# Ulam Stability and Non-Stability of Additive Functional Equation in IFN-Spaces and 2-Banach Spaces by Different Methods

Authors: N. Uthirasamy; K. Tamilvanan; Masho Jima Kabeto
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This paper introduces a new dimension of an additive functional equation and obtains its general solution. The main goal of this study is to examine the Ulam stability of this equation in IFN-spaces (intuitionistic fuzzy normed spaces) with the help of direct and fixed point approaches and 2-Banach spaces. Also, we use an appropriate counterexample to demonstrate that the stability of this equation fails in a particular case.


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## Body

## 1. Introduction

The study of stability problems for functional equations is one of the essential research areas in mathematics, which originated in issues related to applied mathematics. The first question concerning the stability of homomorphisms was given by Ulam [1] as follows.Given a groupG,∗, a metric group G′,· with the metric d, and a mapping f from G and G′, does δ>0 exist such that(1)dfx∗y,fx·fy≤δ,for all x,y∈G. If such a mapping exists, then does a homomorphism h:G⟶G′ exist such that(2)dfx,hx≤ε,for all x∈G? Ulam defined such a problem in 1940 and solved it the following year for the Cauchy functional equation(3)ψu+v=ψu+ψv,by the way of Hyers [2]. The consequence of Hyers becomes stretched out by Aoki [3] with the aid of assuming the unbounded Cauchy contrasts. Hyers theorem for additive mapping was investigated by Rassias [4], and then Rassias results were generalized by Gavruta [5].As of late, Nakmahachalasint [6] gave the overall answer and HUR (briefly, Hyers–Ulam–Rassias) stability of finite variable functional equation; furthermore, Khodaei and Rassias [7] examined the stability of generalized additive functions in several variables. The stability result of additive functional equations was examined by means of Najati and Moghimi [8], Shin et al. [9], and Gordji [10]. Stability problems of various functional equations have been investigated by many researchers, and there are various interesting results about this problem (see [11–14]).Zadeh [15] established the concept of fuzzy sets, which is a tool for demonstrating weakness and ambiguity in several scientific and technological problems. The possibility of IFN-spaces, from the start, has been presented in [16]. Saadati [17] have examined the modified intuitionistic fuzzy metric spaces and proven some fixed point theorems in these spaces.The IFN-spaces and IF2N-spaces (briefly, intuitionistic fuzzy 2-normed spaces) have been studied by a number of researchers [18–20]. Furthermore, several researchers have discussed the generalized Ulam–Hyers stability of various functional equations in IFN-spaces (see [21–24]).In this current work, we present a new kind of additive functional equation:(4)∑1≤a<b<c≤sϕ−va−vb−vc+∑d=1,d≠a≠b≠csvd−s3−9s2+20s−126∑a=1sϕva−ϕ−va2=0,where s>4 is a fixed integer, and obtain its general solution. The main goal of this study is to examine the Ulam–Hyers stability of this equation in IFN-spaces with the help of direct and fixed point approaches and 2-Banach spaces by using the direct approach. Also, we use an appropriate counterexample to demonstrate that the stability of equation (4) fails in a particular case.


## 2. General Solution

Theorem 1.
If a mappingϕbetween two real vector spacesWandFsatisfies functional equation (4), then the functionϕis additive.Proof.
Settingv1=⋯=vs=0 in (4), we have ϕ0=0. Replacing v1,v2,…,vs by v,0,0,…,0︸s−1−times in (4), we get ϕ−v=−ϕv for all v∈W. Hence, ϕ is an odd function. Replacing v1,v2,v3,…,vs by v,v,0,0,…,0︸s−2−times in (4), we have(5)ϕ2v=2ϕv,for all v∈W. Replacing v by 2v in (5), we have(6)ϕ22v=22ϕv,for all v∈W. Again, replacing v with 2v in (6), we get(7)ϕ23v=23ϕv,for all v∈W. In general, for any non-negative integer a>0, we have(8)ϕ2av=2aϕv,for all v∈W. Replacing v1,v2,v3,…,vs by s,t,0,0,…,0︸s−2−times in (4), we obtain (3) for all s,t∈W.Remark 1.
If a mappingϕ between two real vector spaces W and F satisfies functional equation (3), then the function ϕ satisfies additive functional equation (4), for all v1,v2,v3,…,vs∈W.For our notational handiness, we define a mappingϕ:W⟶F by(9)Dϕv1,v2,…,vs=∑1≤a<b<c≤sϕ−va−vb−vc+∑d=1,d≠a≠b≠csvd−s3−9s2+20s−126∑a=1sϕva−ϕ−va2,for all v1,v2,…,vs∈W.


## 3. Stability Results in IFN-Spaces

We can recall some basic notions and preliminaries from [25] and using the alternative fixed point theorem which are important results in fixed point theory [26].Definition 1 (see [25]).
Consider a membership degreeμ and non-membership degree ν of an intuitionistic fuzzy set from W×0,+∞ to 0,1 such that μvt+νvt≤1 for all v∈W and t>0. The triple W,Iμ,ν,Υ is called as an Intuitionistic Fuzzy Normed-space (briefly, IFN-space) if a vector space W, a continuous t-representable Υ and Iμ,ν:W×0,+∞⟶L∗ satisfying v1,v2∈W and t,s>0,(IFN1)Iμ,νv1,0=0L∗.(IFN2)Iμ,νv1,t=1L∗ if and only if v1=0.(IFN3)Iμ,ναv1,t=Iμ,νv1,t/α, for all α≠0.(IFN4)Iμ,νv1+v2,t+s≥L∗YIμ,νv1,t,Iμ,νv2,s.
In this case,Iμ,ν is called an intuitionistic fuzzy norm, where Iμ,νv1,t=μv1t,νv1t.Definition 2 (see [25]).
A sequencevm in W is called as a Cauchy sequence if for every ϵ>0 and t>0, there exists m0 such that(10)Iμ,νvm+p−vm,t>1−ε,m≥m0,for all p>0.Remark 2.
In an intuitionistic fuzzy normed space, every convergent sequence is a Cauchy sequence.
If every Cauchy sequence is convergent, then the intuitionistic fuzzy normed space is called as complete.Definition 3 (see [25]).
A mappingϕ between two IFN-spaces W and F is continuous at v0 if for every vm converging to v0 in W, the sequence ϕvm converges to ϕv0. If ϕ is continuous at each point v0∈W, then the mapping ϕ is called as a continuous mapping on W.Example 1.
LetW,·be a normed space. LetTa,b=a,b,mina2+b2,1for alla=a1,a2;b=b1,b2∈L∗andμ,νbe membership and non-membership degree of an intuitionistic fuzzy set defined by(11)Iμ,νv,t=μvt,νvt=tt+v,vt+v,t∈R+.
Then,W,Iμ,ν,T is an IFN-space.Theorem 2 (see [26]).
LetW,dbe a generalized complete metric space and a strictly contractive mappingM:W⟶Wwith Lipschitz constantL<1. Then, for allv1∈W, either(12)dMmv1,Mm+1v1=∞,m≥m0,or there exists a positive integer m0 such that(i)
dMmv1,Mm+1v1<∞,m≥m0.(ii)
The sequenceMmv1m∈ℕconverges to a fixed pointv1∗ofM.(iii)
v1∗is the unique fixed point ofMinW∗=v2∈W|dMm0v1,v2<∞.(iv)
dv2,v1∗≤1/1−LdMv2,v2, for allv2∈W∗.


### 3.1. Stability Results: Direct Technique

In this section, we assume thatW, Z,Iμ,ν′,Y, and F,Iμ,ν,Y are linear space, IFN-space, and complete IFN-space, respectively.Theorem 3.
If a mappingφ:Ws⟶Zwith0<ς/2<1,(13)Iμ,ν′φ2v,2v,0,…,0,ε≥L∗Iμ,ν′ςφv,v,0,…,0,ε,(14)limk⟶∞Iμ,ν′φ2kv1,2kv2,…,2kvs,2kε=1L∗,for all v,v1,v2,…,vs∈W and all ϵ>0. If a mapping ϕ:W⟶F satisfies(15)Iμ,νDϕv1,v2,…,vs,ε≥L∗Iμ,ν′φv1,v2,…,vs,ε,for all v1,v2,…,vs∈W and all ϵ>0, then the limit(16)Iμ,νA1v−ϕ2kv2k,ϵ⟶1L∗ask⟶∞,exists and there exists a unique additive mapping A1:W⟶F satisfying functional equation (4) and(17)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′φv,v,0,…,0,s3−9s2+20s−126ϵ2−ς,for all v∈W and all ϵ>0.Proof.
Fixv∈W and all ϵ>0. Replacing v1,v2,…,vs by v,v,0,…,0 in (15), we have(18)Iμ,νs3−9s2+20s−126ϕ2v−2s3−9s2+20s−126ϕv,ε≥L∗Iμ,ν′φv,v,0,…,0,ε.
Replacingv by 2kv in (18) and using (IFN3), we obtain(19)Iμ,νϕ2k+1v2−ϕ2kv,6ϵ2s3−9s2+20s−12≥L∗Iμ,ν′φ2kv,2kv,0,…,0,ϵ.
By the inequality (13) and (IFN3) in (19), we have(20)Iμ,νϕ2k+1v2−ϕ2kv,6ϵ2s3−9s2+20s−12≥L∗Iμ,ν′φv,v,0,…,0,ϵςk.
Clearly, we can show from inequality (20) that(21)Iμ,νϕ2k+1v2k+1−ϕ2kv2k,6ϵ2k+1s3−9s2+20s−12≥L∗Iμ,ν′φv,v,0,…,0,ϵςk.
Replacingϵ by ςkϵ in (21), we get(22)Iμ,νϕ2k+1v2k+1−ϕ2kv2k,6ςkϵ2k+1s3−9s2+20s−12≥L∗Iμ,ν′φv,v,0,…,0,ϵ.
Clearly,(23)ϕ2kv2k−ϕv=∑a=0k−1ϕ2a+1v2a+1−ϕ2av2a.
It follows from (22) and (23) that(24)Iμ,νϕ2kv2k−ϕv,∑a=0k−16ςaϵ2a+1s3−9s2+20s−12≥L∗Ya=0k−1Iμ,ν′ϕ2a+1v2a+1−ϕ2av2a,6ςaϵ2a+1s3−9s2+20s−12≥L∗Ya=0k−1Iμ,ν′φv,v,0,…,0,ϵ≥L∗Iμ,ν′φv,v,0,…,0,ϵ,for all v∈W and ϵ>0. Replacing v by 2tv in (24) and with the help of (13), we have(25)Iμ,νϕ2k+tv2k+t−ϕ2tv2t,∑a=0k−16ςaϵ2a+t2s3−9s2+20s−12≥L∗Iμ,ν′φv,v,0,…,0,ϵςt,for every t,k≥0. Replacing ϵ by ςtϵ in (25), we have(26)Iμ,νϕ2k+tv2k+t−ϕ2tv2t,∑a=tk+t−16ςaϵ2a+1s3−9s2+20s−12≥L∗Iμ,ν′φv,v,0,…,0,ϵ.
Using (IFN3) in (26), we obtain(27)Iμ,νϕ2k+tv2k+t−ϕ2tv2t,ϵ≥L∗Iμ,ν′φv,v,0,…,0,ϵ∑a=tk+t−16ςa/2a2s3−9s2+20s−12,for all t,k≥0. Since 0<ς<2 and ∑a=0kς/2a<∞, the Cauchy criterion for convergence in IFNS shows that ϕ2kv/2k is Cauchy sequence in F,Iμ,ν,Υ. Since F,Iμ,ν,Υ is a complete, this sequence converges to some point A1v∈F. Then, we can define the mapping A1:W⟶F by(28)Iμ,νA1v−ϕ2kv2k⟶1L∗ask⟶∞.
Settingt=0 in inequality (29), we obtain(29)Iμ,νϕ2kv2k−ϕv,ϵ≥L∗Iμ,ν′φv,v,0,…,0,ϵ∑a=0k−16ςa/2a2s3−9s2+20s−12.
Taking the limit ask⟶∞ in (29), we obtain(30)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′φv,v,0,…,0,s3−9s2+20s−126ϵ2−ς.
Next, we want to prove that the functionA1 satisfies functional equation (4); replacing v1,v2,…,vs by 2kv1,2kv2,…,2kvs in (15), we have(31)Iμ,ν12kDϕ2kv1,…,2kvs,ϵ≥L∗Iμ,ν′φ2kv1,…,2kvs,2kϵ,for all v1,v2,…,vs∈W and all ϵ>0. Since(32)limk⟶∞Iμ,ν′φ2kv1,2kv2,…,2kvs,2kε=1L∗,the function A1 satisfies functional equation (4). Thus, the function A1 is additive. Finally, we want to prove that the function A1 is unique; consider another additive mapping A2:W⟶F satisfying functional equations (4) and (17). Hence,(33)Iμ,νA1v−A2v,ϵ=Iμ,νA12kv2k−A22kv2k,ϵ≥L∗ΥIμ,νA12kv2k−ϕ2kv2k,ϵ2,Iμ,νϕ2kv2k−A22kv2k,ϵ2≥L∗Iμ,ν′φ2kv,2kv,0,…,0,s3−9s2+20s−122kϵ2−ς12≥L∗Iμ,ν′φv,v,0,…,0,s3−9s2+20s−122kϵ2−ς12ςk,for all v∈W and all ϵ>0. As(34)lims⟶∞s3−9s2+20s−122kϵ2−ς12ςk=∞,we obtain(35)limk⟶∞Iμ,ν′φv,v,0,…,0,s3−9s2+20s−122kϵ2−ς12ςk=1L∗.
Thus,Iμ,νA1v−A2v,ϵ=1L∗.
Therefore,A1v=A2v. Thus, the additive function A1v is unique. This ends the proof.Theorem 4.
If a mappingφ:Ws⟶Zwith0<2/ς<1,(36)Iμ,ν′φ2−1v,2−1v,0,…,0,ϵ≥L∗Iμ,ν′1ςφv,v,0,…,0,ϵ,(37)limk⟶∞Iμ,ν′φ2−kv1,2−kv2,…,2−kvs,2−kϵ=1L∗,for allv,v1,v2,…,vs∈Wand allϵ>0. If a mappingϕ:E⟶Fsatisfies (15), then the limitIμ,νA1v−2kϕv/2k,ϵ⟶1L∗ask⟶∞ exists and there exists a unique additive mapping A1:W⟶F satisfying functional equation (4) and(38)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′φv,v,0,…,0,s3−9s2+20s−126ϵς−2,for all v∈W and all ϵ>0.Proof.
Fixv∈W and all ϵ>0. Replacing v1,v2,…,vs by v,v,0,…,0 in (15), we have(39)Iμ,νs3−9s2+20s−126ϕ2v−2s3−9s2+20s−126ϕv,ϵ≥L∗Iμ,ν′φv,v,0,…,0,ϵ.
From (39), we obtain that(40)Iμ,νϕ2v−2ϕv,6ϵ2s3−9s2+20s−12≥L∗Iμ,ν′φv,v,0,…,0,ϵ.
Replacingv by v/2 in (40), we get(41)Iμ,νϕv−2ϕv2,6ϵ2s3−9s2+20s−12≥L∗Iμ,ν′φv2,v2,0,…,0,ϵ.
Replacingv by v/2k in (41) and using (IFN3), we have(42)Iμ,νϕv2k−2ϕv2k+1,6ϵ2s3−9s2+20s−12≥L∗Iμ,ν′φv2k+1,v2k+1,0,…,0.
With the help of inequality (36) and (IFN3) in (42), we obtain that(43)Iμ,νϕv2k−2ϕv2k+1,6ϵ2s3−9s2+20s−12≥L∗Iμ,ν′φv,v,0,…,0,ϵςk+1.
The remaining part of the proof can be proven in the same way as Theorem3.Corollary 1.
Letθ∈ℝ+. If a mappingϕ:W⟶Fsuch that(44)Iμ,νDϕv1,v2,…,vs,ϵ≥L∗Iμ,ν′θ,ϵ,for allv1,v2,…,vs∈Wand allϵ>0, then there exists a unique additive mappingA1:W⟶Fsatisfying(45)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′6θ,2−1ϵs3−9s2+20s−12,for all v∈W and all ϵ>0.Proof.
The proof holds from Theorems3 and 4 by letting φv1,v2,…,vs=θ and ς=20.Corollary 2.
Letθ,ξ∈ℝ+withξ∈0,1∪1,+∞. If a mappingϕ:W⟶Fsuch that(46)Iμ,νDϕv1,v2,…,vs,ϵ≥L∗Iμ,ν′θ∑a=1svaξ,ϵ,for allv1,v2,…,vs∈Wand allϵ>0, then there exists a unique additive mappingA1:W⟶Fsatisfying(47)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′12θvξ,2−2ξs3−9s2+20s−12ϵ,for all v∈W and all ϵ>0.Proof.
The proof holds from Theorems3 and 4 by setting φv1,v2,…,vs=θ∑a=1svaξ and ς=2ξ.Corollary 3.
Letθ,ξ,γ,τ∈ℝ+withsξ,sτ∈0,1∪1,+∞. If a mappingϕ:W⟶Fsuch that(48)Iμ,νDϕv1,v2,…,vs,ϵ≥L∗Iμ,ν′θ∑a=1svasξ+γ∏a=1svaτ,ϵ,for allv1,v2,…,vs∈Wand allϵ>0, then there exists a unique additive mappingA1:W⟶Fsatisfying(49)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′12θvsξ,2−2sξs3−9s2+20s−12ϵ,for all v∈W and all ϵ>0.Proof.
The proof holds from Theorems3 and 4 by setting φv1,v2,…,vs=θ∑a=1svasξ+γ∏a=1svaτ and ς=2sξ.Corollary 4.
Letγ,τ∈ℝ+with0<sτ≠1. If a mappingϕ:W⟶Fsuch that(50)Iμ,νDϕv1,v2,…,vs,ϵ≥L∗Iμ,ν′γ∏a=1svaτ,ϵ,for all v1,v2,…,vs∈W and all ϵ>0, then the mapping ϕ is additive.Proof.
The proof is valid from Theorems3 and 4 by setting φv1,v2,…,vs=γ∏a=1svaτ.


### 3.2. Stability Results: Fixed Point Technique

Before we begin, let us consider a constantβa such that(51)βa=2,ifa=0,12,ifa=1,and Ψ is the set such that Ψ=n1|n1:W⟶F,n10=0.Theorem 5.
Consider a mappingϕ:W⟶Ffor which there is a mappingφ:Ws⟶Zwith(52)liml⟶∞Iμ,ν′φ2lv1,2lv2,…,2lvs,2lϵ=1L∗,satisfying functional inequality (15). If there isL=Lasuch thatv⟶ηv=6/s3−9s2+20s−12φv/2,v/2,0,…,0has the property(53)Iμ,ν′L1βaηβav,ϵ=Iμ,ν′ηv,ϵ,then there exists a unique additive mapping A1:W⟶Fsatisfying functional equation (4) and(54)Iμ,νϕv−A1v,ϵn≥L∗Iμ,ν′L1−a1−Lηv,ϵ,for all v∈W and all ϵ>0.Proof.
Letς be a general metric on Ψ:(55)ςn1,n2=inft∈0,∞|Iμ,νn1v−n2v,ϵ≥L∗Iμ,ν′tηv,ϵ,v∈W,ϵ>0.
Clearly,Ψ,ς is complete. Define a mapping Υ:Ψ⟶Ψ by Υn1v=1/βan1βav, for all v∈W. For n1,n2∈Ψ, we have(56)ςn1,n2≤t,⇒Iμ,νn1v−n2v,ε≥L∗Iμ,ν′tηv,ε⇒Iμ,νn1βavβa−n2βavβa,ε≥L∗Iμ,ν′tηβavβa,ε⇒Iμ,νΥn1v−Υn2v,ε≥L∗Iμ,νtLηv,ε⇒ςΥn1v,Υn2v≤tL⇒ςΥn1,Υn2≤Lςn1,n2.
Thus, the functionΥ is strictly contractive on Ψ with L (Lipschitz constant). Replacing v1,v2,…,vs by v,v,0,…,0 in (15), we have(57)Iμ,νs3−9s2+20s−126ϕ2v−2s3−9s2+20s−126ϕv,ϵ≥L∗Iμ,ν′φv,v,0,…,0,ϵ.
Using (IFN3) in (57), we have(58)Iμ,νϕ2v2−ϕv,ϵ≥L∗Iμ,ν′62s3−9s2+20s−12φv,v,0,…,0,ϵ.
Using equation (53) for the case a=0, we have(59)Iμ,νϕ2v2−ϕv,ϵ≥L∗Iμ,ν′Lηv,ϵ⇒ςΥϕ,ϕ≤L=L1=L1−a.
Replacingv by v/2 in (57), we have(60)Iμ,νϕv−2ϕv2,ϵ≥L∗Iμ,ν′6s3−9s2+20s−12φv2,v2,0,…,0,ϵ,for all v∈W and all ϵ>0; using (53) for the case a=1, we have(61)Iμ,νϕv−2ϕv2,ϵ≥L∗Iμ,ν′ηv,ϵ⇒ςϕ,Υϕ≤1=L0=L1−a.
We can conclude from equations (59) and (61) that(62)ςϕ,Υϕ≤L1−a<∞.
By the fixed point alternative in both cases, there is a fixed pointA1 of Υ in Ψ such that(63)limk⟶∞Iμ,νϕβakvβak−A1v,ϵ⟶1L∗,v∈W,ϵ>0.
Replacingv1,v2,…,vs by βav1,βav2,…,βavs in (15), we obtain(64)Iμ,ν1βaDϕβav1,βav2,…,βavs,ϵ≥L∗Iμ,ν′φβav1,βav2,…,βavs,βaϵ,for all v1,v2,…,vs∈W and all ϵ>0. By same manner of Theorem 3, we can show that the function A1 satisfies functional equation (4). By Theorem 2, as A1 is a unique fixed point of Υ in Δ=ϕ∈Ψ|ςϕ,A1<∞, the function A1 is unique such that(65)Iμ,νA1v−ϕv,ε≥L∗Iμ,ν′tηv,ε,t>0.
Using fixed point alternative, we reach(66)ςϕ,A1≤11−Lςϕ,Υϕ⇒ςϕ,A1≤L1−a1−L⇒Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′L1−a1−Lηv,ϵ,for all v∈W and all ϵ>0. Hence, the proof of the theorem is now completed.Corollary 5.
Letθ,ξ∈ℝ+withθ>0. If a mappingϕ:W⟶Fsuch that(67)Iμ,νDϕv1,v2,…,vs,ϵ≥L∗Iμ,ν′θ,ϵ,Iμ,ν′θ∑j=1svjξ,ϵ,Iμ,ν′θ∏j=1svjξ+∑j=1svjsξ,ϵ,for allv1,v2,…,vs∈Wandϵ>0, then there exists a unique additive mappingA1:W⟶Fsatisfying(68)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′6θ,s3−9s2+20s−12ϵ,Iμ,ν′12θvs,s3−9s2+20s−122−2ξϵ,ξ<1orξ>1,Iμ,ν′12θvs,s3−9s2+20s−122−2sξϵ,ξ<1sorξ>1s,for all v∈W and all ϵ>0.Proof.
Set(69)φv1,v2,…,vs=θ,θ∑j=1svjξ,θ∏j=1svjξ+∑j=1svjsξ.
Then,(70)Iμ,ν′φβalv1,βalv2,…,βalvs,βalϵ=Iμ,ν′θ,βalϵ,Iμ,ν′θ∑j=1svjξ,βa1−ξlϵ,Iμ,ν′θ∏j=1svjξ+∑j=1svjsξ,βa1−sξlϵ,=⟶1L∗asl⟶∞,⟶1L∗asl⟶∞,⟶1L∗asl⟶∞.
Thus, (52) holds. But we have that(71)ηv=6s3−9s2+20s−12φv2,v2,0,…,0,has the property(72)Iμ,ν′L1βaηβav,ϵ≥L∗Iμ,ν′ηv,ϵ,v∈W,ϵ>0.
Hence,(73)Iμ,ν′ηv,ϵ=Iμ,ν′6s3−9s2+20s−12φv2,v2,0,…,0,ϵ=Iμ,ν′6θ,s3−9s2+20s−12ϵ,Iμ,ν′12θ2ξvξ,s3−9s2+20s−12ϵ,Iμ,ν′12θ2sξvsξ,s3−9s2+20s−12ϵ.
Now,(74)Iμ,ν′1βaηβav,ϵ=Iμ,ν′6θβa,s3−9s2+20s−12ϵ,Iμ,ν′12θ2ξβaβavξ,s3−9s2+20s−12ϵ,Iμ,ν′12θ2sξβaβavsξ,s3−9s2+20s−12ϵ,=Iμ,ν′βa−1ηv,ϵ,Iμ,ν′βaξ−1ηv,ϵ,Iμ,ν′βasξ−1ηv,ϵ.
From inequality (53), we can verify the following cases for conditions of βa.Case 1.
L=2−1 if a=0.(75)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′2−11−2−1ηv,ϵ=Iμ,ν′6θ,s3−9s2+20s−12ϵ.Case 2.
L=2 if a=1.(76)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′11−2ηv,ϵ=Iμ,ν′−6θ,s3−9s2+20s−12ϵ.Case 3.
L=2ξ−1 for ξ<1 if a=0.(77)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′2ξ−11−2ξ−1ηv,ϵ=Iμ,ν′12θvξ,s3−9s2+20s−122−2ξϵ.Case 4.
L=21−ξ for ξ>1 if a=1.(78)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′11−21−ξηv,ϵ=Iμ,ν′12θvξ,s3−9s2+20s−122ξ−2ϵ.Case 5.
L=2sξ−1 for ξ<1/s if a=0.(79)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′2sξ−11−2sξ−1ηv,ϵ=Iμ,ν′12θvsξ,s3−9s2+20s−122−2sξϵ.Case 6.
L=21−sξ for ξ<1/s if a=1.(80)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′11−21−sξηv,ϵ=Iμ,ν′12θvsξ,s3−9s2+20s−122sξ−2ϵ.


## 3.1. Stability Results: Direct Technique

In this section, we assume thatW, Z,Iμ,ν′,Y, and F,Iμ,ν,Y are linear space, IFN-space, and complete IFN-space, respectively.Theorem 3.
If a mappingφ:Ws⟶Zwith0<ς/2<1,(13)Iμ,ν′φ2v,2v,0,…,0,ε≥L∗Iμ,ν′ςφv,v,0,…,0,ε,(14)limk⟶∞Iμ,ν′φ2kv1,2kv2,…,2kvs,2kε=1L∗,for all v,v1,v2,…,vs∈W and all ϵ>0. If a mapping ϕ:W⟶F satisfies(15)Iμ,νDϕv1,v2,…,vs,ε≥L∗Iμ,ν′φv1,v2,…,vs,ε,for all v1,v2,…,vs∈W and all ϵ>0, then the limit(16)Iμ,νA1v−ϕ2kv2k,ϵ⟶1L∗ask⟶∞,exists and there exists a unique additive mapping A1:W⟶F satisfying functional equation (4) and(17)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′φv,v,0,…,0,s3−9s2+20s−126ϵ2−ς,for all v∈W and all ϵ>0.Proof.
Fixv∈W and all ϵ>0. Replacing v1,v2,…,vs by v,v,0,…,0 in (15), we have(18)Iμ,νs3−9s2+20s−126ϕ2v−2s3−9s2+20s−126ϕv,ε≥L∗Iμ,ν′φv,v,0,…,0,ε.
Replacingv by 2kv in (18) and using (IFN3), we obtain(19)Iμ,νϕ2k+1v2−ϕ2kv,6ϵ2s3−9s2+20s−12≥L∗Iμ,ν′φ2kv,2kv,0,…,0,ϵ.
By the inequality (13) and (IFN3) in (19), we have(20)Iμ,νϕ2k+1v2−ϕ2kv,6ϵ2s3−9s2+20s−12≥L∗Iμ,ν′φv,v,0,…,0,ϵςk.
Clearly, we can show from inequality (20) that(21)Iμ,νϕ2k+1v2k+1−ϕ2kv2k,6ϵ2k+1s3−9s2+20s−12≥L∗Iμ,ν′φv,v,0,…,0,ϵςk.
Replacingϵ by ςkϵ in (21), we get(22)Iμ,νϕ2k+1v2k+1−ϕ2kv2k,6ςkϵ2k+1s3−9s2+20s−12≥L∗Iμ,ν′φv,v,0,…,0,ϵ.
Clearly,(23)ϕ2kv2k−ϕv=∑a=0k−1ϕ2a+1v2a+1−ϕ2av2a.
It follows from (22) and (23) that(24)Iμ,νϕ2kv2k−ϕv,∑a=0k−16ςaϵ2a+1s3−9s2+20s−12≥L∗Ya=0k−1Iμ,ν′ϕ2a+1v2a+1−ϕ2av2a,6ςaϵ2a+1s3−9s2+20s−12≥L∗Ya=0k−1Iμ,ν′φv,v,0,…,0,ϵ≥L∗Iμ,ν′φv,v,0,…,0,ϵ,for all v∈W and ϵ>0. Replacing v by 2tv in (24) and with the help of (13), we have(25)Iμ,νϕ2k+tv2k+t−ϕ2tv2t,∑a=0k−16ςaϵ2a+t2s3−9s2+20s−12≥L∗Iμ,ν′φv,v,0,…,0,ϵςt,for every t,k≥0. Replacing ϵ by ςtϵ in (25), we have(26)Iμ,νϕ2k+tv2k+t−ϕ2tv2t,∑a=tk+t−16ςaϵ2a+1s3−9s2+20s−12≥L∗Iμ,ν′φv,v,0,…,0,ϵ.
Using (IFN3) in (26), we obtain(27)Iμ,νϕ2k+tv2k+t−ϕ2tv2t,ϵ≥L∗Iμ,ν′φv,v,0,…,0,ϵ∑a=tk+t−16ςa/2a2s3−9s2+20s−12,for all t,k≥0. Since 0<ς<2 and ∑a=0kς/2a<∞, the Cauchy criterion for convergence in IFNS shows that ϕ2kv/2k is Cauchy sequence in F,Iμ,ν,Υ. Since F,Iμ,ν,Υ is a complete, this sequence converges to some point A1v∈F. Then, we can define the mapping A1:W⟶F by(28)Iμ,νA1v−ϕ2kv2k⟶1L∗ask⟶∞.
Settingt=0 in inequality (29), we obtain(29)Iμ,νϕ2kv2k−ϕv,ϵ≥L∗Iμ,ν′φv,v,0,…,0,ϵ∑a=0k−16ςa/2a2s3−9s2+20s−12.
Taking the limit ask⟶∞ in (29), we obtain(30)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′φv,v,0,…,0,s3−9s2+20s−126ϵ2−ς.
Next, we want to prove that the functionA1 satisfies functional equation (4); replacing v1,v2,…,vs by 2kv1,2kv2,…,2kvs in (15), we have(31)Iμ,ν12kDϕ2kv1,…,2kvs,ϵ≥L∗Iμ,ν′φ2kv1,…,2kvs,2kϵ,for all v1,v2,…,vs∈W and all ϵ>0. Since(32)limk⟶∞Iμ,ν′φ2kv1,2kv2,…,2kvs,2kε=1L∗,the function A1 satisfies functional equation (4). Thus, the function A1 is additive. Finally, we want to prove that the function A1 is unique; consider another additive mapping A2:W⟶F satisfying functional equations (4) and (17). Hence,(33)Iμ,νA1v−A2v,ϵ=Iμ,νA12kv2k−A22kv2k,ϵ≥L∗ΥIμ,νA12kv2k−ϕ2kv2k,ϵ2,Iμ,νϕ2kv2k−A22kv2k,ϵ2≥L∗Iμ,ν′φ2kv,2kv,0,…,0,s3−9s2+20s−122kϵ2−ς12≥L∗Iμ,ν′φv,v,0,…,0,s3−9s2+20s−122kϵ2−ς12ςk,for all v∈W and all ϵ>0. As(34)lims⟶∞s3−9s2+20s−122kϵ2−ς12ςk=∞,we obtain(35)limk⟶∞Iμ,ν′φv,v,0,…,0,s3−9s2+20s−122kϵ2−ς12ςk=1L∗.
Thus,Iμ,νA1v−A2v,ϵ=1L∗.
Therefore,A1v=A2v. Thus, the additive function A1v is unique. This ends the proof.Theorem 4.
If a mappingφ:Ws⟶Zwith0<2/ς<1,(36)Iμ,ν′φ2−1v,2−1v,0,…,0,ϵ≥L∗Iμ,ν′1ςφv,v,0,…,0,ϵ,(37)limk⟶∞Iμ,ν′φ2−kv1,2−kv2,…,2−kvs,2−kϵ=1L∗,for allv,v1,v2,…,vs∈Wand allϵ>0. If a mappingϕ:E⟶Fsatisfies (15), then the limitIμ,νA1v−2kϕv/2k,ϵ⟶1L∗ask⟶∞ exists and there exists a unique additive mapping A1:W⟶F satisfying functional equation (4) and(38)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′φv,v,0,…,0,s3−9s2+20s−126ϵς−2,for all v∈W and all ϵ>0.Proof.
Fixv∈W and all ϵ>0. Replacing v1,v2,…,vs by v,v,0,…,0 in (15), we have(39)Iμ,νs3−9s2+20s−126ϕ2v−2s3−9s2+20s−126ϕv,ϵ≥L∗Iμ,ν′φv,v,0,…,0,ϵ.
From (39), we obtain that(40)Iμ,νϕ2v−2ϕv,6ϵ2s3−9s2+20s−12≥L∗Iμ,ν′φv,v,0,…,0,ϵ.
Replacingv by v/2 in (40), we get(41)Iμ,νϕv−2ϕv2,6ϵ2s3−9s2+20s−12≥L∗Iμ,ν′φv2,v2,0,…,0,ϵ.
Replacingv by v/2k in (41) and using (IFN3), we have(42)Iμ,νϕv2k−2ϕv2k+1,6ϵ2s3−9s2+20s−12≥L∗Iμ,ν′φv2k+1,v2k+1,0,…,0.
With the help of inequality (36) and (IFN3) in (42), we obtain that(43)Iμ,νϕv2k−2ϕv2k+1,6ϵ2s3−9s2+20s−12≥L∗Iμ,ν′φv,v,0,…,0,ϵςk+1.
The remaining part of the proof can be proven in the same way as Theorem3.Corollary 1.
Letθ∈ℝ+. If a mappingϕ:W⟶Fsuch that(44)Iμ,νDϕv1,v2,…,vs,ϵ≥L∗Iμ,ν′θ,ϵ,for allv1,v2,…,vs∈Wand allϵ>0, then there exists a unique additive mappingA1:W⟶Fsatisfying(45)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′6θ,2−1ϵs3−9s2+20s−12,for all v∈W and all ϵ>0.Proof.
The proof holds from Theorems3 and 4 by letting φv1,v2,…,vs=θ and ς=20.Corollary 2.
Letθ,ξ∈ℝ+withξ∈0,1∪1,+∞. If a mappingϕ:W⟶Fsuch that(46)Iμ,νDϕv1,v2,…,vs,ϵ≥L∗Iμ,ν′θ∑a=1svaξ,ϵ,for allv1,v2,…,vs∈Wand allϵ>0, then there exists a unique additive mappingA1:W⟶Fsatisfying(47)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′12θvξ,2−2ξs3−9s2+20s−12ϵ,for all v∈W and all ϵ>0.Proof.
The proof holds from Theorems3 and 4 by setting φv1,v2,…,vs=θ∑a=1svaξ and ς=2ξ.Corollary 3.
Letθ,ξ,γ,τ∈ℝ+withsξ,sτ∈0,1∪1,+∞. If a mappingϕ:W⟶Fsuch that(48)Iμ,νDϕv1,v2,…,vs,ϵ≥L∗Iμ,ν′θ∑a=1svasξ+γ∏a=1svaτ,ϵ,for allv1,v2,…,vs∈Wand allϵ>0, then there exists a unique additive mappingA1:W⟶Fsatisfying(49)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′12θvsξ,2−2sξs3−9s2+20s−12ϵ,for all v∈W and all ϵ>0.Proof.
The proof holds from Theorems3 and 4 by setting φv1,v2,…,vs=θ∑a=1svasξ+γ∏a=1svaτ and ς=2sξ.Corollary 4.
Letγ,τ∈ℝ+with0<sτ≠1. If a mappingϕ:W⟶Fsuch that(50)Iμ,νDϕv1,v2,…,vs,ϵ≥L∗Iμ,ν′γ∏a=1svaτ,ϵ,for all v1,v2,…,vs∈W and all ϵ>0, then the mapping ϕ is additive.Proof.
The proof is valid from Theorems3 and 4 by setting φv1,v2,…,vs=γ∏a=1svaτ.


## 3.2. Stability Results: Fixed Point Technique

Before we begin, let us consider a constantβa such that(51)βa=2,ifa=0,12,ifa=1,and Ψ is the set such that Ψ=n1|n1:W⟶F,n10=0.Theorem 5.
Consider a mappingϕ:W⟶Ffor which there is a mappingφ:Ws⟶Zwith(52)liml⟶∞Iμ,ν′φ2lv1,2lv2,…,2lvs,2lϵ=1L∗,satisfying functional inequality (15). If there isL=Lasuch thatv⟶ηv=6/s3−9s2+20s−12φv/2,v/2,0,…,0has the property(53)Iμ,ν′L1βaηβav,ϵ=Iμ,ν′ηv,ϵ,then there exists a unique additive mapping A1:W⟶Fsatisfying functional equation (4) and(54)Iμ,νϕv−A1v,ϵn≥L∗Iμ,ν′L1−a1−Lηv,ϵ,for all v∈W and all ϵ>0.Proof.
Letς be a general metric on Ψ:(55)ςn1,n2=inft∈0,∞|Iμ,νn1v−n2v,ϵ≥L∗Iμ,ν′tηv,ϵ,v∈W,ϵ>0.
Clearly,Ψ,ς is complete. Define a mapping Υ:Ψ⟶Ψ by Υn1v=1/βan1βav, for all v∈W. For n1,n2∈Ψ, we have(56)ςn1,n2≤t,⇒Iμ,νn1v−n2v,ε≥L∗Iμ,ν′tηv,ε⇒Iμ,νn1βavβa−n2βavβa,ε≥L∗Iμ,ν′tηβavβa,ε⇒Iμ,νΥn1v−Υn2v,ε≥L∗Iμ,νtLηv,ε⇒ςΥn1v,Υn2v≤tL⇒ςΥn1,Υn2≤Lςn1,n2.
Thus, the functionΥ is strictly contractive on Ψ with L (Lipschitz constant). Replacing v1,v2,…,vs by v,v,0,…,0 in (15), we have(57)Iμ,νs3−9s2+20s−126ϕ2v−2s3−9s2+20s−126ϕv,ϵ≥L∗Iμ,ν′φv,v,0,…,0,ϵ.
Using (IFN3) in (57), we have(58)Iμ,νϕ2v2−ϕv,ϵ≥L∗Iμ,ν′62s3−9s2+20s−12φv,v,0,…,0,ϵ.
Using equation (53) for the case a=0, we have(59)Iμ,νϕ2v2−ϕv,ϵ≥L∗Iμ,ν′Lηv,ϵ⇒ςΥϕ,ϕ≤L=L1=L1−a.
Replacingv by v/2 in (57), we have(60)Iμ,νϕv−2ϕv2,ϵ≥L∗Iμ,ν′6s3−9s2+20s−12φv2,v2,0,…,0,ϵ,for all v∈W and all ϵ>0; using (53) for the case a=1, we have(61)Iμ,νϕv−2ϕv2,ϵ≥L∗Iμ,ν′ηv,ϵ⇒ςϕ,Υϕ≤1=L0=L1−a.
We can conclude from equations (59) and (61) that(62)ςϕ,Υϕ≤L1−a<∞.
By the fixed point alternative in both cases, there is a fixed pointA1 of Υ in Ψ such that(63)limk⟶∞Iμ,νϕβakvβak−A1v,ϵ⟶1L∗,v∈W,ϵ>0.
Replacingv1,v2,…,vs by βav1,βav2,…,βavs in (15), we obtain(64)Iμ,ν1βaDϕβav1,βav2,…,βavs,ϵ≥L∗Iμ,ν′φβav1,βav2,…,βavs,βaϵ,for all v1,v2,…,vs∈W and all ϵ>0. By same manner of Theorem 3, we can show that the function A1 satisfies functional equation (4). By Theorem 2, as A1 is a unique fixed point of Υ in Δ=ϕ∈Ψ|ςϕ,A1<∞, the function A1 is unique such that(65)Iμ,νA1v−ϕv,ε≥L∗Iμ,ν′tηv,ε,t>0.
Using fixed point alternative, we reach(66)ςϕ,A1≤11−Lςϕ,Υϕ⇒ςϕ,A1≤L1−a1−L⇒Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′L1−a1−Lηv,ϵ,for all v∈W and all ϵ>0. Hence, the proof of the theorem is now completed.Corollary 5.
Letθ,ξ∈ℝ+withθ>0. If a mappingϕ:W⟶Fsuch that(67)Iμ,νDϕv1,v2,…,vs,ϵ≥L∗Iμ,ν′θ,ϵ,Iμ,ν′θ∑j=1svjξ,ϵ,Iμ,ν′θ∏j=1svjξ+∑j=1svjsξ,ϵ,for allv1,v2,…,vs∈Wandϵ>0, then there exists a unique additive mappingA1:W⟶Fsatisfying(68)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′6θ,s3−9s2+20s−12ϵ,Iμ,ν′12θvs,s3−9s2+20s−122−2ξϵ,ξ<1orξ>1,Iμ,ν′12θvs,s3−9s2+20s−122−2sξϵ,ξ<1sorξ>1s,for all v∈W and all ϵ>0.Proof.
Set(69)φv1,v2,…,vs=θ,θ∑j=1svjξ,θ∏j=1svjξ+∑j=1svjsξ.
Then,(70)Iμ,ν′φβalv1,βalv2,…,βalvs,βalϵ=Iμ,ν′θ,βalϵ,Iμ,ν′θ∑j=1svjξ,βa1−ξlϵ,Iμ,ν′θ∏j=1svjξ+∑j=1svjsξ,βa1−sξlϵ,=⟶1L∗asl⟶∞,⟶1L∗asl⟶∞,⟶1L∗asl⟶∞.
Thus, (52) holds. But we have that(71)ηv=6s3−9s2+20s−12φv2,v2,0,…,0,has the property(72)Iμ,ν′L1βaηβav,ϵ≥L∗Iμ,ν′ηv,ϵ,v∈W,ϵ>0.
Hence,(73)Iμ,ν′ηv,ϵ=Iμ,ν′6s3−9s2+20s−12φv2,v2,0,…,0,ϵ=Iμ,ν′6θ,s3−9s2+20s−12ϵ,Iμ,ν′12θ2ξvξ,s3−9s2+20s−12ϵ,Iμ,ν′12θ2sξvsξ,s3−9s2+20s−12ϵ.
Now,(74)Iμ,ν′1βaηβav,ϵ=Iμ,ν′6θβa,s3−9s2+20s−12ϵ,Iμ,ν′12θ2ξβaβavξ,s3−9s2+20s−12ϵ,Iμ,ν′12θ2sξβaβavsξ,s3−9s2+20s−12ϵ,=Iμ,ν′βa−1ηv,ϵ,Iμ,ν′βaξ−1ηv,ϵ,Iμ,ν′βasξ−1ηv,ϵ.
From inequality (53), we can verify the following cases for conditions of βa.Case 1.
L=2−1 if a=0.(75)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′2−11−2−1ηv,ϵ=Iμ,ν′6θ,s3−9s2+20s−12ϵ.Case 2.
L=2 if a=1.(76)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′11−2ηv,ϵ=Iμ,ν′−6θ,s3−9s2+20s−12ϵ.Case 3.
L=2ξ−1 for ξ<1 if a=0.(77)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′2ξ−11−2ξ−1ηv,ϵ=Iμ,ν′12θvξ,s3−9s2+20s−122−2ξϵ.Case 4.
L=21−ξ for ξ>1 if a=1.(78)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′11−21−ξηv,ϵ=Iμ,ν′12θvξ,s3−9s2+20s−122ξ−2ϵ.Case 5.
L=2sξ−1 for ξ<1/s if a=0.(79)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′2sξ−11−2sξ−1ηv,ϵ=Iμ,ν′12θvsξ,s3−9s2+20s−122−2sξϵ.Case 6.
L=21−sξ for ξ<1/s if a=1.(80)Iμ,νϕv−A1v,ϵ≥L∗Iμ,ν′11−21−sξηv,ϵ=Iμ,ν′12θvsξ,s3−9s2+20s−122sξ−2ϵ.


## 4. Stability Results in 2-Banach Spaces

In 1960, Gahler [27, 28] developed the concept of linear 2-normed spaces.Definition 4.
Consider a linear spaceW over ℝ with dimension W>1 and consider a mapping ·,·:W2⟶ℝ with the following conditions:(a)
r,s=0 if and only if r and s are linearly dependent.(b)
r,s=s,r,(c)
λr,s=λr,s,(d)
r,s+w≤r,s+r,w,for all r,s,w∈W and λ∈ℝ.
Then, the function·,· is called as a 2-norm on W and the pair W,·,· is called as a linear 2-normed space. A typical example of 2-normed space is ℝ2 with 2-norm defined as r,s= the area of the triangle with the vertices 0, r, and s is a typical example of a 2-normed space.As a result of (d), it follows that(81)r+s,w≤r,w+s,wandr,w−s,w≤r−s,w.Thus,r⟶r,s are continuous mappings of W into ℝ for any fixed s∈W.Definition 5.
A sequencerj in a linear 2-normed space W is known as a Cauchy sequence if there exist two points s,w∈W such that s and w are linearly independent.(82)limi,j⟶∞ri−rj,s=0,limi,j⟶∞ri−rj,w=0.Definition 6.
A sequencerj in a linear 2-normed space W is called as a convergent sequence if there exists an element r∈W such that(83)limi,j⟶∞rj−r,s=0,for all s∈W. If rj converges to r, then we denote rj⟶r as j⟶∞ and say that r is the limit point of rj. We also write in this instance(84)limj⟶∞rj=r.Definition 7.
A 2-Banach space is a linear 2-normed space in which every Cauchy sequence is convergent.Lemma 1 (see [29]).
LetW,·,·be a linear 2-normed space. Ifr∈Wand‖r,s‖=0for everys∈W, thenr=0.Lemma 2 (see [29]).
For a convergent sequencerjin a linear 2-normed spaceW,(85)limj⟶∞rj,w=limj⟶∞rj,s,for every s∈W.Park studied approximate additive mappings, approximate Jensen mappings, and approximate quadratic mappings in 2-Banach spaces in his paper [29]. In [30], Park examined the superstability of the Cauchy functional inequality and the Cauchy–Jensen functional inequality in 2-Banach spaces under certain conditions.In this section, we letW be a normed linear space and F be a 2-Banach space.Theorem 6.
Letφ:W⟶0,+∞be a function such that(86)limi⟶∞12iφ2iv1,2iv2,…,2ivs,w=0,for all v1,v2,…,vs,w∈W. If a mapping ϕ:W⟶F such that ϕ0=0 and (87)Dϕv1,v2,…,vs,w≤φv1,v2,…,vs,w,(88)φ^v,w≕∑j=0∞12jφ2jv,2jv,0,…,0,w<∞, for all v1,v2,…,vs,w∈W. Then, there exists a unique additive mapping A1:W⟶F satisfying(89)ϕv−A1v,w≤62s3−9s2+20s−12φ^v,w,for all v,w∈W.Proof.
Replacingv1,v2,…,vs by v,v,0,…,0 in (87), we get(90)s3−9s2+20s−126ϕ2v−2s3−9s2+20s−126ϕv,w≤φv,v,0,…,0,w,for all v,w∈W. Replacing v by 2nv in (90) and dividing both sides by 2n−1, we have(91)12n+1ϕ2n+1v−12nϕ2nv,w≤62n+1s3−9s2+20s−12φ2iv,2iv,0,…,0,w,for all v,w∈W and all non-negative integers i. Hence,(92)12n+1ϕ2n+1v−12mϕ2mv,w≤∑j=mi12j+1ϕ2j−1v−12jϕ2jv,w≤62s3−9s2+20s−12∑j=mi12jφ2jv,2jv,0,…,0,w,for all v,w∈W and all non-negative integers m and i with i≥m. Therefore, it follows from (15) and (19) that the sequence 1/2iϕ2iv is Cauchy in F for every v∈W. Since F is complete, the sequence 1/2iϕ2iv converges in F for all v∈W. Thus, we may define a mapping A1:W⟶F by(93)A1v:limi⟶∞12iϕ2iv,for all v∈W. Therefore,(94)limi⟶∞12iϕ2iv−A1v,w=0,for all v,w∈W. Letting m=0 and taking the limit as i⟶∞ in (94), we have (89). Next, we want to prove that the function A1 is additive. From inequalities (86), (87), and (94) and Lemma 2,(95)Dϕv1,v2,…,vs,w=limi⟶∞Dϕ2iv1,2iv2,…,2ivs,w≤limi⟶∞12iφ2iv1,2iv2,…,2ivs,w=0,for all v1,v2,…,vs,w∈W. By Lemma 1,(96)DA1v1,v2,…,vs=0,for all v1,v2,…,vs∈W. Hence, according to Theorem 1, the mapping A1:W⟶F is additive.
To prove that the functionA1 is unique, we consider another additive mapping A1′:W⟶F satisfying (89). Then,(97)A1v−A1′v,w=limi⟶∞12iA12iv−ϕ2iv+ϕ2iv−A1′2iv,w≤6s3−9s2+20s−12limi⟶∞12iφ^2iv,w=0,for all v,w∈W. By Lemma 1, A1v−A1′v=0 for all v∈W. Therefore, A1=A1′.Remark 3.
A theorem analogous to (93) can be formulated, in which the sequence(98)A1v≔limi⟶∞2iϕv2iis defined with appropriate assumptions for φ.Corollary 6.
Letλ:0,∞⟶0,∞be a mapping such thatλ0=0and(i)
λpq≤λpλs.(ii)
λp<pfor allp>1.If a mappingϕ:W⟶F with ϕ0=0 and(99)Dϕv1,v2,…,vs,w≤∑i=1sλvi+λw,for all v1,v2,…,vs,w∈W, then there exists a unique additive mapping A1:W⟶F satisfying(100)ϕv−A1v,w≤6s3−9s2+20s−122λv2−λ2+λw,for all v,w∈W.Proof.
Let(101)φv1,v2,…,vs,w=∑i=1sλvi+λw,for all v1,v2,…,vs,w∈W. It follows from (i) that(102)λ2i≤λ2i,φ2iv1,2iv2,…,2ivs,w≤λ2i∑i=1sλvi+λw.
By using Theorem6, we obtain (96).Corollary 7.
Letqbe a positive real number such thatq<1and letH:0,∞×0,∞⟶0,∞be a homogeneous mapping with degreeq. If a mappingϕ:W⟶Fwithϕ0=0and(103)Dϕv1,v2,…,vs,w≤Hv1,v2,…,vs+w,for all v1,v2,…,vs,w∈W, then there exists a unique additive mapping A1:W⟶F satisfying(104)ϕv−A1v,w≤6s3−9s2+20s−12Hv,v,0,…,0+w2−q,for all v,w∈W.Proof.
Let(105)φv1,v2,…,vs,w=Hv1,v2,…,vs+w,for all v1,v2,…,vs,w∈W. By using Theorem 6, we have (104).Corollary 8.
Letq∈ℝ+such thatq<1and letH:0,∞×0,∞⟶0,∞be a homogeneous mapping with degreeq. If a mappingϕ:W⟶Fwithϕ0=0and(106)Dϕv1,v2,…,vs,w≤Hv1,v2,…,vsw,for all v1,v2,…,vs,w∈W, then there exists a unique additive mapping A1:W⟶F satisfying(107)ϕv−A1v,w≤62s3−9s2+20s−12Hv,v,0,…,0w2−2q,for all v,w∈W.Proof.
Let(108)φv1,v2,…,vs,w=Hv1,v2,…,vsw,for all v1,v2,…,vs,w∈W. By using Theorem 6, we have (110).Corollary 9.
Letp∈ℝ+such thatp<1. If a mappingϕ:W⟶Fwithϕ0=0and(109)Dϕv1,v2,…,vs,w≤∑i=1svip+w,for all v1,v2,…,vs,w∈W, then there exists a unique additive mapping A1:W⟶F satisfying(110)ϕv−A1v,w≤6s3−9s2+20s−122vp+w2−p,for all v,w∈W.We use an appropriate example to demonstrate that the stability of the functional equation (4) fails in the singular case. We provide the following counterexample, which shows the instability in a particular condition p=2 in Corollary 9 of functional equation (4), inspired by Gajda’s excellent example in [31].Remark 4.
If a mappingϕ:ℝ⟶W satisfies (4), then the following assertions hold:(1)
ϕmcv=mcϕv,v∈ℝ,m∈ℚ, and c∈ℤ.(2)
ϕv=vϕ1,v∈ℝ if the function ϕ is continuous.Example 2.
Let a mappingϕ:ℝ⟶ℝbe defined by(111)ϕv=∑p=0∞ψ2pv2p,where(112)ψv=λv,−1<v<1,λ,else.
Then, the mappingϕ:ℝ⟶ℝ satisfies(113)Dϕv1,v2,…,vs≤n4−8n3+5n2+34n−32443λ∑j=1svj,for all v1,v2,…,vs∈ℝ, but there does not exist an additive mapping A1:ℝ⟶ℝ satisfying(114)ϕv−A1v≤δv,v∈ℝ,where λ and δ are constants.


## 5. Conclusion

In this work, a new dimensional additive functional (equation (4)) has been introduced. We primarily found its solution and examined Hyers–Ulam stability in IFN-spaces using the direct approach in Section 3.1 and the fixed point approach in Section 3.2. In Section 4, we investigated the Hyers–Ulam stability in 2-Banach space by using the direct method. Also, we provided the counterexample, which shows the instability in a particular condition p=2 in Corollary 9 of equation (4), by the way of Gajda.



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