2020, Volume 8, Issue 2, Pages: 582-588
J. Environ. Treat. Tech.
Journal web link: http://www.jett.dormaj.com
Stability and Super Stability of Fuzzy Approximately
Ring Homomorphisms and Fuzzy Approximately
Department of Mathematics, Facualty of Mathematical Sciences, University of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran
In this paper, we establish the Hyers-Ulam-Rassias stability of ring homomorphisms and ring derivations in the uniform case on fuzzy
Keywords: Fuzzy normed space; Approximately ring homomorphism; Stability
It seems that the stability problem of functional equations had
for every 푎,푏∈퐴;
for every 푎,푏∈퐴.
been first raised by Ulam . An answer to this problem has
been given at first by Hyers  and then by Th. M. Rassias as
follows . Suppose 퐸 and 퐸 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∈퐸, 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 푓 which is not an exact ring
derivation? If such a mapping 푓do exist, then it seems natural to
consider the following stability problem: does there exist ring
derivation near to 푓? The purpose of this paper is to prove the
stability of fuzzy approximately ring derivations. In fact, under a
mild assumption that 퐴is without order, we show the Bourgin-
type  super stability result.
such that ||푓(푥)−푇(푥)||≤ꢂ훿||푥|| /|ꢂ−ꢂ |for every 푥∈퐸.
In 1991, Gajda  gave a solution to this question for 푝>ꢀ. For
the case 푝=ꢀ, Th. M. Rassias and Šemrl  showed that there
exists a continuous real-valued function 푓:ℝ→ℝsuch that 푓can
not be approximated with an additive map. In 1992, Gavruta 
generalized the result of Rassias for the admissible control
functions. Moreover the approximately mappings have been
studied extensively in several papers. (See for instance , ).
Fuzzy notion introduced firstly by Zadeh  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 푋to [0,ꢀ]. Later, in 1984 Katsaras  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.
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 푋 be a real linear space. A
function 푁:푋×ℝ→[0,ꢀ]is said to be a fuzzy norm on 푋if for
all 푥,푦∈푋and all 푡,푠∈ℝ,
(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;
(N5) 푁(푥,.) is non-decreasing function on ℝ and
To answer this question, we use here the definition of fuzzy
normed spaces given in  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 푓from an algebra 푋into a
fuzzy Banach algebra 푌can be approximated in a fuzzy sense by
a ring homomorphism 푇from 푋to 푌. Let 퐴be a real or complex
Banach algebra. A mapping 퐷:퐴→퐴 is said to be a ring
(N6) for 푥≠0, 푁(푥,.)is (upper semi) continuous on ℝ.
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: firstname.lastname@example.org,email@example.com.