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Agarwal et al. Journal of Inequalities and Applications (2017) 2017:204 DOI 10.1186/s13660-017-1478-9

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Approximation of the multiplicatives on random multi-normed space Ravi P Agarwal1 , Reza Saadati2* and Ali Salamati2 *

Correspondence: [email protected] Department of Mathematics, Iran University of Science and Technology, Tehran, Iran Full list of author information is available at the end of the article 2

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 applications, one can see [–]. Definition . Let (E, μ, ∗) be a random normed space. ∗ is a continuous t-norm. A multirandom norm on {Ek , k ∈ N} is sequence {Nk } such that Nk is a random norm on Ek (k ∈ N), μx (t) = μx (t) for each x ∈ E and t ∈ R and the following axioms are satisfied for each k ∈ N with k ≥ : (NF) μkAσ (x) (t) = μkx (t), for each σ ∈ σk , x ∈ Ek , t ∈ R, (NF) μkMα (x) (t) ≥ μkmaxi∈N |αi |x (t), for each α = (α , . . . , αk ) ∈ Rk , x ∈ Ek , t ∈ R, k

k (NF) μk+ (x ,...,xk ,) (t) = μ(x ,...,xk ) (t), for each x , . . . , xk ∈ E and t ∈ R,

k (NF) μk+ (x ,...,xk ,xk ) (t) = μ(x ,...,xk ) (t), for each x , . . . , xk ∈ E and t ∈ R. In this case {(Ek , μk , ∗), k ∈ N} is called a random multi-normed space. Moreover, if axiom (NF) is replaced by the following axiom: k (DF) μk+ (x ,...,xk ,xk ) (t) = μ(x ,...,xk ) (t), for each x , . . . , xk ∈ E and t ∈ R, k then {μ } 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 metric d on non-empty set ϒ with range [, ∞] is called a generalized metric. © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Lemma . ([, ]) Let k ∈ N, and let E and F be linear spaces such that (F k , 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 set ϒ := {η : E −→ F : η() = }, and define d : ϒ × ϒ on [, ∞] by d(η, ζ )  = inf c >  : μ(η(x )–ζ (x ),...,η(xk )–ζ (xk )) (ct) ≥

 t , x , . . . , xk ∈ E . t + ψ(x , . . . , xk )

Then (ϒ, d) is a complete generalized metric space, and the mapping J : ϒ −→ ϒ defined (x ∈ ϒ) is a strictly contractive mapping. by (Jg)(x) := g(λx) λ Theorem . Let E be a linear space and let ((F n , μn , ∗) : n ∈ N) be a complete random multi-normed space. Let k ∈ N and let there exist  ≤ M <  and a function ϕ : 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 (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 (xk yk )–f (xk )f (yk )) (t) ≥

t , t + ϕ(x , y , . . . , xk , yk )

(.)

for all λ ∈ T := {λ ∈ C : |λ| = } and x , y , . . . , xk , yk ∈ E, t > . Then 

x H(x) := lim  f n n→∞ 



n

(.)

exists for any x , . . . , xk ∈ E and defines a random homomorphism H : E −→ F such that ( – M )t , ( – M )t + M ψ(x , . . . , xk )   xk xk x x ψ(x , . . . , xk ) = ϕ , ,..., , ,    

μ(f (x )–H(x ),...,f (xk )–H(xk )) (t) ≥

(.) (.)

for all x , . . . , xk ∈ E and t > . Proof Let x =

x , . . . , xk 

=

xk , y 

=

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

y , . . . , yk 

=

yk 

in (.). We get

 xk yk x y , ,..., , ,    

(.)

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since f is odd, f () = . So μf () ( t ) = . Letting λ =  and y = x, we get μk(f (x )–f (x ),...,f (xk )–f (xk )) (t) ≥

t t + ϕ(x , x , . . . , xk , xk )

(.)

for all x , y , . . . , xk , yk ∈ E. Consider the following set: s =: {g : E −→ F} and introduce the generalized metric on s: d(g, h)  = inf ν ∈ R+ : μk(g(x )–h(x ),...,g(xk )–h(xk )) (νt) ≥

 t , x , . . . , xk ∈ E, t >  , t + ϕ(x , . . . , xk )

where, as usual, inf φ = +∞. It is easy to show (s, d) is complete. Now, we consider the linear mapping J : s −→ s such that     x J g(x) := g  for all x ∈ E. Let g, h ∈ s be given such that d(g, h) = ε. Then we have μk(g(x )–h(x ),...,g(xk )–h(xk )) (εt) ≥

t , t + ϕ(x , x , . . . , xk , xk )

for all x , . . . , xk ∈ E and all t >  and hence we have μk(Jg(x )–Jh(x ),...,Jg(xk )–Jh(xk )) (M εt) = μkg( x )–h( x ),...,g( xk )–h( xk )) (M εt) 









= μkg( x )–h( x ),...,g( xk )–h( xk ))     ≥ ≥ =

M εt 



M t  M x x + ϕ(  ,  , . . . , xk , xk ) 

M 

+

M t  M ϕ(x , x , . . . , xk , xk ) 

t t + ϕ(x , x , . . . , xk , xk )

for all x , . . . , xk ∈ E and t > . Then d(g, h) = ε implies that d(Jg, Jh) ≤ M ε. This means that d(Jg, Jh) ≤ M ε for all g, h ∈ s. It follows that  μ(f (x )–f ( x ),...,f (x 

xk k )–f (  ))

 t M t ≥  t + ϕ(x , x , . . . , xk , xk )

for all x , . . . , xk ∈ E and t > . So d(f , Jf ) ≤

M . 

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Now, there exists a mapping H : E −→ F satisfying the following: () H is a fixed point of J, i.e., H

   x = H(x)  

(.)

for all x ∈ E. Since f : E −→ E is odd, H : E −→ F is an odd mapping. The mapping H is a unique fixed point of J in the set  M = g ∈ s : d(f , g) < ∞ . This implies that H is 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 all x , . . . , xk ∈ E, () d(J n f , H) →  as n → ∞. This implies that 

x n

lim n f

n→∞

 = H(x)

for all x ∈ E,  () d(f , H) ≤ –M d(f , Jf ), which implies  d(f , H) ≤

M .  – M

Put λ =  in (.). Then μk(n (f ( x + y )–f ( x )–f ( y )),...,n (f ( xk + yk )–f ( xk )–f ( yk ))) (t) n

≥

n

n

n

n

n

n

n

t n t n

+

Mn n

ϕ(x , y , . . . , xk , yk )

for all x , . . . , xk , y , . . . , yk ∈ E, t >  and n ≥ . Since t n

lim

n→∞ t n

+

Mn ϕ(x , y , . . . , xk , yk ) n

=

for all x , . . . , xk , y , . . . , yk ∈ E, t > . It follows that μk(H(x +y )–H(x )–H(y ),...,H(xk +yk )–H(xk )–H(yk )) (t) =  for all x , . . . , xk , y , . . . , yk ∈ E, t > . So mapping H : E −→ F is Cauchy additive. Let y = x , . . . , yk = xk in (.). Then we have μk n

 βx

βx

βx

βx

 (f ( n )–f ( n ),...,f ( nk )–f ( nk ))

 n t ≥

t t

+ ϕ( xn , xn , . . . , xnk , xnk )

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for all λ, β ∈ T, λ = β , x , . . . , xk ∈ E, t >  and n ≥ . So we have μk n

βx

βx

βx

βx

 (f ( n )–f ( n ),...,f ( nk )–f ( nk ))

(t) ≥

t n t n

+

Mn ϕ(x , x , . . . , xk , xk ) n

for all β ∈ T, x , . . . , xk ∈ E, t >  and n ≥ . We have t n

lim

n→∞ t n

+

Mn n

=

ϕ(x , x , . . . , xk , xk )

for all x , . . . , xk ∈ E, t > , and μk(H(βx )–βH(x ),...,H(βxk )–βH(xk )) (t) =  for all β ∈ T, x , . . . , xk ∈ E, t > . Thus, the additive mapping H : E −→ F is R-linear. From (.), we have 

μk(n f ( x

x y x y y x y )–n f ( n )n f ( n ),...,n f ( nk kn )–n f ( nk )n f ( kn )) n n

≥

n t



t t

+ ϕ( xn , . . . , xnk , yn , . . . , ykn )

for all x , . . . , xk ∈ E, t >  and n ≥ . Then we have 

μk(n f ( x

x y x y y x y )–n f ( n )n f ( n ),...,n f ( nk kn )–n f ( nk )n f ( kn )) n n

≥

n t



t n t n

+

Mn tn

ϕ(x , . . . , xk , y , . . . , yk )

for all x , . . . , xk ∈ E, t >  and n ≥ . Since t n

lim

n→∞ t n

+

Mn tn

=

ϕ(x , . . . , xk , y , . . . , yk )

for all x , . . . , xk ∈ E, t > , we have μk(H(x y )–H(x )H(y ),...,H(xk yk )–H(xk )H(yk )) (t) =  for all x , . . . , xk , y , . . . , yk ∈ E, t > . Thus, the mapping H : E −→ F is multiplicative. Therefore, there exists a unique random homomorphism H : E −→ F satisfying (.), and this completes the proof. 

3 Approximation in dual random multi-normed space The following lemma is an immediate result of the definition of 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 ∈ E and λ , . . . , λk be real numbers of absolute value . Then we have: (i) μk(λ x ,...,λk xk ) (t) = μk(x ,...,xk ) (t), (ii) μk(x ,...,xk ) (t) ≥ μk+ (x ,...,xk ,xk+ ) (t), k+n (iii) μ(x ,...,xk ,xk+ ,...,xk+n ) (t) ≥ TM (μk(x ,...,xk ) (αt), μn(xk+ ,...,xk+n ) (βt)), where α, β ≥  and α + β = , (iv) mini∈Nk μxi (t) ≥ μk(x ,...,xk ) (t) ≥ mini∈Nk μxi (αi t),

where α , . . . , αk ≥  and ki= αi = . In particular, we have μk(x ,...,xk ) (t) ≥ min μkxi (t). i∈Nk

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   α t ≥– , x +y f (x ) f (y ) x +y f (x ) f (y ) (f (    )–  –  ,...,f ( k  k )– k – k ) s t

μk

(.)

where x , . . . , xk , y , . . . , yk ∈ E and t, s ∈ N with 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 ∈ N with (t, s) = . Proof Replacing x , . . . , xk and y , . . . , yk by x , . . . , xk and , . . . ,  in (.), respectively, yields  μk(f (x )–f (x ),...,f (xk )–f (xk ))

t s

 ≥–

α . t

(.)

Replacing x , . . . , xk , t, s by x , . . . , xk , t, s, respectively, in (.) and repeating this process for n-time (n ∈ N), it follows that  μkf (n– x (

n– x ) f (n x ) n k – k )  ) – f ( x ) ,..., f ( n n n– n–



t n– s

≥–

α n– t

(.)

for n, m ∈ N with n > m. Using (.) and (RN) we get

k

μ

(

 n– f (m x ) f (n x ) f (m x ) f (n x ) – n ,..., m k – n k ) m

i=m



–i t

s

≥–

α . m t

Then μkf (m x ) (

m

f (m x ) f (n x ) f (n x ) – n  ,..., m k – n k )

  t α ≥– m s  t

(.)

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for x ∈ E. Then, replacing x , . . . , xk by x, x, . . . , k– x in (.), we have   t μ f (m x) f (n x) f (m+k– x) f (n+k– x) ( m – n ,..., m+k– – n+k– ) s   k

α m t α ≥– m. 

≥–

(.)

Let ε >  be given. Then there exists n ∈ N such that αn < ε. Now we substitute m, n with n, n + p (p ∈ N), respectively, in (.), for each n ≥ n , and we get μkf (n x) ( n

  α t ≥– n f (n+p x) f (n+k– x) f (n+p+k– x)  t – n+p ,..., n+k– – n+p+k– ) s  



>  – ε. By Lemma ., we have μ

f (n x) f (n+p x) – n+p n 

  t >–ε s

(.)

for all n > n and p ∈ N. The density of rational numbers in R is useful in checking correctness of (.) with positive real number r instead of ts . Then we have μ f (n x) – f (n+p x) (r) >  – ε n

n+p

n

for each x ∈ E, r ∈ R+ , n ≥ n and p ∈ N. Then { f (n x) } is a Cauchy sequence, so it is convern gent in the random multi-Banach space {(Ek , μk , ∗), k ∈ N}. Setting T(x) := limn→∞ f (n x) and applying again Lemma ., for each r > , we have   r μ f (n x ) , (r) ≥ min μ f (n xi ) –T(x ) f (n x ) i n i∈Nk ( n –T(x ),..., n k –T(xk )) k  k

and f (n xk ) = T(xk ). n→∞ n lim

We put m =  in (.), and we get μk

f (n x ) f (n x ) (f (x )– n  ,...,f (xk )– n k )

  t α ≥– . s t

Then  μkf(x )–T(x ),...,f (xk )–T(xk )  ≥ TM μk

t s



f (n x ) f (n x ) (f (x )– n  ,...,f (xk )– n k )

  t , s

(.)

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  t μ f (n x ) f (n x ) ( n –T(x ),..., n k –T(xk )) s α ≥– t k

(.)

by (.) and when n → ∞, which implies that (.). Now, we show that T is additive. Let x, y ∈ E and replace x , . . . , xk by n x, y , . . . , yk by n y, and t by n t in (.). We get  μk n x+y f (n x) f (n y) x+y f (n x) f (n y) (f (  )–  –  ,...,f (n  )–  –  )

n t s

 ≥–

α . n t

Using (NF), we conclude that   t α μ f n ( x+y )  f (n x)  f (n y) ≥– n .  – s  t  n –  n n

(.)

On the other hand, we obtain that  μ

x+y T(  )–  T(x)–  T(y)

t s



 ≥ TM μ

n x+y x+y f ( (  )) T(  )– n

  t ,   n s   t , μ T(y) –  f (n y)   n s

  t , s

μ T(x) –  f (n x)

μ

x+y f (n (  ))  f (n )  f (n y) –  n x –  n n

≥–

  t s

α n

(.)

for each x, y ∈ E, t, s ∈ N with (t, s) = . Utilizing again the density of Q in R, we find that (.) remains true if ts is substituted with a positive real number r. Consequently, μT( x+y )–  T(x)–  T(y) (r) ≥  – 





α n

for each x, y ∈ E and r ∈ R. Letting n → ∞ reveals that T complies with Jensen, and using the fact that T() = , we conclude that T is additive [, Theorem ]. It remains to show the uniqueness of T. Suppose that T  is another additive mapping satisfying (.). Then, for each t, s ∈ N, sufficiently large n in N and x ∈ E, μT  (x)–T(x)

    t t = μ T  (n x) – T(n x) n n s s     n–   n–   t  t , μT(n x)–f (n x) ≥ TM μT  (n x)–f (n x) s s α ≥  – n–  t α ≥  – n– . 

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This inequality holds for each r ∈ R+ instead of ts , too. Therefore, for each r ∈ R+ , n ∈ N, α μT  (x)–T(x) (r) ≥  – n– , letting n → ∞, it follows that T = T  . 

4 Conclusion In this paper, we consider multi-Banach spaces, approximate by multiplicatives, and provide 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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