Approximation of the multiplicatives on random multi normed space

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R E S E A R C H

Open Access

Approximation of the multiplicatives on

random multi-normed space

Ravi P Agarwal

1

, Reza Saadati

2*

and Ali Salamati

2

*Correspondence: rsaadati@eml.cc 2Department of Mathematics, Iran

University of Science and Technology, Tehran, Iran Full list of author information is available at the end of the article

Abstract

In this paper, we consider random multi-normed spaces introduced by Dales and Polyakov (Multi-Normed Spaces, 2012). Next, by the fixed point method, we approximate the multiplicatives on these spaces.

MSC: Primary 39A10; 39B52; 39B72; 46L05; 47H10; 46B03; secondary 54E40; 54E35; 54H25

Keywords: approximation; homomorphisms; random multi-normed space; dual

random multi-normed space

1 Introduction

The concept of random normed spaces and their properties are discussed in []. Also, the concept of multi-normed spaces was introduced by Dales and Polyakov. In this paper we combine the mentioned concepts and introduce random multi-normed spaces. Next, we get an approximation for homomorphisms in these spaces. For more results and applica-tions, one can see [–].

Definition . Let (E,μ,∗) be a random normed space.∗is a continuous t-norm. A multi-random norm on{Ek,kN}is sequence{N

k}such thatNkis a random norm onEk(k∈N),

μx(t) =μx(t) for eachxEandt∈Rand the following axioms are satisfied for eachk∈N

withk≥:

(NF) μkAσ(x)(t) =μk

x(t), for eachσσk,xEk,t∈R,

(NF) μkMα(x)(t)≥μkmax

i∈Nk|αi|x(t), for eachα= (α, . . . ,αk)∈R

k,xEk,tR,

(NF) μk(x+

,...,xk,)(t) =μ

k

(x,...,xk)(t), for eachx, . . . ,xkEandt∈R,

(NF) μk(x+

,...,xk,xk)(t) =μ

k

(x,...,xk)(t), for eachx, . . . ,xkEandt∈R.

In this case{(Ek,μk,∗),k∈N}is called a random multi-normed space. Moreover, if ax-iom (NF) is replaced by the following axax-iom:

(DF) μk+

(x,...,xk,xk)(t) =μ

k

(x,...,xk)(t), for eachx, . . . ,xkEandt∈R,

then{μk} is called a dual random multi-normed and{(Ek,μk,),kN}is called a dual

random multi-normed space.

2 Approximation of the multiplicatives

We apply fixed point theory [] to get an approximation for multiplicatives. A metricd on non-empty setϒwith range [,∞] is called a generalized metric.

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Lemma .([, ]) Let k∈N,and let E and F be linear spaces such that(Fk,μk,)is

a complete random multi-normed space.Let there exist≤M< ,λ> ,and a function ψ:Ek−→[,)such that

ψ(λx, . . . ,λxk)≤λMψ(x, . . . ,xk) (x, . . . ,xkE). (.)

We setϒ:={η:E−→F:η() = },and define d:ϒ×ϒon[,∞]by

d(η,ζ)

=inf

c>  :μ(η(x)–ζ(x),...,η(xk)–ζ(xk))(ct)≥

t t+ψ(x, . . . ,xk)

,x, . . . ,xkE

.

Then(ϒ,d)is a complete generalized metric space,and the mapping J:ϒ−→ϒdefined by(Jg)(x) :=g(λλx) (x∈ϒ)is a strictly contractive mapping.

Theorem . Let E be a linear space and let((Fn,μn,∗) :n∈N)be a complete random multi-normed space.Let k∈Nand let there exist≤M< and a functionϕ:Ek−→

[,∞)satisfying

ϕ(x, y, . . . , xk, yk)≤Mϕ(x,y, . . . ,xk,yk) (.)

for all x,y, . . . ,xk,ykE.Suppose that f :E−→F is a mapping with f() = and

μk(f(λx

+λy)–λf(x)–λf(y),...,f(λxk+λyk)–λf(xk)–λf(yk))(t)

t

t+ϕ(x,y, . . . ,xk,yk)

, (.)

μk(f(x

y)–f(x)f(y),...,f(xkyk)–f(xk)f(yk))(t)≥

t

t+ϕ(x,y, . . . ,xk,yk)

, (.)

for allλ∈T:={λ∈C:|λ|= }and x,y, . . . ,xk,ykE,t> .

Then

H(x) := lim

n→∞ nf

xn

(.)

exists for any x, . . . ,xkE and defines a random homomorphism H:E−→F such that

μ(f(x)–H(x),...,f(xk)–H(xk))(t)≥

( –M)t

( –M)t+Mψ(x, . . . ,xk)

, (.)

ψ(x, . . . ,xk) =ϕ

x ,

x , . . . ,

xk

, xk

, (.)

for all x, . . . ,xkE and t> .

Proof Letx=x

, . . . ,xk=

xk

,y=

y

, . . . ,yk=

yk

 in (.). We get

ϕ(x,y, . . . ,xk,yk)≤Mϕ

x ,

y , . . . ,

xk

, yk

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sincef is odd,f() = . Soμf()(t) = . Lettingλ=  andy=x, we get

μk(f(x)–f(x),...,f(x

k)–f(xk))(t)≥

t

t+ϕ(x,x, . . . ,xk,xk)

(.)

for allx,y, . . . ,xk,ykE. Consider the following set:

s=:{g:E−→F}

and introduce the generalized metric ons:

d(g,h)

=inf

ν∈R+:μk(g(x)–h(x),...,g(x

k)–h(xk))(νt)

t t+ϕ(x, . . . ,xk)

,x, . . . ,xkE,t> 

,

where, as usual,infφ= +∞. It is easy to show (s,d) is complete. Now, we consider the linear mappingJ:s−→ssuch that

Jg(x):= g

x

for allxE. Letg,hsbe given such thatd(g,h) =ε. Then we have

μk(g(x)–h(x),...,g(x

k)–h(xk))(εt)

t

t+ϕ(x,x, . . . ,xk,xk)

,

for allx, . . . ,xkEand allt>  and hence we have

μk(Jg(x

)–Jh(x),...,Jg(xk)–Jh(xk))(Mεt) =μ

k

g(x )–h(

x

),...,g(xk)–h(xk))

(Mεt)

=μk

g(x )–h(

x

),...,g(xk)–h(xk))

M  εt

M

t

M

 +ϕ(

x

,

x

, . . . ,

xk

,

xk

)

M

t

M

 +

M

ϕ(x,x, . . . ,xk,xk)

= t

t+ϕ(x,x, . . . ,xk,xk)

for allx, . . . ,xkEandt> . Thend(g,h) =εimplies thatd(Jg,Jh)Mε. This means

that

d(Jg,Jh)Mε

for allg,hs. It follows that

μ(f(x

)–f(x),...,f(xk)–f(xk))

Mt

t

t+ϕ(x,x, . . . ,xk,xk)

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Now, there exists a mappingH:E−→Fsatisfying the following: () His a fixed point ofJ,i.e.,

H

x

= 

H(x) (.)

for allxE. Sincef :E−→Eis odd,H:E−→Fis an odd mapping. The mappingH is a unique fixed point ofJin the set

M=gs:d(f,g) <∞ .

This implies thatHis a unique mapping satisfying (.) such that there exists a ν∈(,∞)satisfying

μk(f(x

)–H(x),...,f(xk)–H(xk))(νt)

t t+ϕ(x, . . . ,xk)

for allx, . . . ,xkE,

() d(Jnf,H)→asn→ ∞. This implies that

lim

n→∞

nf

xn

=H(x)

for allxE, () d(f,H)–M

d(f,Jf), which implies

d(f,H)M  – M

.

Putλ=  in (.). Then

μk (n(f(x

n+ yn)–f(

xn)–f(

y

n)),...,n(f(xkn+ykn)–f(xkn)–f(ykn)))

(t)

t

n

t

n+

Mn

(x,y, . . . ,xk,yk)

for allx, . . . ,xk,y, . . . ,ykE,t>  andn≥. Since

lim

n→∞

t

n

t

n +

Mn

(x,y, . . . ,xk,yk)

= 

for allx, . . . ,xk,y, . . . ,ykE,t> . It follows that

μk(H(x+y)–H(x)–H(y),...,H(x

k+yk)–H(xk)–H(yk))(t) = 

for allx, . . . ,xk,y, . . . ,ykE,t> . So mappingH:E−→Fis Cauchy additive.

Lety=x, . . . ,yk=xkin (.). Then we have

μkn(f(βx

n)–f(βxn),...,f(

βxk

n)–f(

βxk

n))

ntt

t+ϕ(x

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for allλ,β∈T,λ=β,x, . . . ,xkE,t>  andn≥. So we have

μkn(f(βx

n)–f(

βx n),...,f(

βxkn)–f(

βxkn))

(t)≥

t

n

t

n+

Mn

(x,x, . . . ,xk,xk)

for allβ∈T,x, . . . ,xkE,t>  andn≥. We have

lim

n→∞

t

n

t

n +

Mn

(x,x, . . . ,xk,xk)

= 

for allx, . . . ,xkE,t> , and

μk(H(βx

)–βH(x),...,H(βxk)–βH(xk))(t) = 

for allβ∈T,x, . . . ,xkE,t> . Thus, the additive mappingH:E−→FisR-linear. From

(.), we have

μk (nf(x

nyn)–nf(xn)nf(yn),...,nf(xknykn)–nf(xkn)nf(ykn))

nt

t

t+ϕ(x

n, . . . ,xkn,yn, . . . ,ykn)

for allx, . . . ,xkE,t>  andn≥.

Then we have

μk (nf(x

n yn)–nf(

xn)nf(

y

n),...,nf(xknykn)–nf(xkn)nf(ykn))

nt

t

n

t

n+

Mn

tn ϕ(x, . . . ,xk,y, . . . ,yk)

for allx, . . . ,xkE,t>  andn≥.

Since

lim

n→∞

t

n

t

n+

Mn

tnϕ(x, . . . ,xk,y, . . . ,yk)

= 

for allx, . . . ,xkE,t> , we have

μk(H(x

y)–H(x)H(y),...,H(xkyk)–H(xk)H(yk))(t) = 

for allx, . . . ,xk,y, . . . ,ykE,t> . Thus, the mappingH:E−→Fis multiplicative.

There-fore, there exists a unique random homomorphismH:E−→Fsatisfying (.), and this

completes the proof.

3 Approximation in dual random multi-normed space

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Lemma . Let {(Ek,μk,),k N} be a dual random multi-normed space, k,n N,

x,x, . . . ,xk,xk+, . . . ,xk+n∈Eandλ, . . . ,λkbe real numbers of absolute value.Then we

have: (i) μk(λ

x,...,λkxk)(t) =μ

k

(x,...,xk)(t),

(ii) μk

(x,...,xk)(t)≥μ

k+

(x,...,xk,xk+)(t),

(iii) μk(x+n

,...,xk,xk+,...,xk+n)(t)≥TM(μ

k

(x,...,xk)(αt),μ

n

(xk+,...,xk+n)(βt)),whereα,β≥and

α+β= ,

(iv) mini∈Nkμxi(t)≥μ

k

(x,...,xk)(t)≥mini∈Nkμxi(αit),

whereα, . . . ,αk≥and

k

i=αi= .In particular,we have

μk(x,...,x

k)(t)≥miniNkμkxi(t).

Theorem . Let E be a linear space,and{(Ek,μk,),kN}be a random multi space.Let

α∈(, )and f :E−→F is a mapping satisfying f() = and

μk (f(x+y

 )– f(x)

 – f(y)

 ,...,f(xk +yk  )–

f(xk)  –

f(yk)  )

t s

≥ –α

t, (.)

where x, . . . ,xk,y, . . . ,ykE and t,s∈Nwith the greatest common divisor(t,s) = .

Then there exists a unique additive mapping T:E−→F such that

μk(f(x

)–T(x),...,f(xk)–T(xk))

t s

≥ –α

t (.)

for all x, . . . ,xkE and t,s∈Nwith(t,s) = .

Proof Replacingx, . . . ,xk andy, . . . ,yk by x, . . . , xk and , . . . ,  in (.), respectively,

yields

μk(f(x)–f(x),...,f(x

k)–f(xk))

t s

≥ –α

t. (.)

Replacingx, . . . ,xk,t,s by x, . . . , xk, t, s, respectively, in (.) and repeating this

process for n-time (n∈N), it follows that

μk (f(n–x)

n–f(nx)

n ,..., f(n–xk)

n–f(nxk)

n )

tn–s

≥ – α

n–t (.)

forn,m∈Nwithn>m. Using (.) and (RN) we get

μk (f(mx)

mf(nx)

n ,..., f(mxk)

mf(nxk)

n ) n–

i=m

–it s

≥ – αmt.

Then

μk

(f(mxm)–f(nxn),..., f(mxk)

mf(nxk)

n )

t s

≥ – α

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forxE. Then, replacingx, . . . ,xkbyx, x, . . . , k–xin (.), we have

μk

(f(mx)mf(nx)n ,...,f( m+k–x)

m+k–f(n+k–x)

n+k– )

t s

≥ – αmt

≥ – α

m. (.)

Letε>  be given. Then there existsn∈Nsuch that αn <ε. Now we substitutem,n

withn,n+p(p∈N), respectively, in (.), for eachnn, and we get

μk (f(nx)nf(

n+px)n+p ,...,

f(n+k–x)n+k–

f(n+p+k–x)n+p+k– )

t s

≥ – αnt

>  –ε.

By Lemma ., we have

μf(nx) nf(

n+px)

n+p

t s

>  –ε (.)

for alln>nandp∈N. The density of rational numbers inRis useful in checking cor-rectness of (.) with positive real numberrinstead of ts. Then we have

μf(nx) nf(

n+px)

n+p

(r) >  –ε

for eachxE,r∈R+,nnandp∈N. Then{f(nnx)}is a Cauchy sequence, so it is

conver-gent in the random multi-Banach space{(Ek,μk,),kN}. SettingT(x) :=lim n→∞f(

nx)

n

and applying again Lemma ., for eachr> , we have

μk (f(nx)

nT(x),..., f(nxk)

nT(xk))

(r)≥min

i∈Nkμf(nxin )–T(xi)

r k

,

and

lim

n→∞ f(nx

k)

n =T(xk).

We putm=  in (.), and we get

μk

(f(x)–f(nxn),...,f(xk)–f(nxnk))

t s

≥ –α

t. (.)

Then

μkf(x

)–T(x),...,f(xk)–T(xk)

t s

TM

μk

(f(x)–f(nxn),...,f(xk)– f(nxk)

n )

t s

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μk (f(nx)

nT(x),..., f(nxk)

nT(xk))

t s

≥ –α

t (.)

by (.) and whenn→ ∞, which implies that (.).

Now, we show thatT is additive. Letx,yEand replacex, . . . ,xk by nx,y, . . . ,ykby

ny, andtby ntin (.). We get

μk (f(n x+y

 )– f(nx)

 – f(ny)

 ,...,f(n x +y  )– f(nx)  – f(ny)  )

nt

s

≥ – αnt.

Using (NF), we conclude that

μfn(x+y )n

 

f(nx) n

  f(ny) n t s

≥ – α

nt. (.)

On the other hand, we obtain that

μT(x+y

 )–T(x)–T(y)

t s

TM

μ

T(x+y )–f(

n(x+y  )) n t s , μT(x)  – f(nx) n t s , μT(y)  – f(ny) n t s ,

μf(n(x+y  )) n

 

f(n) nx–

f(ny) n t s

≥ – α

n (.)

for eachx,yE,t,s∈Nwith (t,s) = . Utilizing again the density ofQinR, we find that (.) remains true if st is substituted with a positive real numberr.

Consequently,

μT(x+y

 )–T(x)–T(y)(r)≥ –

αn

for eachx,yEandr∈R. Lettingn→ ∞reveals thatTcomplies with Jensen, and using the fact thatT() = , we conclude thatT is additive [, Theorem ].

It remains to show the uniqueness ofT. Suppose thatTis another additive mapping satisfying (.). Then, for eacht,s∈N, sufficiently largeninNandxE,

μT(x)–T(x)

t s

=μT(nx) n

T(nx)n t s

TM

μT(nx)–f(nx)

n–t s

,μT(nx)–f(nx)

n–t s

≥ – αn–t

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This inequality holds for eachr∈R+instead oft

s, too. Therefore, for eachr∈R+,n∈N,

μT(x)–T(x)(r)≥ –n–α , lettingn→ ∞, it follows thatT=T.

4 Conclusion

In this paper, we consider multi-Banach spaces, approximate by multiplicatives, and pro-vide some controlled mappings, which are stable by control functions.

Acknowledgements

The authors are grateful to the reviewer(s) for their valuable comments and suggestions.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Author details

1Department of Mathematics, Texas A & M University - Kingsville, Kingsville, TX 78363, USA.2Department of Mathematics,

Iran University of Science and Technology, Tehran, Iran.

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Received: 15 May 2017 Accepted: 23 August 2017 References

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