# Hyers–Ulam–Rassias Stability of Additive Mappings in Fuzzy Normed Spaces

Authors: Jianrong Wu; Lingxiao Lu
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, the Hyers–Ulam–Rassias stabilities of two functional equations,fax+by=rfx+sfy and fx+y+z=2fx+y/2+fz, are investigated in the framework of fuzzy normed spaces.


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## Body

## 1. Introduction

The classical stability problem of functional equations is as follows: when is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation? It originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms. In the succeeding year, Hyers [2] gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces. In 1978, Rassias [3] proved a generalization of Hyers’ theorem for additive mappings. Since then, many 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, such as the Cauchy equation fx+y=fx+fy, the Jensen equation 2fx+y/2=fx+fy, the quadratic equation, the cubic equation, and the mixed-type equation.The reader is referred to [4–8] and references therein for detailed information on the stability of functional equations.Studies on fuzzy normed spaces are relatively recent in the field of fuzzy functional analysis. In 1984, Katsaras [9] first introduced the idea of fuzzy norm on a linear space to construct a fuzzy vector topological structure on the space. After that, some mathematicians have defined fuzzy norms on a vector space from various points of view. In particular, Cheng and Mordeson [10] defined another type of fuzzy norm on a linear space in such a manner that the corresponding fuzzy metric is in the sense of Kramosil and Michalek [11].With the development of fuzzy analysis, it has been involved in the Ulam stability. In 2008, Mirmostafaee and Moslehian [12] established some stability results concerning the cubic functional equation f2x+y+f2x−y=2fx+y+2fx−y+12fx in fuzzy normed spaces. In the same year, Mirmostafaee et al. [13] proved a generalized Hyers–Ulam–Rassias stability theorem in the fuzzy sense.Later, several various fuzzy versions of stability concerning Jensen, cubic, quadratic, and mixed additive-cubic functional equations were investigated (see, e.g., [14–24] and the references therein).The main purpose of this paper is to establish the stability results concerning the following functional equations:(1)fax+by=rfx+sfy,where constants a,b>0 and r,s∈R with a+b=r+s≠1, and(2)fx+y+z=2fx+y2+fz,in the setting of fuzzy normed spaces introduced in [10]. The achieved results via this paper improve and extend some recent well-known pertinent results.The organization of the paper is as follows: Section2 comprises the basic notions on fuzzy normed spaces and some preliminary results. In Section 3, we prove Theorems 1 and 2 to show the stability of Cauchy equation (1) and Jensen equation (2), respectively. Finally, in Section 4, we give a brief conclusion.In this paper,ℝ and ℕ stand for the set of all real numbers and the set of all positive integers, respectively.


## 2. Terminology and Basic Notions

In this section, we recall some necessary notions and fundamental results which are useful in this paper.Definition 1 (see [10]).
A fuzzy norm on a real vector spaceX is a fuzzy set N in X×0,∞ satisfying the following conditions for all x,y∈X:(FN1)Nx,0=0(FN2)Nx,t=1 for all t>0 if and only if x=0(FN3)Ncx,t=Nx,t/c for all t>0 and c∈ℝ\0(FN4)minNx,t,Ny,s≤Nx+y,t+s for all s,t≥0(FN5)limt⟶∞Nx,t=1(FN6)Nx,⋅: 0,∞⟶0,1 is left continuous
LetX,N be a fuzzy normed vector space. A sequence xn in X is said to be convergent or said to converge if there exists an x∈X such that limn⟶∞Nxn−x,t=1 for all t>0. In this case, x is called the limit of the sequence xn and is denoted by N−limn⟶∞xn=x.Remark 1.
IfN−limn⟶∞xn=x, then N−limn⟶∞λxn=λx for any λ∈ℝ.
A sequencexn in X is said to be Cauchy if for each ε>0 and each δ>0, there exists an n0∈ℕ such that Nxm−xn,δ>1−εm,n≥n0.
It is well known that every convergent sequence in a fuzzy normed space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed space is called a fuzzy Banach space.Example 1.
LetX, be a normed vector space. Then,(3)Nx,t=0,t=0,tt+x,t>0,is a fuzzy norm on X, and it is called the standard fuzzy norm on X,. It is obvious that X, is complete if X, is complete.


## 3. Stability of Functional Equations (1) and (2)

Throughout this section, we assume thatX is a real vector space, the pair Y,N is a fuzzy Banach space, and Z,N′ is a fuzzy normed space.First, we show the stability of functional equation (1). For notational convenience, given a mapping f:X⟶Y of (1), we define a difference operator Df:X2⟶Y by(4)Dfx,y=fax+by−rfx−sfy,for allx,y∈X.Theorem 1.
Supposef:X⟶Y satisfies the inequality(5)NDfx,y,t≥N′φx,y,t,for all x,y∈X, t>0, where constants a,b>0 and r,s∈ℝ with a+b=r+s≠1; φ:X2⟶Z is a mapping for which there is a constant 0<α<a+b such that(6)N′φa+bx,a+bx,t≥N′αφx,x,t,and limn⟶∞N′φa+bnx,a+bny,a+bnt=1 for all x,y∈X. Then, one can find a unique additive mapping T:X⟶Y satisfying the equation DTx,y=0 and the inequality(7)NTx−fx,t≥N′φx,x,a+b−αt2.Proof.
Lettingy=x in (5), we obtain(8)Nfa+bx−r+sfx,t≥N′φx,x,t.
Replacingx by a+bnxn∈ℕ in (8), we obtain(9)Nfa+bn+1x−r+sfa+bnx,t≥N′φa+bnx,a+bnx,t.
This together with (6) implies that(10)Nfa+bn+1xa+bn+1−fa+bnxa+bn,ta+bn+1≥N′φa+bnx,a+bnx,t≥N′αnφx,x,t=N′φx,x,tαn.
By replacingt with αnt, we obtain(11)Nfa+bn+1xa+bn+1−fa+bnxa+bn,αnta+bn+1≥N′φx,x,t.
Then,(12)Nfa+bn+pxa+bn+p−fa+bnxa+bn,∑i=nn+p−1αita+bi+1=N∑i=nn+p−1fa+bi+1xa+bi+1−fa+bixa+bi,∑i=nn+p−1αita+bi+1≥minn≤i≤n+p−1Nfa+bi+1xa+bi+1−fa+bixa+bi,αita+bi+1≥N′φx,x,t.
Hence,(13)Nfa+bn+pxa+bn+p−fa+bnxa+bn,t≥N′φx,x,t∑i=nn+p−1αi/a+bi+1.
Since∑k=0∞αk/a+bk+1=1/a+b−α<∞, the Cauchy criterion for convergence shows that fnx is a Cauchy sequence in Y,N, where f0x=fx and fnx=fa+bnx/a+bnn∈ℕ. Thanks to Y,N being a fuzzy Banach space, this sequence converges to a point Tx∈Y. That is, Tx=N−limn⟶∞fnx.
From (11), we know that(14)Nfnx−fx,∑k=0n−1αkta+bk+1=N∑k=0n−1fk+1x−fkx,∑k=0n−1αkta+bk+1≥min0≤k≤n−1Nfk+1x−fkx,αkta+bk+1≥N′φx,x,t.
Therefore,(15)Nfnx−fx,t≥N′φx,x,t∑k=0n−1αk/a+bk+1≥N′φx,x,t∑k=0∞αk/a+bk+1=N′φx,x,a+b−αt.
Hence,(16)NTx−fx,t≥minNTx−fnx,t2,Nfnx−fx,t2≥N′φx,x,a+b−αt2,for n large enough.
Now, we proveT is an additive mapping. Replace x and y by a+bnx and a+bny, respectively, in (5) to obtain(17)Nfa+bnax+bya+bn−rfa+bnxa+bn−sfa+bnya+bn,t≥N′φa+bnx,a+bny,a+bnt,for all x, y∈X and all t>0. Since limn⟶∞N′φa+bnx,a+bny,a+bnt=1, we conclude that DTx,y=0.
To prove the uniqueness of the additive mappingT, assume that there exists another additive mapping S:X⟶Y which satisfies (7). Since DTx,y=0, we have Tax+by=rTx+sTy. Let x=y, and then,(18)Ta+bx=r+sTx=a+bTx.
Consequently,Ta+bkx=a+bkTx. Similarly, Sa+bkx=a+bkSx for all x∈X and k∈ℕ. From (7) and (6), we have(19)NTx−Sx,t=NTa+bnxa+bn−Sa+bnxa+bn,t=NTa+bnx−Sa+bnx,a+bnt≥minNTa+bnx−fa+bnx,a+bnt2,Nfa+bnx−Sa+bnx,a+bnt2≥N′φa+bnx,a+bnx,≥N′φx,x,a+b−αa+bnt4αn.
Sincelimn⟶∞a+b−αa+bnt/4αn=∞, we obtain NTx−Sx,t=1 for all t>0. Therefore, Tx−Sx=0. The proof ends.Example 2.
LetX be a Banach space and Z be a normed space. Denote by N and N′ the standard fuzzy norms on X and Z, respectively. Then, X,N is a fuzzy Banach space. Let φ:X2⟶Z be denoted by(20)φx,y=x22+y26z0,where z0 is a fixed unit vector in Z. Define f:X⟶X by(21)fx=x+x2x0,where x0 is a fixed unit vector in X. Set a=b=1/3, s=1/6, and r=1/2; then, a+b=r+s≠1 and(22)Dfx,y=fax+by−rfx−sfy=x+y9−x2−y6x0.
Therefore,(23)NDfx,y,t=tt+x+y/9−x/2−y/6x0=18t18t+9x2−2x+y2+3y2≥18t18t+9x2+3y2=N′φx,y,t,N′φa+bx,a+bx,t=tt+φ2x/3,2x/3=tt+αφx,x=N′αφx,x,t,where α=4/9<2/3=a+b. Moreover,(24)limn⟶∞N′φa+bnx,a+bny,a+bnt=limn⟶∞N′φ23nx,23ny,23nt=limn⟶∞2/3nt2/3nt+2/32nφx,y=1.
Hence, the conditions of Theorem1 are fulfilled. Therefore, there is a unique additive mapping T:X⟶X satisfying the inequality(25)NTx−fx,t≥N′φx,x,a+b−αt2=N′46x2,t9.
Indeed,(26)Tx=N−limn⟶∞fnx=N−limn⟶∞f2/3nx2/3n=N−limn⟶∞2/3nx+2/3nx2x02/3n=N−limn⟶∞x+23nx2x0=x,∀x∈X.
Now, we show the stability of functional equation (2). For the mapping f:X⟶Y of (2), we define a difference operator Df:X3⟶Y by(27)Dfx,y,z=fx+y+z−2fx+y2−fz.Theorem 2.
Letφ:X3⟶Z be a mapping for which there is a constant 0<α<3, such that(28)N′φ3x,3x,3x,t≥N′αφx,x,x,t,and limn⟶∞N′φ3nx,3ny,3nz,3nt=1 for all x,y,z∈X. Assume that a mapping f:X⟶Y satisfies the inequality(29)NDfx,y,z,t≥N′φx,y,z,t,for all x,y,z∈X and t>0. Then, one can find a unique additive mapping T:X⟶Y satisfying the equation DTx,y,z=0 and the inequality(30)NTx−fx,t≥N′φx,x,x,3−αt6.Proof.
The techniques are completely similar to that of Theorem1. Hence, we present a sketch of proof. Put x=y=z in (29) to obtain(31)Nf3x−3fx,t≥N′φx,x,x,t.
Replacingx and t by 3nx and αnt, respectively, we obtain(32)Nf3n+1x3n+1−f3nx3n,αnt3n≥N′φx,x,x,t.
Denotef0x=fx and fnx=f3nx/3n, where x∈X and n∈ℕ; we know(33)Nfn+1x−fnx,αnt3n≥N′φx,x,x,t.
Then,(34)Nfn+px−fnx,∑i=nn+p−1αit3i≥N′φx,x,x,t.
Hence,(35)Nfn+px−fnx,t≥N′φx,x,x,t∑i=nn+p−1αi/3i.
Since∑k=0∞αk/3k=3/3−α<∞, N−limn⟶∞fnx=Tx∈Y.
It follows from (33) that(36)Nfnx−fx,∑k=0n−1αkt3k≥N′φx,x,x,t.
Therefore,(37)Nfnx−fx,t≥N′φx,x,x,3−αt3.Hence,(38)NTx−fx,t≥minNTx−fnx,t2,Nfnx−fx,t2≥N′φx,x,x,3−αt6,for n large enough.
To proveT is an additive mapping, we replace x and y by 3nx and 3ny, respectively, in (29), and we obtain(39)Nf3nx+3ny+3nz3n−23nf3nx+3ny2−13nf3nz,t≥N′φ3nx,3ny,3nz,3nt,for all x,y,z∈X and all t>0. Since limn⟶∞N′φ3nx,3ny,3nz,3nt=1, we conclude that DTx,y,z=0.
To prove the uniqueness of the additive mappingT, assume that there exists another additive mapping S:X⟶Y which satisfies (30). It is easy to see that T3kx=3kTx and S3kx=3kSx for all x∈X and k∈ℕ. This together with (30) and (28) implies that(40)NTx−Sx,t≥N′φx,x,x,3−α3nt12αn.
Since0<α<3, we obtain NTx−Dx,t=1 for all t>0. Therefore, Tx−Sx=0. The proof ends.


## 4. Conclusion

By using the direct method, we establish the Hyers–Ulam–Rassias stability of the functional equations in the framework of fuzzy normed spaces. We show that the existence and uniqueness of the solutions for these equations. These circumstances can be applied to other significant functional equations in the set of fuzzy normed spaces.



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