑i=0k‍fi((l+1)nx)(l+1)nfk-i(y),t3)⋄ν(∑i=0k‍fi((l+1)nx)(l+1)nfk-i(y)-∑i=0k‍Li(x)fk-i(y),t3)
for all x,y∈𝒜 and t>0. On account of (2.26), we see that for each k=0,1,…,
(2.32)μ(fk((l+1)nx⋅y)(l+1)n-∑i=0k‍fi((l+1)nx)(l+1)nfk-i(y),t3)=μ(fk((l+1)nx⋅y)-∑i=0k‍fi((l+1)nx)fk-i(y),(l+1)nt3)≥max⁡{μ′′(Φ(x),(l+1β)nt6),μ′′(Φ(y),(l+1)nt6)},ν(fk((l+1)nx⋅y)(l+1)n-∑i=0k‍fi((l+1)nx)(l+1)nfk-i(y),(l+1)nt3)=ν(fk((l+1)nx⋅y)-∑i=0k‍fi((l+1)nx)fk-i(y),(l+1)nt3)≤min⁡{μ′′(Φ(x),(l+1β)nt6),ν′′(Φ(y),(l+1)nt6)},
for all x,y∈𝒜 and t>0. Due to additivity of ℒk, for each k=0,1,…,
(2.33)μ(∑i=0k‍fi((l+1)nx)(l+1)nfk-i(y)-∑i=0k‍Li(x)fk-i(y),t3)≥∏i=0k‍μ(fi((l+1)nx)fk-i(y)-(l+1)nLi(x)fk-i(y),(l+1)nt3(k+1)),ν(∑i=0k‍fi((l+1)nx)(l+1)nfk-i(y)-∑i=0k‍Li(x)fk-i(y),t3)≤∐i=0k‍ν(fi((l+1)nx)fk-i(y)-(l+1)nLi(x)fk-i(y),(l+1)nt3(k+1))
for all x,y∈𝒜 and t>0. In addition, we feel that
(2.34)μ(fi((l+1)nx)fk-i(y)-(l+1)nLi(x)fk-i(y),(l+1)nt3(k+1))≥max⁡{μ(fi((l+1)nx)-(l+1)nLi(x),(l+1)nt6(k+1)),μ(fk-i(y),(l+1)nt6(k+1))},ν(fi((l+1)nx)fk-i(y)-(l+1)nLi(x)fk-i(y),(l+1)nt3(k+1))≤min⁡{ν(fi((l+1)nx)-(l+1)nLi(x),(l+1)nt6(k+1)),ν(fk-i(y),(l+1)nt6(k+1))}.
Letting n→∞ in (2.31), (2.32), (2.33), and (2.34), we get μ(ℒk(xy)-∑i=0k‍ℒi(x)fk-i(y),t)=1 and ν(ℒk(xy)-∑i=0k‍ℒi(x)fk-i(y),t)=0. This implies that
(2.35)Lk(xy)=∑i=0k‍Li(x)fk-i(y),
for each k=0,1,… and all x,y∈𝒜.
Using additivity ofℒk and (2.35), we find that
(2.36)(l+1)n∑i=0k‍Li(x)fk-i(y)=Lk((l+1)nx⋅y)=Lk(x⋅(l+1)ny)=∑i=0k‍L(x)f((l+1)ny).
So we obtain ∑i=0k‍ℒi(x)fk-i(y)=∑i=0k‍ℒi(x)(fk-i((l+1)ny)/(l+1)n). Hence for each k=0,1,…,
(2.37)μ(∑i=0k‍Li(x)fk-i(y)-∑i=0k‍Li(x)fk-i((l+1)ny)(l+1)n,t)=1,ν(∑i=0k‍Li(x)fk-i(y)-∑i=0k‍Li(x)fk-i((l+1)ny)(l+1)n,t)=0,
for all x,y∈𝒜 and t>0. This relation yields that for each k=0,1,…,
(2.38)μ(∑i=0k‍Li(x)Lk-i(y)-∑i=0k‍Li(x)fk-i(y),t)≥μ(∑i=0k‍Li(x)Lk-i(y)-∑i=0k‍Li(x)fk-i((l+1)ny)(l+1)n,t2)*μ(∑i=0k‍Li(x)fk-i((l+1)ny)(l+1)n-∑i=0k‍Li(x)fk-i(y),t2)≥∏i=0k‍μ(Li(x)Lk-i(y)-Li(x)fk-i((l+1)ny)(l+1)n,t2(k+1)),(2.39)ν(∑i=0k‍Li(x)Lk-i(y)-∑i=0k‍Li(x)fk-i(y),t)≤ν(∑i=0k‍Li(x)Lk-i(y)-∑i=0k‍Li(x)fk-i((l+1)ny)(l+1)n,t2)⋄ν(∑i=0k‍Li(x)fk-i((l+1)ny)(l+1)n-∑i=0k‍Li(x)fk-i(y),t2)≤∐i=0k‍ν(Li(x)Lk-i(y)-Li(x)fk-i((l+1)ny)(l+1)n,t2(k+1)),
for all x,y∈𝒜 and t>0. On the other hand, we see that
(2.40)μ(Li(x)Lk-i(y)-Li(x)fk-i((l+1)ny)(l+1)n,t2(k+1))≥max⁡{μ(Li(x),(l+1)nt4(k+1)),μ(Lk-i(y)-fk-i((l+1)ny)(l+1)n,t4(k+1))},μ(Li(x)Lk-i(y)-Li(x)fk-i((l+1)ny)(l+1)n,t2(k+1))≤min⁡{ν(Li(x),(l+1)nt4(k+1)),ν(Lk-i(y)-fk-i((l+1)ny)(l+1)n,t4(k+1))}.
Sending n→∞ in (2.38) and (2.40), we have that for each k=0,1,…,
(2.41)μ(∑i=0k‍Li(x)Lk-i(y)-∑i=0k‍Li(x)fk-i(y),t)=1,ν(∑i=0k‍Li(x)Lk-i(y)-∑i=0k‍Li(x)fk-i(y),t)=0,
for all x,y∈𝒜 and t>0. Thus, we conclude that
(2.42)∑i=0k‍Li(x)Lk-i(y)=∑i=0k‍Li(x)fk-i(y),
for each k=0,1,… and all x,y∈𝒜.
Therefore, by combining (2.35) and (2.42), we get the required result, which completes the proof.As a consequence of Theorem2.2, we get the following superstability.Corollary 2.3.
Let(ℬ,μ,ν,*,⋄) be an intuitionistic fuzzy Banach algebra with unit, and let a sequence of operators F={f0,f1,…,fk,…} on 𝒜 satisfy fk(0)=0 for each k=0,1,…, where f0 is an identity operator. Suppose that φ is a function from 𝒜 to an intuitionistic fuzzy normed algebra (C,μ′,ν′,*,⋄) satisfying  (2.25) and (2.14) and that Φ is a function from 𝒜 to an intuitionistic fuzzy normed space (D,μ′′,ν′′,*,⋄) satisfying (2.26). If l>1 is a fixed integer, φ((l+1)x)=αφ(x), and Φ((l+1)x)=βΦ(x) for some real numbers α and β with 0<|α|<l+1 and 0<|β|<l+1, then F is a strong higher ring derivation on 𝒜.Proof.
According to (2.30), we have ℒ0(x)=x for all x∈𝒜, and so ℒ0(=f0) is an identity operator on 𝒜. By induction, we get the conclusion. If k=1, then it follows from (2.29) that f1(x)=ℒ1(x) holds for all x∈𝒜 since 𝒜 contains the unit element. Let us assume that fm(x)=ℒm(x) is valid for all x∈𝒜 and m<k. Then (2.29) implies that x{ℒm(y)-fm(y)}=0 for all x,y∈𝒜. Since 𝒜 has the unit element, fk(y)=ℒk(y) for all x∈𝒜. Hence we conclude that fk(y)=ℒk(y) for each k=0,1,2,… and all x∈𝒜. So this tells us that F is a higher ring derivation of any rank from 𝒜 and ℬ. The proof of the corollary is complete.We remark that we can prove the preceding result for the case whenα>l+1 and β>l+1.



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*Source: 503671-2012-07-16.xml*