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: [email protected]} 2_{Department 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_,_k_∈_N_}_{is sequence}_{_N

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

μ_x(t) =μx(t) for eachx∈Eandt∈Rand the following axioms are satisﬁed for eachk∈N

withk≥:

(NF) μk_A_σ(_x₎(t) =μk

x(t), for eachσ∈σk,x∈Ek,t∈R,

(NF) μk_M_α(_x₎(t)≥μk_max

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

k_,_x_∈_Ek_,_t_∈_R_,

(NF) μk₍_x+

,...,xk,)(t) =μ

k

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

(NF) μk₍_x+

,...,xk,xk)(t) =μ

k

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

then{μk} is called a dual random multi-normed and{(Ek_,_μk_,_∗_),_k_∈_N_}_{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, . . . ,xk∈E). (.)

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, . . . ,xk∈E

.

Then(ϒ,d)is a complete generalized metric space,and the mapping J:ϒ−→ϒdeﬁned by(Jg)(x) :=g(λ_λx) (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:Ek_−→

[,∞)satisfying

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

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

μk₍_f₍_λ_x

+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,yk∈E,t> .

Then

H(x) := lim

n→∞ n_f

x n

(.)

exists for any x, . . . ,xk∈E 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, . . . ,xk∈E 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,yk∈E. 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, . . . ,xk∈E,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 allx∈E. Letg,h∈sbe 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, . . . ,xk∈Eand 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, . . . ,xk∈Eandt> . Thend(g,h) =εimplies thatd(Jg,Jh)≤Mε. This means

that

d(Jg,Jh)≤Mε

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

M=g∈s: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, . . . ,xk∈E,

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

lim

n→∞

n_f

x n

=H(x)

for allx∈E, () d(f,H)≤_–_M

d(f,Jf), which implies

d(f,H)≤ M  – M

.

Putλ=  in (.). Then

μk (n₍_f₍x

n+ y_ n)–f(

x_ n)–f(

y_

n)),...,n(f(xkn+ykn)–f(xkn)–f(ykn)))

(t)

≥

t

n

t

n+

Mn_

nϕ(x,y, . . . ,xk,yk)

for allx, . . . ,xk,y, . . . ,yk∈E,t>  andn≥. Since

lim

n→∞

t

n

t

n +

Mn

nϕ(x,y, . . . ,xk,yk)

= 

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

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

μk n₍_f₍βx

n)–f(βxn),...,f(

β_xk

n)–f(

β_xk

n))

nt≥ t

t+ϕ(x

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for allλ,β∈T,λ=β_,x, . . . ,xk∈E,t>  andn≥. So we have

μk n₍_f₍βx

n)–f(

βx n),...,f(

βxk n)–f(

βxk n))

(t)≥

t

n

t

n+

Mn 

nϕ(x,x, . . . ,xk,xk)

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

lim

n→∞

t

n

t

n +

M_n

nϕ(x,x, . . . ,xk,xk)

= 

for allx, . . . ,xk∈E,t> , and

μk₍_H₍_β_x

)–βH(x),...,H(βxk)–βH(xk))(t) = 

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

(.), we have

μk (n_f₍x

nyn)–nf(xn)nf(yn),...,nf(xknykn)–nf(xkn)nf(ykn))

nt

≥ t

t+ϕ(x

n, . . . ,x_kn,_yn, . . . ,_ykn)

for allx, . . . ,xk∈E,t>  andn≥.

Then we have

μk (n_f₍x

n y_ n)–nf(

x_ n)nf(

y_

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

nt

≥

t

n

t

n+

Mn_

tn ϕ(x, . . . ,xk,y, . . . ,yk)

for allx, . . . ,xk∈E,t>  andn≥.

Since

lim

n→∞

t

n

t

n+

Mn_

tnϕ(x, . . . ,xk,y, . . . ,yk)

= 

for allx, . . . ,xk∈E,t> , we have

μk₍_H₍_x

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

for allx, . . . ,xk,y, . . . ,yk∈E,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)≥min_i_∈_Nkμkxi(t).

Theorem . Let E be a linear space,and{(Ek_,_μk_,_∗_),_k_∈_N_}_{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, . . . ,yk∈E 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, . . . ,xk∈E 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 )

t n–_s

≥ – α

n–_t (.)

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

μk (f(mx)

m – f(nx)

n ,..., f(mx_k)

m – f(nx_k)

n ) _n_–

i=m

–it s

≥ – α m_t.

Then

μk

(f(mx_m)–f(nxn),..., f(mxk)

m – f(nxk)

n )

t s

≥ – α

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

μk

(f(mx)_m –f(nx)_n ,...,f( m+k–x)

m+k– – f(n+k–x)

n+k– )

t s

≥ – α m_t

≥ – α

m. (.)

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

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

μk (f(nx)_n –f(

n+_px) n+p ,...,

f(n+k–x) n+k– –

f(n+p+k–x) n+p+k– )

t s

≥ – α n_t

>  –ε.

By Lemma ., we have

μf(nx) n –f(

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 t_s. Then we have

μf(nx) n –f(

n+_px)

n+p

(r) >  –ε

for eachx∈E,r∈R+,n≥nandp∈N. Then{f(_nnx)}is a Cauchy sequence, so it is

conver-gent in the random multi-Banach space{(Ek_,_μk_,_∗_),_k_∈_N_}_{. Setting}_T_{(x) :=}_lim n→∞f(

n_x₎

n

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

μk (f(nx)

n –T(x),..., f(nx_k)

n –T(xk))

(r)≥min

i∈Nkμf(nxin )–T(xi)

r k

,

and

lim

n→∞ f(n_x

k)

n =T(xk).

We putm=  in (.), and we get

μk

(f(x)–f(nx_n),...,f(xk)–f(nx_nk))

t s

≥ –α

t. (.)

Then

μk_f₍_x

)–T(x),...,f(xk)–T(xk)

t s

≥TM

μk

(f(x)–f(nx_n),...,f(xk)– f(nxk)

n )

t s

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μk (f(nx)

n –T(x),..., f(nx_k)

n –T(xk))

t s

≥ –α

t (.)

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

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

n_{y, and}_t_{by }n_t_{in (.). We get}

μk (f(n x+y

 )– f(nx)

 – f(ny)

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

n_t

s

≥ – α n_t.

Using (NF), we conclude that

μfn(x+y_{ )} n –

 

f(nx) n –

  f(ny) n t s

≥ – α

n_t. (.)

On the other hand, we obtain that

μ_T₍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 ,

μ_f_(n₍x+y  )) n –

 

f(n) nx–

f(ny) n t s

≥ – α

n (.)

for eachx,y∈E,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,

μ_T₍x+y

 )–T(x)–T(y)(r)≥ –

α n

for eachx,y∈Eandr∈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 largeninNandx∈E,

μT(x)–T(x)

t s

=μT(nx) n –

T(nx) n t s

≥TM

μT(n_x_)–_f_(n_x₎

n–t s

,μT(n_x_)–_f_(n_x₎

n–t s

≥ – α n–_t

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This inequality holds for eachr∈R+_{instead of}t

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

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