Skip to main content

Ideal convergence of double sequences in random 2-normed spaces

Abstract

Quite recently, Alotaibi and Mohiuddine (Adv. Differ. Equ. 2012:39, 2012) studied the idea of a random 2-normed space to determine some stability results concerning the cubic functional equation. In this paper, we define and study the concepts of I-convergence and I -convergence for double sequences in random 2-normed spaces and establish the relationship between these types of convergence, i.e., we show that I -convergence implies I-convergence in random 2-normed spaces. Furthermore, we have also demonstrated through an example that, in general, I-convergence does not imply I -convergence in random 2-normed spaces.

MSC:40A05, 46A70.

1 Introduction

Menger [1] generalized the metric axioms by associating a distribution function with each pair of points of a set. This system, called a probabilistic metric space, originally a statistical metric space, has been developed extensively by Schweizer and Sklar [2]. The idea of Menger was to use distribution functions instead of nonnegative real numbers. An important family of probabilistic metric spaces are probabilistic normed spaces. Probabilistic normed spaces are real linear spaces in which the norm of each vector is an appropriate probability distribution function rather than a number. Such spaces were first introduced by S̆erstnev in 1963 [3]. In [4], Alsina et al. gave a new definition of PN spaces that includes S̆erstnev’s and leads naturally to the identification of the principal class of PN spaces, the Menger spaces. Recently, the concept of probabilistic normed spaces was extended to random/probabilistic 2-normed spaces by Golet [5] using the concept of 2-norm of Gähler [6].

The concept of statistical convergence for sequences of real numbers was introduced by Fast [7] in 1951, and since then several generalizations and applications of this notion have been investigated by various authors (see [821]). The notion of statistical convergence is a very useful functional tool for studying the convergence problems of numerical sequences/matrices (double sequences) through the concept of density. The concept of I-convergence, which is a generalization of statistical convergence, was introduced by Kastyrko, Salat and Wilczynski [22] by using the ideal I of subsets of the set of natural numbers N and further studied in [2327] and references therein. Recently, Şahiner et al.[28], and Gürdal and Acik [29] studied ideal convergence and I-Cauchy sequences respectively in 2-normed spaces, and also Gürdal and Pehlivan [30] studied statistical convergence in a 2-Banach space. Most recently, Alotaibi and Mohiuddine [31] determined the stability of a cubic functional equation in a random 2-normed space.

In this paper, we study the concept of I-convergence and I -convergence in a more general setting, i.e., in a random 2-normed space. We discuss the relationship between I-convergence and I -convergence and prove some interesting results.

2 Definitions, notations and preliminary results

In this section, we recall some basic definitions and notations which form the background of the present work.

A distribution function is an element of Δ + , where Δ + ={f:R[0,1];f is left-continuous, nondecreasing,f(0)=0 and f(+)=1}, and the subset D + Δ + is the set D + ={f Δ + ; l f(+)=1}. Here l f(+) denotes the left limit of the function f at the point x. The space Δ + is partially ordered by the usual pointwise ordering of functions, i.e., fg if and only if f(x)g(x) for all xR. For any aR, ε a is a distribution function defined by

ε a (x)={ 0 if  x a ; 1 if  x > a .

The set Δ, as well as its subsets, can be partially ordered by the usual pointwise order: in this order, ε 0 is the maximal element in Δ + .

A triangle function is a binary operation on Δ + , namely a function τ: Δ + × Δ + Δ + , that is associative, commutative, nondecreasing, and has ε 0 as unit; that is, for all f,g,h Δ + , we have:

  1. (i)

    τ(τ(f,g),h)=τ(f,τ(g,h)),

  2. (ii)

    τ(f,g)=τ(g,f),

  3. (iii)

    τ(f,g)=τ(g,f) whenever fg,

  4. (iv)

    τ(f, ε 0 )=f.

A t-norm is a binary operation :[0,1]×[0,1][0,1] such that for all a,b,c,d[0,1] we have:

  1. (i)

    is associative and commutative, (ii) is continuous, (iii) a1=a, (iv) abcd whenever ac and bd.

The concept of a 2-normed space was first introduced by Gähler [6].

Let X is a linear space of a dimension d, where 2d<. A 2-norm on X is a function ,:X×XR satisfying the following conditions: for every x,yX, (i) x,y=0 if and only if x and y are linearly dependent; (ii) x,y=y,x; (iii) αx,y=|α|x,y, αR; (iv) x+y,zx,z+y,z. In this case (X,,) is called a 2-normed space.

Example 2.1 Take X= R 2 being equipped with the 2-norm x,y = the area of the parallelogram spanned by the vectors x and y, which may be given explicitly by the formula

x,y=| x 1 y 2 x 2 y 1 |,where x=( x 1 , x 2 ),y=( y 1 , y 2 ).

Recently, Goleţ [5] introduced the notion of a random 2-normed space as follows.

Let X be a linear space of a dimension greater than one, τ a triangle function, and F:X×X Δ + . Then F is called a probabilistic 2-norm on X and (X,F,τ) a probabilistic 2-normed space if the following conditions are satisfied:

  1. (i)

    F x , y (t)= ε 0 (t) if x and y are linearly dependent, where F x , y (t) denotes the value of F x , y at tR,

  2. (ii)

    F x , y ε 0 if x and y are linearly independent,

  3. (iii)

    F x , y = F y , x for every x, y in X,

  4. (iv)

    F α x , y (t)= F x , y ( t | α | ) for every t>0, α0 and x,yX,

  5. (v)

    F x + y , z τ( F x z , F y z ) whenever x,y,zX.

If (v) is replaced by

(v) F x + y , z ( t 1 + t 2 ) F x , z ( t 1 ) F y , z ( t 2 ), for all x,y,zX and t 1 , t 2 R 0 + ,

then triple (X,F,) is called a random 2-normed space (for short, RTN-space).

Example 2.2 Let (X,,) be a 2-normed space with x,z= x 1 z 2 x 2 z 1 , x=( x 1 , x 2 ), z=( z 1 , z 2 ) and ab=ab for a,b[0,1]. For all xX, t>0 and nonzero zX, consider

F x , z (t)={ t t + x , z if  t > 0 , 0 if  t 0 .

Then (X,F,) is a random 2-normed space.

Remark 2.3 Note that every 2-normed space (X,,) can be made a random 2-normed space in a natural way, by setting F x , y (t)= ε 0 (tx,y), for every x,yX, t>0 and ab=min{a,b}, a,b[0,1].

In 1900, Pringsheim [32] introduced the notion of convergence of double sequences as follows: A double sequence x=( x j k ) is said to converge to the limit L in Pringsheim’s sense (shortly, P-convergent to L) if for every ε>0 there exists an integer N such that | x j k L|<ε whenever j,k>N. In this case L is called the P-limit of x.

Let KN×N be a two-dimensional set of positive integers and let K m , n ={(j,k):jm,kn}. Then the two-dimensional analogue of natural density can be defined as follows.

In case the sequence (K(m,n)/mn) has a limit in Pringsheim’s sense, then we say that K has a double natural density and is defined as

P- lim m , n K ( m , n ) m n = δ 2 (K).

For example, let K={( i 2 , j 2 ):i,jN}. Then

δ 2 (K)=P- lim m , n K ( m , n ) m n P- lim m , n m n m n =0,

i.e., the set K has double natural density zero, while the set {(i,2j):i,jN} has double natural density 1 2 .

Statistical convergence for double sequences x=( x j k ) of real numbers was introduced and studied by Mursaleen and Edely [17] and in intuitionistic fuzzy normed spaces by Mursaleen and Mohiuddine [18].

A real double sequence x=( x j k ) is said to be statistically convergent to the number if for each ϵ>0 the set

{ ( j , k ) , j m  and  k n : | x j k | ϵ }

has double natural density zero. In this case we write s t 2 - lim j , k x j k = and denote the set of all statistically convergent double sequences.

If X is a non-empty set, then a family of subsets of X is called an ideal in X if and only if

  1. (a)

    I,

  2. (b)

    A,BI implies ABI,

  3. (c)

    for each AI and BA we have BI,

I is called a nontrivial ideal if XI and P(X) is the power set of X.

Let X be a non-empty set. A non-empty family of sets FP(X) is called a filter on X if and only if

  1. (a)

    F,

  2. (b)

    A,BF implies ABF,

  3. (c)

    for each AF and BA we have BF.

A nontrivial ideal I in X is called an admissible ideal if it is different from P(N) and contains all singletons, i.e., {x}I for each xX.

Let IP(X) be a nontrivial ideal. Then a class F(I)={MX:M=XA, for some AI} is a filter on X, called the filter associated with the ideal I.

An admissible ideal IP(N) is said to satisfy the condition (AP) if for every sequence ( A n ) n N of pairwise disjoint sets from I there are sets B n N, nN such that the symmetric difference A n B n is a finite set for every n and n N B n I.

Let I be a nontrivial ideal of N. A sequence x=( x k ) is said to be I-convergent[22] to LX if, for each ϵ>0, the set {kN:| x k |ϵ}I. In this case we write I-limx=L.

3 Main results

In this section, we study the concept of I-convergence and I -convergence of double sequences in a random 2-normed space. We shall assume throughout this paper that I as a nontrivial ideal in N×N and (X,F,) is a random 2-normed space.

We define:

Definition 3.1 A double sequence x=( x j k ) is convergent in (X,F,) or simply (F)-convergent to if, for every ϵ>0, θ(0,1), there exists a positive integer N such that F x j k , z (ϵ)>1θ whenever j,kN and nonzero zX. In this case we write (F)- lim j , k x j k = and is called the F-limit of x=( x j k ).

Definition 3.2 Let I a be nontrivial ideal of N×N. A double sequence x=( x j k ) is said to be I-convergent in (X,F,) or simply I F -convergent to if for every ϵ>0, t(0,1) and nonzero zX, we have A(ϵ,t)I, where

A(ϵ,t)= { ( j , k ) N × N : F x j k , z ( ϵ ) 1 t } .

In this case we write I F -limx=ξ.

Definition 3.3 We say that a double sequence x=( x j k ) is said to be I -convergent in (X,F,) or simply I F -convergent to if there exists a subset K={( j m , k m ): j 1 < j 2 <; k 1 < k 2 <} of N×N such that KF(I) (i.e., N×NKI) and F- lim m x j m k m =. In this case we write I F -limx= and is called the I F -limit of the double sequence x=( x j k ).

Theorem 3.4 If a double sequencex=( x j k )is I F -convergent, then I F -limit is unique.

Proof Suppose that I F -limx= 1 and I F -limx= 2 . Given ϵ>0, choose s>0 such that (1s)(1s)>1ϵ. Then, for any t>0 and nonzero zX, define the following sets as:

Since I F -limx= 1 , we have D F (s,t)I. Similarly I F -limx= 2 implies that D F (s,t)I. Now let D F (s,t)= D F (s,t) D F (s,t). Then from the definition of I, D F (s,t)I, and hence its compliment D F C (s,t) is a non-empty set which belongs to F(I). Now, if (j,k)N×N D F (s,t), then we have

F 1 2 , z (t) F x j k 1 , z (t/2) F x j k 2 , z (t/2)>(1s)(1s)>1ϵ.

Since ϵ>0 was arbitrary, we get F 1 - 2 , z (t)=1 for all t>0 and nonzero zX. Hence 1 = 2 .

This completes the proof of the theorem. □

Theorem 3.5 LetIbe an admissible ideal. If a double sequencex=( x j k )is(F)-convergent to , then it is I F -convergent to the same limit. But the converse need not be true.

Proof Let (F)-limx=. Then for every ϵ>0, t>0 and nonzero zX, there is a positive integer N such that

F x j k , z (t)>1ϵ

for all j,kN. Since the set

A(ϵ):= { ( j , k ) N × N : F x j k , z ( ϵ ) 1 ϵ }

is contained in S×S, where S={1,2,3,,N1} and the ideal I is admissible, A(ϵ)I. Hence I F -limx=.

For the converse, we construct the following example.

Example 3.6 Let X= R 2 , with the 2-norm x,z=| x 1 z 2 x 2 z 1 |, x=( x 1 , x 2 ), z=( z 1 , z 2 ), and ab=ab for all a,b[0,1]. Let F x , z (t)= t t + x , z for all x,yX, t>0. Now we define a double sequence x=( x j k ) by

x j k ={ ( j k , 0 ) if  j  and  k  are squares ; ( 0 , 0 ) otherwise .

Write

K m , n (ϵ,t):= { j m , k n : F x j k , z ( t ) 1 ϵ } ,0<ϵ<1;=(0,0).

We see that

F x j k , z (t)={ t t + j k z 2 if  j  and  k  are squares ; 1 otherwise .

Taking limit j,k, we get

lim j , k F x j k , z (t)={ 0 if  j  and  k  are squares ; 1 otherwise .

Hence, a double sequence x=( x j k ) is not convergent in (X,F,). But if we take I=I(δ)={AN×N: δ 2 (A)=0}, then since K m , n (ϵ,t){(1,0),(4,0),(9,0),(16,0),}, δ 2 ( K m , n (ϵ,t))=0, that is, I F -limx=.

This completes the proof of the theorem. □

Theorem 3.7 LetIbe an admissible ideal andx=( x j k )be a double sequence. If I F -limx=then I F -limx=.

Proof Suppose that I F -limx=. Then there exists a subset K={( j m , k m ): j 1 < j 2 <; k 1 < k 2 <} of N×N such that KF(I) (i.e., N×NKI) and F- lim m x j m k m =. But, for each ϵ>0 and t>0 there exists NN such that F x j m k m , z (t)>1ϵ for all m>N. Since {( j m , k m )K: F x j m k m , z (t)1ϵ} is contained in { j 1 < j 2 << j N 1 }{ k 1 < k 2 << k N 1 } and the ideal I is admissible, we have

{ ( j m , k m ) K : F x j m k m , z ( t ) 1 ϵ } I.

Thus

{ ( j , k ) : F x j k , z ( t ) 1 ϵ } H{ j 1 < j 2 << j N 1 }{ k 1 < k 2 << k N 1 }I

for all ϵ>0 and t>0. Therefore, we conclude that I F -limx=.

This completes the proof of the theorem. □

The following example shows that the converse of the above theorem need not be true.

Example 3.8 Let X= R 2 , with the 2-norm x,z=| x 1 z 2 x 2 z 1 |, x=( x 1 , x 2 ), z=( z 1 , z 2 ), and ab=ab for all a,b[0,1]. Define F x , z (t)= t t + x , z for all x,yX and t>0. Then (X,F,) is a RTN-space. Let N×N= i , j i j be a decomposition of N×N such that, for any (m,n)N×N each i j contains infinitely many (i,j)’s, where im, jn and i j m n = for (i,j)(m,n). Now we define a double sequence x=( x m n ) by x m n =( 1 i j ,0) if (m,n) i j . Then F x m n , z (t)1 as m,n. Hence I F - lim m , n x m n =0.

On the other hand, suppose that I F - lim m , n x m n =0. Then there exists a subset K={( m j , n j ): m 1 < m 2 <; n 1 < n 2 <} of N×N such that KF(I) and F- lim j x m j n j =0. Since KF(I), there is a set HI such that K=N×NH. Now, from the definition of I, there exists say p,qN such that

H ( m = 1 p ( n = 1 m n ) ) ( n = 1 q ( m = 1 m n ) ) .

But p + 1 , q + 1 K, therefore x m j n j =( 1 ( p + 1 ) ( q + 1 ) ,0)>0 for infinitely many ( m j , n j )’s from K which contradicts F- lim j x m j n j =0. Therefore, the assumption I - lim m , n x m n =0 leads to the contradiction. Hence the converse of the Theorem 3.7 need not be true.

Now the question arises under what condition the converse may hold. The following theorem shows that the converse holds if the ideal I satisfies the condition (AP).

An admissible ideal IP(N×N) is said to satisfy the condition (AP) if, for every sequence ( A n ) n N of pairwise disjoint sets from I, there are sets B n N, nN such that the symmetric difference A n B n is a finite set for every n and n N B n I.

Theorem 3.9 Suppose the idealIsatisfies the condition (AP). Ifx=( x j k )is a double sequence in X such that I F -limx=, then I F -limx=.

Proof Suppose I satisfies the condition (AP) and I F -limx=. Then, for each ϵ>0, t>0 and nonzero zX, we have

{ ( j , k ) N × N : F x j k , z ( t ) 1 ϵ } I.

We define the set A p for pN, t>0 and zX as

A p = { ( j , k ) N × N : 1 1 p F x j k , z ( t ) < 1 1 p + 1 } .

Obviously, { A 1 , A 2 ,} is countable and belongs to I, and A i A j = for ij. By the condition (AP), there is a countable family of sets { B 1 , B 2 ,}I such that the symmetric difference A i B i is a finite set for each iN and B= i = 1 B i I. From the definition of the associated filter F(I) there is a set KF(I) such that K=N×NB. To prove the theorem it is sufficient to show that the subsequence ( x j k ) ( j , k ) K is convergent to in (X,F,). Let s>0 and t>0. Choose qN such that 1 q <s. Then

{ ( j , k ) N × N : F x j k , z ( t ) 1 s } { ( j , k ) N × N : F x j k , z ( t ) 1 1 q } i = 1 q + 1 A i .

Since A i B i , i=1,2,,q+1 are finite, there exists ( j 0 , k 0 )N×N such that

( i = 1 q + 1 B i ) { ( j , k ) : j j 0  and  k k 0 } = ( i = 1 q + 1 A i ) { ( j , k ) : j j 0  and  k k 0 } .
(3.1)

If j j 0 , k k 0 and (j,k)K, then (j,k) i = 1 q + 1 B i . Therefore by (3.1), we have (j,k) i = 1 q + 1 A i . Hence, for every j j 0 , k k 0 and (j,k)K, we have F x j k , z (t)>1s. Since s was arbitrary, we have I F -limx=.

This completes the proof of the theorem. □

References

  1. 1.

    Menger K: Statistical metrics. Proc. Natl. Acad. Sci. USA 1942, 28: 535–537. 10.1073/pnas.28.12.535

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Schweizer B, Sklar A: Statistical metric spaces. Pac. J. Math. 1960, 10: 313–334. 10.2140/pjm.1960.10.313

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    S̆erstnev AN: On the notion of a random normed space. Dokl. Akad. Nauk SSSR 1963, 149: 280–283.

    MathSciNet  Google Scholar 

  4. 4.

    Alsina C, Schweizer B, Sklar A: Continuity properties of probabilistic norms. J. Math. Anal. Appl. 1997, 208(2):446–452. 10.1006/jmaa.1997.5333

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Goleţ I: On probabilistic 2-normed spaces. Novi Sad J. Math. 2006, 35(1):95–102.

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Gähler S: 2-metrische Räume und ihre topologeische Struktur. Math. Nachr. 1963, 26: 115–148. 10.1002/mana.19630260109

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Fast H: Sur la convergence statistique. Colloq. Math. 1951, 2: 241–244.

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Çolak R, Bektaş CA: λ -statistical convergence of order α . Acta Math. Sci., Ser. B 2011, 31(3):953–959.

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Fridy JA: On statistical convergence. Analysis 1985, 5: 301–313.

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Karakaya V: Some geometric properties of sequence spaces involving lacunary sequence. J. Inequal. Appl. 2007., 2007: Article ID 81028

    Google Scholar 

  11. 11.

    Mohiuddine SA, Danish Lohani QM: On generalized statistical convergence in intuitionistic fuzzy normed space. Chaos Solitons Fractals 2009, 42: 1731–1737. 10.1016/j.chaos.2009.03.086

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Mohiuddine SA, Şevli H, Cancan M: Statistical convergence in fuzzy 2-normed space. J. Comput. Anal. Appl. 2010, 12(4):787–798.

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Mohiuddine SA, Aiyub M: Lacunary statistical convergence in random 2-normed spaces. Appl. Math. Inf. Sci. 2012, 6(3):581–585.

    MathSciNet  Google Scholar 

  14. 14.

    Mohiuddine SA, Alotaibi A, Mursaleen M: Statistical convergence of double sequences in locally solid Riesz spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 719729

    Google Scholar 

  15. 15.

    Mohiuddine SA, Savaş E: Lacunary statistically convergent double sequences in probabilistic normed spaces. Ann. Univ. Ferrara 2012. doi:10.1007/s11565–012–0157–5

    Google Scholar 

  16. 16.

    Moricz F: Statistical convergence of multiple sequences. Arch. Math. 2003, 81: 82–89. 10.1007/s00013-003-0506-9

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Mursaleen M, Edely OHH: Statistical convergence of double sequences. J. Math. Anal. Appl. 2003, 288: 223–231. 10.1016/j.jmaa.2003.08.004

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Mursaleen M, Mohiuddine SA: Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 2009, 41: 2414–2421. 10.1016/j.chaos.2008.09.018

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Mursaleen M, Mohiuddine SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math. 2009, 233: 142–149. 10.1016/j.cam.2009.07.005

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Mursaleen M, Çakan C, Mohiuddine SA, Savaş E: Generalized statistical convergence and core of double sequences. Acta Math. Sin. Engl. Ser. 2010, 26(9):2131–2144.

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Savaş E, Mohiuddine SA: λ ¯ -statistically convergent double sequences in probabilistic normed spaces. Math. Slovaca 2012, 62(1):99–108. 10.2478/s12175-011-0075-5

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Kastyrko P, S̆alát T, Wilczyński W: I -convergence. Real Anal. Exch. 2000–2001, 26: 669–686.

    Google Scholar 

  23. 23.

    Das P, Kastyrko P, Wilczyński W, Malik P: I and I -convergence of double sequences. Math. Slovaca 2008, 58(5):605–620. 10.2478/s12175-008-0096-x

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Mursaleen M, Mohiuddine SA, Edely OHH: On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces. Comput. Math. Appl. 2010, 59: 603–611. 10.1016/j.camwa.2009.11.002

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Mursaleen M, Mohiuddine SA: On ideal convergence of double sequences in probabilistic normed spaces. Math. Rep. 2010, 12(4):359–371.

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Mursaleen M, Alotaibi A: On I -convergence in random 2-normed spaces. Math. Slovaca 2011, 61(6):933–940. 10.2478/s12175-011-0059-5

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Mursaleen M, Mohiuddine SA: On ideal convergence in probabilistic normed spaces. Math. Slovaca 2012, 62(1):49–62. 10.2478/s12175-011-0071-9

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Şahiner A, Gürdal M, Saltan S, Gunawan H: Ideal convergence in 2-normed spaces. Taiwan. J. Math. 2007, 11(5):1477–1484.

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Gürdal M, Acik I: On I -Cauchy sequences in 2-normed spaces. Math. Inequal. Appl. 2008, 11(2):349–354.

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Gürdal M, Pehlivan S: The statistical convergence in 2-Banach spaces. Thai J. Math. 2004, 2(1):107–113.

    MATH  Google Scholar 

  31. 31.

    Alotaibi A, Mohiuddine SA: On the stability of a cubic functional equation in random 2-normed spaces. Adv. Differ. Equ. 2012., 2012: Article ID 39

    Google Scholar 

  32. 32.

    Pringsheim A: Zur theorie der zweifach unendlichen Zahlenfolgen. Math. Z. 1900, 53: 289–321.

    MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Syed Abdul Mohiuddine.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Mohiuddine, S.A., Alotaibi, A. & Alsulami, S.M. Ideal convergence of double sequences in random 2-normed spaces. Adv Differ Equ 2012, 149 (2012). https://doi.org/10.1186/1687-1847-2012-149

Download citation

Keywords

  • double sequences
  • t-norm
  • random 2-normed spaces
  • ideal convergence