2020, Volume 8, Issue 2, Pages: 582-588  
J. Environ. Treat. Tech.  
ISSN: 2309-1185  
Journal web link: http://www.jett.dormaj.com  
Stability and Super Stability of Fuzzy Approximately  
Ring Homomorphisms and Fuzzy Approximately  
Ring Derivations  
N. Eghbali  
Department of Mathematics, Facualty of Mathematical Sciences, University of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran  
Received: 05/08/2014  
Accepted: 15/02/2020  
Published: 20/05/2020  
Abstract  
In this paper, we establish the Hyers-Ulam-Rassias stability of ring homomorphisms and ring derivations in the uniform case on fuzzy  
Banach algebras.  
Keywords: Fuzzy normed space; Approximately ring homomorphism; Stability  
1
Introduction1  
It seems that the stability problem of functional equations had  
퐷(푎+푏)=퐷(푎)+퐷(푏),  
퐷(푎푏)=퐷(푎)푏+퐷(푏)푎,  
for every 푎,푏∈퐴;  
for every 푎,푏∈퐴.  
been first raised by Ulam [12]. An answer to this problem has  
been given at first by Hyers [5] and then by Th. M. Rassias as  
follows [10]. Suppose  and  are two real Banach spaces and  
푓:퐸 →퐸 is a mapping. If there exist 훿≥0and 0≤푝<ꢀsuch  
that ||푓(푥+푦)−푓(푥)−푓(푦)||≤훿(||푥|| +||푦|| ) for all  
푥,푦∈, then there is a unique additive mapping 푇:퐸 →퐸  
It is of interest to consider an approximately ring derivation  
on a Banach algebra. First of all, does there exist an  
approximately ring derivation  which is not an exact ring  
derivation? If such a mapping do exist, then it seems natural to  
consider the following stability problem: does there exist ring  
derivation near to ? The purpose of this paper is to prove the  
stability of fuzzy approximately ring derivations. In fact, under a  
mild assumption that is without order, we show the Bourgin-  
type [1] super stability result.  
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2
1
2
1
1
2
such that ||푓(푥)−푇(푥)||≤ꢂ훿||푥|| /|ꢂ−ꢂ |for every 푥∈퐸.  
1
In 1991, Gajda [2] gave a solution to this question for 푝>ꢀ. For  
the case 푝=ꢀ, Th. M. Rassias and Šemrl [11] showed that there  
exists a continuous real-valued function 푓:ℝ→such that can  
not be approximated with an additive map. In 1992, Gavruta [3]  
generalized the result of Rassias for the admissible control  
functions. Moreover the approximately mappings have been  
studied extensively in several papers. (See for instance [6], [7]).  
Fuzzy notion introduced firstly by Zadeh [13] that has been  
widely involved in different subjects of mathematics. Zadeh’s  
definition of a fuzzy set characterized by a function from a  
nonempty set to [0,ꢀ]. Later, in 1984 Katsaras [8] defined a  
fuzzy norm on a linear space to construct a fuzzy vector  
topological structure on the space. Defining the class of  
approximately solutions of a given functional equation one can  
ask whether every mapping from this class can be somehow  
approximated by an exact solution of the considered equation in  
the fuzzy Banach algebra.  
2 Preliminaries  
In this section, we provide a collection of definitions and  
related results which are essential and used in the next  
discussions. Definition 2.1 Let  be a real linear space. A  
function 푁:푋×ℝ→[0,ꢀ]is said to be a fuzzy norm on if for  
all 푥,푦∈and all 푡,푠∈,  
(N1) 푁(푥,푐)=0for 푐≤0;  
(N2) 푥=0if and only if 푁(푥,푐)=ꢀfor all 푐>0;  
(N3) 푁(푐푥,푡)=푁(푥, )if 푐≠0;  
|ꢄ|  
(N4) 푁(푥+푦,푠+푡)≥푚푖푛{푁(푥,푠),푁(푦,푡)};  
(N5) 푁(푥,.) is non-decreasing function on  and  
푙푖푚 푁(푥,푡)=ꢀ;  
a
ꢃ→∞  
To answer this question, we use here the definition of fuzzy  
normed spaces given in [8] to exhibit some reasonable notions of  
fuzzy approximately ring homomorphism in fuzzy normed  
algebras and we will prove that under some suitable conditions an  
approximately ring homomorphism from an algebra into a  
fuzzy Banach algebra can be approximated in a fuzzy sense by  
a ring homomorphism from to . Let be a real or complex  
Banach algebra. A mapping 퐷:퐴→ is said to be a ring  
derivative if  
(N6) for 푥≠0, 푁(푥,.)is (upper semi) continuous on .  
The pair (푋,푁)is called a fuzzy normed linear space.  
Example 2.2 Let (푋,||.||)be a normed linear space. Then  
Corresponding author: N. Eghbali, Department of Mathematics, Facualty of Mathematical Sciences, University of Mohaghegh Ardabili,  
6199-11367, Ardabil, Iran. E-mail: nasrineghbali@gmail.com,eghbali@uma.ac.ir.  
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82  
Journal of Environmental Treatment Techniques  
2020, Volume 8, Issue 2, Pages: 582-588  
0
,
ꢈ푡≤ꢈꢈ0;ꢈ  
ꢊꢏꢋ  
ꢊ ꢐꢊ ꢐꢊ  
ꢂ 휑(ꢂ 푥,ꢂ 푦)<ꢎ.  
,
ꢈ0<푡≤ꢈꢈꢉ|푥|ꢉ;ꢈ  
푁(푥,푡)=  
ꢉ|푥|ꢉ  
In particular, the similar results hold for 휑(푥,푦)=||푥|| +  
|
|푦|| , where 푞>ꢀ. Stability and super stability of fuzzy  
,
ꢈ푡>ꢉ|푥|ꢉ.  
approximately ring homomorphisms and fuzzy approximately  
ring derivations (N. Eghbali Tue Dec 24 00:16:25 2019).  
is a fuzzy norm on . Definition 2.3 Let (푋,푁)be a fuzzy  
normed linear space and {푥 }be a sequence in . Then {푥 }is  
3
Stability of fuzzy approximately ring  
homomorphism  
said to be convergent if there exists 푥∈푋 such that  
푙푖푚 푁(푥 −푥,푡)=ꢀfor all 푡>0. In that case, is called  
We start our work with definition of fuzzy approximately ring  
ꢊ→∞  
homomorphism. Definition 3.1 Let be a linear algebra, (푌,푁)  
a fuzzy Banach algebra and 휃≥0. We say that 푓:푋→is a  
fuzzy approximately ring homomorphism map if  
the limit of the sequence {푥 } and we denote it by 푁−  
푙푖푚 푥 =푥. Definition 2.4 A sequence {푥 }in  is called  
ꢊ→∞ ꢊ  
Cauchy if for each 휀>0and each 푡>0there exists such that  
for all 푛≥푛 and all 푝>0, we have 푁(푥 −푥 ,푡)>ꢀ−휀.  
ꢊꢌꢁ  
푙푖푚 푁(푓(푥푦)−푓(푥)푓(푦),푡휃||푥|| ||푦|| )=ꢀ,  
ꢃ→∞  
It is known that every convergent sequence in a fuzzy normed  
space is Cauchy and if each Cauchy sequence is convergent, then  
the fuzzy norm is said to be complete and furthermore the fuzzy  
normed space is called a fuzzy Banach space. Let be an algebra  
and (푋,푁)be complete fuzzy normed space. The pair (푋,푁)is  
said to be a fuzzy Banach algebra if for every 푥,푦∈푋and 푠,푡∈  
 we have 푁(푥푦,푠푡)≥푚푖푛{푁(푥,푠),푁(푦,푡)}. Example 2.5  
Let(푋,||.||)be a Banach algebra. Define,  
uniformly on 푋×푋. Theorem 3.2 Let  be a normed linear  
algebra and (푌,푁)a fuzzy Banach algebra. Let 휃≥0and 푞≥  
,푞≠. Suppose that 푓:푋→is a function such that  
0
푙푖푚 푁(푓(푥+푦)−푓(푥)−푓(푦),푡휃(||푥|| +||푦|| ))=ꢀ,  
ꢃ→∞  
uniformly on 푋×푋and  
푙푖푚 푁(푓(푥푦)−푓(푥)푓(푦),푡휃||푥|| ||푦|| )=ꢀ,  
(3.1)  
ꢃ→∞  
0
, ꢈ푎≤ꢈꢈ||푥||;ꢈ  
, ꢈ푎>||푥||.  
푁(푥,푎)=ꢍ  
uniformly on 푋×푋. Then there is a unique ring homomorphism  
푇:푋→such that  
Then (푋,푁)is a fuzzy Banach algebra. Theorem 2.6 Let 푋  
be a linear space and (푌,푁)be a fuzzy Banach space. Let 휑:푋×  
푋→[0,ꢎ)be a control function such that  
2ꢕꢃ||ꢓ||ꢖ  
푙푖푚 푁(푇(푥)−푓(푥), 12|)=ꢀ  
ꢃ→∞  
|
ꢐꢊ  
휑̃ (푥,푦)=∑ ꢂ 휑(ꢂ 푥,ꢂ 푦)<ꢎ,  
ꢊꢏꢋ  
uniformly on . Proof. Theorem 2.6 and Corollary 2.7 show that  
there exists a unique additive mapping such that  
for all 푥,푦∈. Let 푓:푋→ be a uniformly approximately  
additive function with respect to in the sense that  
2
ꢕꢃ||ꢓ||ꢖ  
|)=,  
|12ꢖꢗꢘ  
푙푖푚 푁(푇(푥)−푓(푥),  
(3.2)  
푙푖푚 푁(푓(푥+푦)−푓(푥)−푓(푦),푡휑(푥,푦))=ꢀ  
ꢃ→∞  
ꢃ→∞  
ꢑ(2 ꢓ)  
2ꢒ  
where 푥∈푋. Now we only need to show that