# Approximation of an Additiveϱ1,ϱ2-Random Operator Inequality

Authors: Sun Young Jang; Reza Saadati
Journal: Journal of Function Spaces
 (2020)

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## Abstract

We solve the additiveϱ1,ϱ2-random operator inequality ξtTω,u+v−Tω,u−Tω,v≥κMξtϱ1Tω,u+v+Tω,u−v−2Tω,u,ξtϱ22Tω,u+v/2−Tω,u−Tω,v, in which ϱ1,ϱ2∈ℂ are fixed and max2ϱ1,ϱ2<1. Finally, we get an approximation of the mentioned additive ϱ1,ϱ2-random operator inequality by direct technique.


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## Body

## 1. Introduction

Park [1] introduced and solved the following additive ρ1,ρ2-functional inequality:(1)fx+y−fx−fy≤ρ1fx+y+fx−y−2fx+ρ22fx+y2−fx−fy,where ρ1 and ρ2 are fixed nonzero complex numbers with 2ρ1+ρ2<1. Next, he proved the Hyers–Ulam stability of the additive ρ1,ρ2-functional inequality (1) in Banach spaces.In this paper, we get a generalization from Park’s results [1] in MB-spaces. Let Ω,U,μ be a probability measure space. Assume that U,BU and V,BV are Borel measureable spaces, in which U and V are MB-spaces and T:Ω×U2⟶V is a random operator. In MB-spaces, first we solve the additive ϱ1,ϱ2-random operator inequality:(2)ξtTω,u+v−Tω,u−Tω,v≥κMξtϱ1Tω,u+v+Tω,u−v−2Tω,u,ξtϱ22Tω,u+v/2−Tω,u−Tω,v,in which ϱ1,ϱ2∈ℂ are fixed and max2ϱ1,ϱ2<1.Next, we get a random approximation of the additive-random operator (2).


## 2. Preliminaries

LetΞ+ be the set of distribution mappings, i.e., the set of all mappings G:ℝ∪−∞,∞⟶0,1, writing Gt for Gt, such that G is left continuous and increasing on ℝ. O+⊆Ξ+ includes all mappings G∈Ξ+ for which ℓ−G+∞ is one and ℓ−gx is the left limit of the mapping g at the point x, i.e., ℓ−gx=limt⟶x−gt.InΞ+, we define “≤” as follows:(3)H≤F⇔Hs≤Fs,for each s in ℝ (partially ordered). Note that the function ϑu defined by(4)ϑsu=0,ifs≤u,1,ifs>u,is an element of Ξ+, and ϑ0 is the maximal element in this space (for some more details, see [2–4]).Definition 1 (see [2, 5]). Let I=0,1.
A continuous triangular norm (shortly, act-norm) is a continuous mapping κ from I×I to I such that(a)
κς,τ=κτ,ς and κς,κτ,υ=κκς,τ,υ for all ς,τ,υ∈0,1(b)
κς,1=ς for all ς∈I(c)
κς,τ≤κυ,ι whenever ς≤υ and τ≤ι for all ς,τ,υ,ι∈I
Some examples of thet-norms are as follows:(1)
κPς,τ=ςτ(2)
κMς,τ=minς,τ(3)
κLς,τ=maxς+τ−1,0 (: the Lukasiewicz t-norm)Definition 2.
(see [3]). Suppose that κ is a ct-norm, V is a linear space and ξ is a mapping from V to O+. In this case, the ordered tuple V,ξ,κ is called a Menger normed space (in short, MN-space) if the following conditions are satisfied:(MN1)ξtv=ϑt0 for all t>0 if and only if v=0(MN2)ξtαv=ξt/αv for all v∈V and α∈ℂ with α≠0(MN3)ξt+su+v≥κξtu,ξsv for all u,v∈V and t,s>0
LetV,⋅ be a linear normed space. Then,(5)ξsv=0,ifs≤0,exp−vs,ifs>0,defines a Menger norm, and the ordered tuple V,ξ,κM is an MN-space.
LetΩ,U,μ be a probability measure space. Assume that U,BU and V,BV are Borel measureable spaces, in which U and V are complete FB-spaces. A mapping T:Ω×U⟶V is said to be a random operator if ω:Tω,u∈B∈U for all u in U and B∈BV. Also, T is a random operator, if Tω,u=vω be a V-valued random variable for every u in U. A random operator T:Ω×U⟶V is called linear if Tω,αu1+βu2=αTω,u1+βTω,u2, almost everywhere for each u1,u2 in U and α,β scalers, and Menger random bounded (in short, MR-bounded) if there exists a nonnegative real-valued random variable Mω such that(6)ξMωtTω,u1−Tω,u2≥ξtu1−u2,almost everywhere for each u1,u2 in U and t>0.
Recently, some authors have published some papers on stability of functional equations in several spaces by the direct method and the fixed point method, for example, Banach spaces [6–8], fuzzy Menger normed algebras [9], fuzzy normed spaces [10], non-Archimedean random Lie C∗-algebras [11], non-Archimedean random normed spaces [12], random multinormed space [13], random lattice normed spaces, and random normed algebras [14, 15]. In [16, 17], the authors studied the stability problem for fractional equations. Next, Cădariu et al. [18–20] applied the fixed point method to solve the stability problem, and their work was continued by Keltouma et al. [21], Park et al. [22, 23], Jung and Lee [24], and Brzdʁk and Ciepliński [25], see also [26, 27].


## 3. Approximation of Additiveϱ1,ϱ2-Random Operator Inequality: Direct Technique

Now, we modify and generalize Park’s results [1]. First, we solve and investigate the additive ϱ1,ϱ2-random operator inequality (2) in MN-spaces.Lemma 1.
LetV,ξ,κM be an MN-space. Let T:Ω×U⟶V be a random operator satisfying Tω,0=0 and (2), and then T is additive.Proof.
Replacingv by u in (2), we get(7)ξtTω,2u−2Tω,u≥ξt/ϱ1Tω,2u−2Tω,u.
Sinceϱ1<1, Tω,2u=2Tω,u for each u∈U and ω∈Ω,(8)Tω,u2=12Tω,u,almost everywhere for each u∈U and ω∈Ω.
(2) and (8) imply that(9)ξtTω,u+v−Tω,u−Tω,v≥κMξt/ϱ1Tω,u+v+Tω,u−v−2Tω,u,ξt/ϱ2Tω,u+v−Tω,u−Tω,v,and so(10)ξtTω,u+v−Tω,u−Tω,v≥ξt/ϱ1Tω,u+v+Tω,u−v−2Tω,u,almost everywhere for each u,v∈U, ω∈Ω, and t>0.
Puttingz=u+v and w=u−v in (10), we get(11)ξtTω,z+w+Tω,z−w−2Tω,z≥ξt/2ϱ1Tω,z+w−Tω,z−Tω,w,almost everywhere for each z,w∈U, ω∈Ω, and t>0.
Now, (10) and (11) imply that(12)ξtTω,u+v−Tω,u−Tω,v≥ξt/2ϱ12Tω,u+v−Tω,u−Tω,v,for all x,y∈X. Since ϱ1<2/2, Tω,u+v=Tω,u+Tω,v, almost everywhere for each u,v∈U and ω∈Ω, which implies that T is additive.
We get an approximation of the additiveϱ1,ϱ2-random operator inequality (2) in MN-spaces, by applying the direct technique.Theorem 1.
LetV,ξ,κM be an MN-space. Assume that ψ:U2⟶O+ be a distribution function such that there exists β<1 with(13)ψβt/2u/2,v/2≥ψtu,v,(14)limp⟶∞ψt/2pu/2p,v/2p=ϑt0,almost everywhere for each u,v∈U, ω∈Ω, and t>0. Suppose that T:Ω×U⟶V be a random operator satisfying Tω,0=0 and(15)ξtTω,u+v−Tω,u−Tω,v≥κMξtϱ1Tω,u+v+Tω,u−v−2Tω,u,ξtϱ22Tω,u+v/2−Tω,u−Tω,v,ψtu,v,in which ϱ1,ϱ2∈ℂ are fixed and max2ϱ1,ϱ2<1. Therefore, there is a unique additive random operator S:Ω×U⟶V such that(16)ξtTω,u−Sω,u≥ψ21−β/βtu,u,almost everywhere for each u∈U, ω∈Ω, and t>0.Proof.
Puttingu=v in (15), we have that(17)ξt2Tω,u/2−Tω,u≥ψtu/2,u/2≥ψ2/βtu,u,almost everywhere for each u∈U, ω∈Ω, and t>0. Replacing u by u/2ℓ in (17) and applying (13), we get(18)ξt2ℓ+1Tω,u/2ℓ+1−2ℓTω,u/2ℓ≥ψ2/βℓ+1tu,u,which implies that(19)ξ∑k=1ℓβk2t2ℓTω,u2ℓ−Tω,u≥ψtu,u.
Replacingu by u/2mπ in (19), we get(20)ξt2ℓ+mTω,u2ℓ+m−2mTω,u2m≥ψt/∑k=m+1ℓ+mβk/2u,u,which tends to ϑ0 when m,ℓ tend to ∞, and so the sequence 2ℓTω,u/2ℓ is Cauchy in the complete MN-space V,ξ,κM and converges to a point Sω,x∈V. Now, for every ς>0, we have that(21)ξt+ςTω,u−Sω,u≥κMξtTω,u−2ℓTω,u/2ℓ,ξςSω,u−2ℓTω,u/2ℓ≥κMψt/∑k=1ℓβk/2u,u,ξςSω,u−2ℓTω,u/2ℓ.
Whenℓ tends to ∞ in (21), we have that(22)ξt+ςTω,u−Sω,u≥ψ21−β/βtu,u.
Sinceς>0 is arbitrary in (22), we have that(23)ξtTω,u−Sω,u≥ψ21−β/βtu,u.
Replacingu and v by u/2m and v/2m in (15) and using (14) imply that S satisfies Lemma 1 and hence is an additive random operator. Now, let S′ be another additive random operator satisfies (16). For an arbitrary u∈U and ω∈Ω, we have that 2mSω,u/2m=Sω,u and 2mS′ω,u/2m=S′ω,u for each natural element m. Using (16), we have that(24)ξtSω,u−S′ω,u=limm⟶∞ξt2mSω,u/2m−2mS′ω,u/2m≥limm⟶∞κMξt/22mSω,u/2m−2mTω,u/2m,ξt/22mTω,u/2m−2mS′ω,u/2m≥limm⟶∞ψ1−β/2mβtu/2m,u/2m≥limm⟶∞ψ1−β/βm+1tu,u⟶ϑt0,which implies that Sω,u=S′ω,u shows the uniqueness.Corollary 1.
LetV,ξ,κM be an MN-space, ρ>1 and τ>0. Suppose that T:Ω×U⟶V be a random operator satisfying Tω,0=0 and(25)ξtTω,u+v−Tω,u−Tω,v≥κMξtϱ1Tω,u+v+Tω,u−v−2Tω,u,ξtϱ22Tω,u+v/2−Tω,u−Tω,v,tt+τuρ+vρ,in which ϱ1,ϱ2∈ℂ are fixed and max2ϱ1,ϱ2<1. Then, there exists a unique additive random operator S:Ω×U⟶V such that(26)ξtTω,u−Sω,u≥2ρ−1−1t2ρ−1−1t+τuρ,for each u∈U, ω∈Ω, and t>0.Proof.
In Theorem1, put ψtu,u=t/t+τuρ+vρ for each u∈U and t>0, and β=21−ρ.Theorem 2.
LetV,ξ,κM be an MN-space. Assume that ψ:U2⟶O+ be a distribution function such that there exists an β<1 with(27)ψ2βt2u,2v≥ψtu,v,(28)limp⟶∞ψ2pt2pu,2pv=ϑt0,almost everywhere for each u,v∈U, ω∈Ω, and t>0. Suppose that T:Ω×U⟶V be a random operator satisfying Tω,0=0 and (15). Therefore, there is a unique additive random operator S:Ω×U⟶V such that(29)ξtTω,u−Sω,u≥ψ21−βtu,u,almost everywhere for each u∈U, ω∈Ω, and t>0.Proof.
Puttingu=v in (15), we have that(30)ξtTω,2u/2−Tω,u≥ψ2tu,u≥ψt/βu/2,u/2,almost everywhere for each u∈U, ω∈Ω, and t>0. Replacing u by 2ℓu in (30) and applying (27), we get(31)ξtTω,2ℓ+1u/2ℓ+1−Tω,2ℓu/2ℓ≥ψ2/βℓtu,u,which implies that(32)ξ∑k=0ℓ−1βk/2tTω,2ℓu/2ℓ−Tω,u≥ψtu,u.
Replacingu by u/2m in (32), we get(33)ξtTω,2ℓ+mu/2ℓ+m−Tω,2mu/2m≥ψt/∑k=mℓ+m−1βk/2u,u,which tends to ϑ0 when m,ℓ tend to ∞, and so the sequence Tω,2ℓu/2ℓ is Cauchy in the complete MN-space V,ξ,κM and converges to a point Sω,x∈V. Now, for every ς>0, we have that(34)ξt+ςTω,u−Sω,u≥κMξtTω,u−Tω,2ℓu/2ℓ,ξςSω,u−Tω,2ℓu/2ℓ≥κMψt/∑k=0ℓ−1βk/2u,u,ξςSω,u−Tω,2ℓu/2ℓ,
Whenℓ tends to ∞ in (34), we have(35)ξt+ςTω,u−Sω,u≥ψ21−βtu,u.
Sinceς>0 is arbitrary in (35), we have(36)ξtTω,u−Sω,u≥ψ21−βtu,u.
Replacingu and v by 2mu and 2mv in (15) and using (14) imply that S satisfies Lemma 1 and hence is an additive random operator. Now, let S′ be another additive random operator satisfies (29). For an arbitrary u∈U and ω∈Ω, we have that Sω,2mu/2m=Sω,u, and for each positive integer m. Using (29), we have(37)ξtSω,u−S′ω,u=limm⟶∞ξtSω,2mu/2m−S′ω,2mu/2m≥limm⟶∞κMξt/2Sω,2mu/2m−Tω,2mu/2m,ξt/2Tω,2mu/2m−S′ω,2mu/2m≥limm⟶∞ψ2m1−βt2mu,2mu≥limm⟶∞ψ1−β/βmtu,u⟶ϑt0,which implies that Sω,u=S′ω,u shows the uniqueness.Corollary 2.
LetV,ξ,κM be an MN-space, ρ<1 and τ>0. Suppose that T:Ω×U⟶V be a random operator satisfying Tω,0=0 and (25). Therefore, there is a unique additive random operator S:Ω×U⟶V such that(38)ξtTω,u−Sω,u≥2−2ρt2−2ρt+2τuρ,for each u∈U, ω∈Ω, and t>0.Proof.
In Theorem2, put ψtu,u=t/t+τuρ+vρ for each u∈U, t>0and β=2ρ−1.



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*Source: 7540303-2020-02-18.xml*