Skip to main content

On lacunary double statistical convergence in locally solid Riesz spaces

Abstract

The concept of statistical convergence is one of the most active areas of research in the field of summability. Most of the new summability methods have relation with this popular method. In this paper, we introduce the concept of double I θ -statistical-τ-convergence which is a more general idea of statistical convergence. We also investigate the ideas of double I θ -statistical-τ-boundedness and double I θ -statistical-τ-Cauchy condition of sequences in the framework of locally solid Riesz space endowed with a topology τ and investigate some of their consequences.

MSC:40G15, 40A35, 46A40.

1 Introduction

The notion of statistical convergence, which is an extension of the idea of usual convergence, was introduced by Fast [1], Steinhaus [2] independently in the same year 1951 and also by Schoenberg [3]. Its topological consequences were studied first by Fridy [4] and Šalát [5]. The notion has also been defined and studied in different steps, for example, in a locally convex space [6]; in topological groups [7, 8]; in probabilistic normed spaces [9, 10], in intuitionistic fuzzy normed spaces [11], in random 2-normed spaces [12]. In [13] Albayrak and Pehlivan studied this notion in locally solid Riesz spaces. Recently, Mohiuddine et al. [14] studied statistically convergent, statistically bounded and statistically Cauchy for double sequences in locally solid Riesz spaces. Also, in [15] Mohiuddine et al. introduced the concept of lacunary statistical convergence, lacunary statistically bounded and lacunary statistically Cauchy in the framework of locally solid Riesz spaces. Quite recently, Das and Savas [16] introduced the ideas of I τ -convergence, I τ -boundedness and I τ -Cauchy condition of nets in a locally solid Riesz space.

The more general idea of lacunary statistical convergence was introduced by Fridy and Orhan in [17]. Subsequently, a lot of interesting investigations have been done on this convergence (see, for example, [1821] where more references can be found).

The idea of statistical convergence was further extended to -convergence in [22] using the notion of ideals of with many interesting consequences. More investigations in this direction and more applications of ideals can be found in [2231] where many important references can be found.

Recently in [24, 30] we used ideals to introduce the concepts of I λ -statistical convergence and -lacunary-statistical convergence and investigated their properties.

The notion of a Riesz space was first introduced by Riesz [32] in 1928, and since then it has found several applications in measure theory, operator theory, optimization and also in economics (see [33]). It is well known that a topology on a vector space that makes the operations of addition and scalar multiplication continuous is called a linear topology and a vector space endowed with a linear topology is called a topological vector space. A Riesz space is an ordered vector space which is also a lattice endowed with a linear topology. Further, if it has a base consisting of solid sets at zero, then it is known as a locally solid Riesz space.

In this paper, we introduce the idea of -double lacunary statistical convergence in a locally solid Riesz space and study some of its properties by using the mathematical tools of the theory of topological vector spaces.

2 Preliminaries

We now recall the following basic facts from [22].

A family of subsets of a non-empty set X is said to be an ideal if (i) A,BI implies ABI, (ii) AI, BA imply BI. is called non-trivial if I{ϕ} and XI. is admissible if it contains all singletons. If is a proper non-trivial ideal, then the family of sets F(I)={MX: M c I} is a filter on X (where c stands for the complement). It is called the filter associated with the ideal .

We also recall some of the basic concepts of Riesz spaces.

Definition 2.1 Let L be a real vector space and let ≤ be a partial order on this space. L is said to be an ordered vector space if it satisfies the following properties:

  1. (i)

    If x,yL and yx, then y+zx+z for each zL.

  2. (ii)

    If x,yL and yx, then λyλx for each λ0.

If in addition L is a lattice with respect to the partial ordering, then L is said to be a Riesz space (or a vector lattice).

For an element x of a Riesz space L, the positive part of x is defined by x + =xθ, the negative part of x by x =(x)θ and the absolute value of x by |x|=x(x), where θ is the element zero of L.

A subset S of a Riesz space L is said to be solid if yS and |x||y| imply xS.

A topology τ on a real vector space L that makes the addition and scalar multiplication continuous is said to be a linear topology, that is, when the mappings

are continuous, where σ is the usual topology on R. In this case, the pair (L,τ) is called a topological vector space.

Every linear topology τ on a vector space L has a base N for the neighborhoods of θ satisfying the following properties:

  1. (a)

    Each VN is a balanced set, that is, λxV holds for all xV and every λR with |λ|1.

  2. (b)

    Each VN is an absorbing set, that is, for every xL, there exists a λ>0 such that λxV.

  3. (c)

    For each VN, there exists some WN with W+WV.

Definition 2.2 A linear topology τ on a Riesz space L is said to be locally solid if τ has a base at zero consisting of solid sets. A locally solid Riesz space (L,τ) is a Riesz space L equipped with a locally solid topology τ.

N sol will stand for a base at zero consisting of solid sets and satisfying the properties (a), (b) and (c) in a locally solid topology.

3 Main results

The notion of statistical convergence depends on the density of subsets of N, the set of natural numbers. A subset E of N is said to have density δ(E) if

δ(E)= lim n 1 n k = 1 n χ E (k)exists.

Note that if KN is a finite set, then δ(K)=0, and for any set KN, δ( K C )=1δ(K).

Definition 3.1 A sequence x=( x k ) is said to be statistically convergent to if for every ε>0,

δ ( { k N : | x k | ε } ) =0.

In another direction, a new type of convergence called lacunary statistical convergence was introduced in [17] as follows. A lacunary sequence is an increasing integer sequence θ= { k r } r N { 0 } such that k 0 =0 and h r = k r k r 1 as r. Let I r =( k r 1 , k r ] and q r = k r k r 1 . A sequence ( x k ) of real numbers is said to be lacunary statistically convergent to L (or, S θ -convergent to L) if for any ϵ>0,

lim r 1 h r | { k I r : | x k L | ϵ } | =0,

where |A| denotes the cardinality of AN. In [17] the relation between lacunary statistical convergence and statistical convergence was established among other things.

We now have the following definitions.

Definition 3.2 (See [22, 25])

Let I 2 N be a proper admissible ideal in . The sequence ( x k ) of elements of is said to be -convergent to LR if for each ϵ>0, the set A(ϵ)={nN:| x k L|ϵ}I. The class of all -statistically convergent sequences will be denoted by S(I).

Definition 3.3 ([24])

Let θ be a lacunary sequence. A sequence x=( x k ) is said to be -lacunary statistically convergent to L or S θ (I)-convergent to L if for any ϵ>0 and δ>0,

{ r N : 1 h r | { k I r : | x k L | ϵ } | δ } I.

In this case, we write x k L( S θ (I)). The class of all -lacunary statistically convergent sequences will be denoted by S θ (I).

It can be checked, as in the case of statistically and lacunary statistically convergent sequences, that both S(I) and S θ (I) are linear subspaces of the space of all real sequences.

Remark 3.1 For I= I f i n ={AN:A is a finite subset}, I S θ -convergence coincides with lacunary statistical convergence which is defined in [17].

Let EN×N be a two-dimensional set of positive integers and let E m , n be the numbers of (i,j) in K such that in and jm. Then the lower asymptotic density δ 2 (E) of E is defined as follows:

lim inf m , n E m , n m n = δ 2 (K).

In the case when the sequence ( E m , n m n ) m , n = 1 , 1 , has a limit, we say that E has a natural density and is defined as follows:

lim m , n E m , n m n = δ 2 (E).

For example, let E={( i 2 , j 2 ):(i,j)N×N}. Then

δ 2 (E)= lim m , n E m , n m n lim m , n m n m n =0

(i.e., the set E has double natural density zero).

Recently, Mursaleen and Edely [34] presented the notion of statistical convergence for a double sequence x=( x k l ) as follows:

A real double sequence x=( x k l ) is said to be statistically convergent to L provided that for each ϵ>0,

lim m , n 1 m n | { ( k , l ) : k m  and  l n , | x k , l L | ϵ } | =0.

The double sequence θ r , s ={( k r , l s )} is called double lacunary if there exist two increasing sequences of integers such that

k 0 =0, h r = k r k k 1 as r

and

l 0 =0, h ¯ s = l s l s 1 as s.

Let us denote k r , s = k r l s , h r , s = h r h ¯ s and θ r , s is determined by I r , s ={(k,l): k r 1 <k k r  and  l s 1 <l l s }.

We have the following.

Definition 3.4 Let ( x k l ) be a sequence in a locally solid Riesz space (L,τ). We say that x is I θ r , s -statistically-τ-convergent to x 0 if for every τ-neighborhood U of zero and for δ>0,

{ ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : x k l x 0 U } | δ } I.

In this case, we write I θ r , s s t τ lim x k l = x 0 (or in brief).

Remark 3.2 For I= I f i n , I θ r , s -statistical-τ-convergence becomes double lacunary statistical τ-convergence in a locally solid Riesz space.

Definition 3.5 Let ( x k l ) be a sequence in a locally solid Riesz space (L,τ). We say that x is I θ r s -statistically-τ-bounded if for every τ-neighborhood U of zero and δ>0, there exists α>0 such that

{ ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : α x k l U } | δ } I.

Definition 3.6 Let ( x k l ) be a sequence in a locally solid Riesz space (L,τ). We say that x is I θ r , s -statistically-τ-Cauchy if for every τ-neighborhood U of zero and δ>0, there exist p,qN such that

{ ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : x k l x p q U } | δ } I.

Now we are ready to present some basic properties of this new convergence in a locally solid Riesz space.

Theorem 3.1 Let (L,τ) be a Hausdorff locally solid Riesz space, x=( x k l ) and y=( y k l ) be two sequences in L. Then the following hold:

  1. (a)

    If I θ r , s s t τ lim x k l = y 0 and I θ r , s s t τ lim x k l = z 0 , then y 0 = z 0 .

  2. (b)

    If I θ r , s s t τ lim x k l = x 0 , then I θ r , s s t τ limα x k l =α x 0 for each αR.

  3. (c)

    If I θ r , s s t τ lim x k l = x 0 and I θ r , s s t τ lim y k l = y 0 , then I θ r , s s t τ lim( x k l + y k l )= x 0 + y 0 .

Proof (a) Let U be any τ-neighborhood of zero. Then there exists a V N sol such that VU. Take a W N sol such that W+WV. Let δ= 1 5 . Since I θ r , s s t τ lim x k l = y 0 and I θ r , s s t τ lim x k l = z 0 , we write

K 1 = { ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : x k l y 0 W } | < δ } F(I)

and

K 2 = { ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : x k l z 0 W } | < δ } F(I).

Then K= K 1 K 2 F(I) and for r,sK,

1 h r s | { ( k , l ) I r s : x k l y 0 W } |<δ,

i.e.,

1 h r s | { ( k , l ) I r s : x k l y 0 W } |>1δ= 4 5 .
(1)

Similarly,

1 h r s | { ( k , l ) I r s : x k l z 0 W } |> 4 5 .
(2)

Now write that {(k,l) I r s : x ( k , l ) y 0 W} and {(k,l) I r s : x k l z 0 W} cannot be disjoint, for then we will have 1 h r s |{(k,l) I r s }|> 8 5 , which is impossible. So, there is a ( k r , l s ) I r s for which

x k r , l s y 0 Wand x k r , l s z 0 W.

Then

x 0 z 0 = y 0 x k r l s + x k r l s z 0 W+WVU.

Thus y 0 z 0 U for every τ-neighborhood U of zero. Since (L,τ) is Hausdorff, the intersection of all τ-neighborhoods of zero is the singleton { θ r , s }, and so y 0 z 0 =θ, i.e., y 0 = z 0 .

  1. (b)

    Let I θ r , s s t τ lim x k = x 0 and let U be an arbitrary τ-neighborhood of zero. Choose V N sol such that VU. For any 1>δ>0,

    K= { ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : x k l x 0 V } | < δ } F(I),

i.e., r,sK,

1 h r s | { ( k , l ) I r s : x k l x 0 V } |>1δ.

First let |α|1. Since V is balanced, x k l x 0 V implies that α( x k l x 0 )V. Therefore

{ ( k , l ) I r s : α x k l α x 0 V } { ( k , l ) I r s : x k l x 0 V } ,

and so r,sK,

1 h r s | { ( k , l ) I r s : α x k l α x 0 V } | 1 h r s | { ( k , l ) I r s : x k l x 0 V } |>1δ,

which implies that

{ ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : α x k l α x 0 V } | < δ } K

and finally

{ ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : α x k l α x 0 V } | < δ } F(I).

If |α|>1 and [|α|] is the smallest integer greater or equal to |α|, choose W N sol such that [|α|]WV. Again, for 1>δ>0, taking

K= { ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : x k l x 0 W } | < δ } F(I)

and in view of the fact that

|α x 0 α x k l |=|α|| x 0 x k l | [ | α | ] | x n m x 0 | [ | α | ] WVUm,

which implies that α x 0 α x k l VU, proceeding as before, we conclude that

{ ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : α x k l α x 0 U } | < δ } F(I).

This proves that I θ r , s s t τ limα x k l =α x 0 .

  1. (c)

    Let U be an arbitrary τ-neighborhood of zero. Then there are V,W N sol such that W+WVU. Since I θ s t τ lim x k l = x 0 and I θ r , s s t τ lim y k l = y 0 , we get, for 0<δ<1,

    K 1 = { ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : x k l x 0 W } | < δ 3 } F(I)

and

K 2 = { ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : y k l y 0 W } | < δ 3 } F(I).

If K= K 1 K 2 , then r,sK,

1 h r s | { ( k , l ) I r s : x k l x 0 W } |< δ 3 ,

i.e.,

1 h r s | { ( k , l ) I r s : x k l x 0 W } |>1 δ 3

and also

1 h r s | { ( k , l ) I r s : y k l y 0 W } |< δ 3 .

But

( x k l + y k l )( x 0 + y 0 )=( x k l x 0 )+( y k l y 0 )W+WVU

(k,l) I r s such that k,lAB when {(k,l) I r s : x k l x 0 W}=A (say) and {(k,l) I r s : y k l y 0 W}=B (say). Note that

|A|=|AB|+|AB||AB|+| B c |,

i.e.,

1 h r s | A | 1 h r s | A B | + 1 h r s | B c | < 1 h r s | A B | + δ 3 ,

i.e.,

1 h r s | A B | = 1 h r s | { ( k , l ) I r s : x k l x 0 W y k l y 0 W } | > 1 h r s | { ( k , l ) I r s : x k l x 0 W } | δ 3 > 1 δ 3 δ 3 > 1 δ .

Since

{ ( k , l ) I r s : ( x k l + y k l ) ( x 0 + y 0 ) U } AB,

so for all r,sK,

1 h r s | { ( k , l ) I r s : ( x k l + y k l ) ( x 0 + y 0 ) U } | 1 h r s |AB|>1δ,

i.e.,

1 h r s | { ( k , l ) I r s : ( x k l + y k l ) ( x 0 + y 0 ) U } |<δ.

Hence

K { ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : ( x k l + y k l ) ( x 0 + y 0 ) U } | < δ }

and so

{ ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : ( x k l + y k l ) ( x 0 + y 0 ) U } | < δ } F(I).

This completes the proof of the theorem. □

Theorem 3.2 Let (L,τ) be a locally solid Riesz space. Let x={ x k l }, y={ y k l } and z={ z k l } be three sequences in L such that x k l y k l z k l for each (k,l)N×N. If I θ r , s s t τ lim x k l =a= I θ r , s s t τ lim z n m , then I θ r , s s t τ lim y k l =a.

Proof Let U be an arbitrary τ-neighborhood of zero. Take V,W N sol such that W+WVU. Since I θ r , s s t τ lim x k l =a= I θ r , s s t τ lim z k l , so for 0<δ<1,

K 1 = { ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : x k l a W } | < δ 3 } F(I)

and

K 2 = { ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : z k l a W } | < δ 3 } F(I).

Hence we observe that r,sK,

1 h r s | { ( k , l ) I r s : x k l a W } |< δ 3 ,

i.e.,

1 h r s | { ( k , l ) I r s : x k l a W } |>1 δ 3

and

1 h r s | { ( k , l ) I r s : z k l a W } |< δ 3 .

Writing A={(k,l) I r s : x k l aW} and B={(k,l) I r s : z k l aW}, we see that k,lAB,

and as V is solid, so

y k l aVU.

Clearly, {(k,l) I r s : y k l aU}AB and as in the previous theorem, we show that r,sK,

1 h r s | { ( k , l ) I r s : y k l a U } | 1 h r s |AB|>1δ,

i.e.,

1 h r s | { ( k , l ) I r s : y k l a U } |<δ.

Hence

{ ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : y k l a U } | < δ } K,

where KF(I) and so

{ ( r , s ) N × N : 1 h r s | { k I r s : y k l a U } | δ } I.

This proves that I θ r , s s t τ lim y k l =a. This completes the proof of the theorem. □

Theorem 3.3 An I θ r , s -statistically τ-convergent sequence ( x k l ) in a locally solid Riesz space (L,τ) is I θ r , s -statistically τ-bounded.

Proof Let ( x k l ) be I θ r , s -statistically τ-convergent to x 0 L. Let U be an arbitrary τ-neighborhood of zero. Choose V,W N sol such that W+WVU. Since W is absorbing, there is a μ>0 such that μ x 0 W. Choose α1 so that αμ. Since W is solid and |λ x 0 ||μ x 0 |, we have α x 0 W. Again, as W is balanced, x k l x 0 W implies that α( x n m x 0 )W. Now, for any 0<δ<1,

K= { ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : x k l x 0 W } | < δ } F(I).

Thus, for all r,sK,

1 h r s | { ( k , l ) I r s : x k l x 0 W } |<δ,

i.e.,

1 h r s | { ( k , l ) I r s : x k l x 0 W } |>1δ.

If B r s ={(k,l) I r s : x k l x 0 W}, then k,l B r s

α x k l =α( x k l x 0 )+α x 0 W+WVU,

and so, for all r,sK,

1 h r s | { ( k , l ) I r s : α x k l W } | 1 h r s | { ( k , l ) I r s : x k l x 0 W } | > 1 δ

i.e.,

1 h r s | { ( k , l ) I r s : α x k l W } |<δ.

Hence

K { ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : α x k l W } | < δ } .

Since KF(I), so the set on the right-hand side also belongs to F(I) and this proves that ( x k l ) is I θ r , s -statistically τ-bounded. □

Theorem 3.4 If a sequence ( x k l ) in a locally solid Riesz space (L,τ) is I θ r , s -statistically τ-convergent, then it is I θ r , s -statistically τ-Cauchy.

Proof Let ( x k l ) be I θ r , s -statistically τ-convergent to x 0 L. Let U be an arbitrary τ-neighborhood of zero. Choose V,W N sol such that W+WVU. Let 0<δ<1. Therefore

K= { ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : x k l x 0 W } | < δ } F(I).

For all r,sK,

1 h r s | { ( k , l ) I r s : x k l x 0 W } |<δ,

i.e.,

1 h r s | { ( k , l ) I r s : x k l x 0 W } |>1δ.

Take r,sK and in view of the above, we can choose p,q{(k,l) I r s : x k l x 0 W} (since this set cannot be empty). Then x p q x 0 W. Now observe that if for (k,l) I r s , x k l x 0 W, then

x k l x p q = x k l x 0 + x 0 x p q W+WVU.

Hence, as in the earlier proofs, we can prove that

K { ( r , s ) N × N : 1 h r s | { ( k , l ) I r s : x k l x p q W } | < δ } ,

which consequently implies that ( x k l ) is I θ r , s -statistically τ-Cauchy.

This completes the proof of the theorem. □

It should be noted that single and double case of I λ -statistical convergence in locally solid Riesz spaces are introduced in [35] and [36] respectively.

References

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

    MathSciNet  Google Scholar 

  2. Steinhaus H: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 1951, 27: 73–74.

    MathSciNet  Google Scholar 

  3. Schoenberg IJ: The integrability of certain functions and related summability methods. Am. Math. Mon. 1959, 66: 361–375. 10.2307/2308747

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Šalát T: On statistically convergent sequences of real numbers. Math. Slovaca 1980, 30: 139–150.

    MathSciNet  Google Scholar 

  6. Maddox IJ: Statistical convergence in locally convex spaces. Math. Proc. Camb. Philos. Soc. 1988, 104: 141–145. 10.1017/S0305004100065312

    Article  MathSciNet  Google Scholar 

  7. Cakalli H: Statistical convergence in topological groups. Pure Appl. Math. Sci. 1996, 43(1–2):27–31.

    MathSciNet  Google Scholar 

  8. Cakalli H, Savas E: Statistical convergence of double sequences in topological groups. J. Comput. Anal. Appl. 2010, 12(2):421–426.

    MathSciNet  Google Scholar 

  9. Mohiuddine SA, Savaş E: Lacunary statistically convergent double sequences in probabilistic normed spaces. Ann. Univ. Ferrara, Sez. 7: Sci. Mat. 2012, 58(2):331–339. 10.1007/s11565-012-0157-5

    Article  Google Scholar 

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

    MathSciNet  Google Scholar 

  11. Mursaleen M, Mohiuddine SA: Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 2009, 41(5):2414–2421. 10.1016/j.chaos.2008.09.018

    Article  MathSciNet  Google Scholar 

  12. Savaş E: On generalized double statistical convergence in random 2-normed space. J. Inequal. Appl. 2012., 2012: Article ID 209

    Google Scholar 

  13. Albayrak H, Pehlivan S: Statistical convergence and statistical continuity on locally solid Riesz spaces. Topol. Appl. 2012, 159: 1887–1893. 10.1016/j.topol.2011.04.026

    Article  MathSciNet  Google Scholar 

  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. Mohiuddine SA, Alghamdi MA: Statistical summability through lacunary sequence in locally solid Riesz spaces. J. Inequal. Appl. 2012., 2012: Article ID 225

    Google Scholar 

  16. Das P, Savaş E: On I -convergence of nets in locally solid Riesz spaces. Filomat 2013, 27(1):84–89.

    Google Scholar 

  17. Fridy JA, Orhan C: Lacunary statistical convergence. Pac. J. Math. 1993, 160: 43–51. 10.2140/pjm.1993.160.43

    Article  MathSciNet  Google Scholar 

  18. Patterson RF, Savaş E: Lacunary statistical convergence of double sequences. Math. Commun. 2000, 10: 55–61.

    Google Scholar 

  19. Savaş E: On some new double lacunary sequences spaces via Orlicz function. J. Comput. Anal. Appl. 2009, 11(3):423–430.

    MathSciNet  Google Scholar 

  20. Savaş E, Patterson RF: Some double lacunary sequence spaces defined by Orlicz functions. Southeast Asian Bull. Math. 2011, 35(1):103–110.

    MathSciNet  Google Scholar 

  21. Savaş E, Patterson RF: Double σ -convergence lacunary statistical sequences. J. Comput. Anal. Appl. 2009, 11(4):610–615.

    MathSciNet  Google Scholar 

  22. Kostyrko P, Šalát T, Wilczynki W: I -convergence. Real Anal. Exch. 2000/2001, 26(2):669–685.

    Google Scholar 

  23. Dems K: On I -Cauchy sequences. Real Anal. Exch. 2004/2005, 30: 123–128.

    MathSciNet  Google Scholar 

  24. Das P, Savaş E, Ghosal SK: On generalizations of certain summability methods using ideals. Appl. Math. Lett. 2011, 24: 1509–1514. 10.1016/j.aml.2011.03.036

    Article  MathSciNet  Google Scholar 

  25. Kostyroko P, Macaj M, Šalát T, Sleziak M: I -convergence and extremal I -limit points. Math. Slovaca 2005, 55: 443–464.

    MathSciNet  Google Scholar 

  26. Lahiri BK, Das P: On I and I convergence of nets. Real Anal. Exch. 2007/2008, 33(2):431–442.

    MathSciNet  Google Scholar 

  27. Savaş E: Δ m -strongly summable sequence spaces in 2-normed spaces defined by ideal convergence and an Orlicz function. Appl. Math. Comput. 2010, 217: 271–276. 10.1016/j.amc.2010.05.057

    Article  MathSciNet  Google Scholar 

  28. Savaş E: A-sequence spaces in 2-normed space defined by ideal convergence and an Orlicz function. Abstr. Appl. Anal. 2011., 2011: Article ID 741382

    Google Scholar 

  29. Savaş E: On some new sequence spaces in 2-normed spaces using ideal convergence and an Orlicz function. J. Inequal. Appl. 2010., 2010: Article ID 482392. doi:10.1155/2010/482392

    Google Scholar 

  30. Savaş E, Das P: A generalized statistical convergence via ideals. Appl. Math. Lett. 2011, 24: 826–830. 10.1016/j.aml.2010.12.022

    Article  MathSciNet  Google Scholar 

  31. Savaş E, Das P, Dutta S: A note on strong matrix summability via ideals. Appl. Math. Lett. 2012, 25(4):733–738. 10.1016/j.aml.2011.10.012

    Article  MathSciNet  Google Scholar 

  32. Riesz F: Sur la decomposition des operations functionelles lineaires. 3. In Alti del Congr. Internaz. del Mat.. Zanichelli, Bologna; 1930:143–148.

    Google Scholar 

  33. Aliprantis CD, Burkinshaw O: Locally Solid Riesz Spaces with Applications to Economics. 2nd edition. Am. Math. Soc., Providence; 2003.

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. Das, P, Savaş, E: On I λ -statistical convergence in locally solid Riesz spaces. Math. Slovaca. Preprint

  36. Savas, E: On generalized double statistical convergence in locally solid Riesz spaces. Miskolc Math. Notes. Preprint

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ekrem Savas.

Additional information

Competing interests

The author declares that they have no competing interests.

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

Savas, E. On lacunary double statistical convergence in locally solid Riesz spaces. J Inequal Appl 2013, 99 (2013). https://doi.org/10.1186/1029-242X-2013-99

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-99

Keywords