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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 507962, 8 pages

http://dx.doi.org/10.1155/2013/507962

## Double Lacunary Density and Some Inclusion Results in Locally Solid Riesz Spaces

^{1}Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia^{2}Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh 791 112, India

Received 21 February 2013; Accepted 13 June 2013

Academic Editor: Alberto Parmeggiani

Copyright © 2013 S. A. Mohiuddine et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We define the notions of double statistically convergent and double lacunary statistically convergent sequences in locally solid Riesz space and establish some inclusion relations between them. We also prove an extension of a decomposition theorem in this setup. Further, we introduce the concepts of double *θ*-summable and double statistically lacunary summable in locally solid Riesz space and establish a relationship between these notions.

#### 1. Introduction

Fast [1] and Steinhaus [2] independently introduced an extension of the usual concept of sequential limits which he called statistical convergence. Actually the idea of statistical convergence was formerly given under the name “almost convergence" by Zygmund in the first edition (Warsaw, 1935) of his celebrated monograph [3]. Schoenberg [4] and Šalát [5] gave some basic properties of statistical convergence. In 1985, Fridy [6] introduced the notion of statistically Cauchy sequence and proved that it is equivalent to the concept of statistical convergence. The notion of statistical convergence is a very useful functional tool for studying the convergence problems of numerical sequences/matrices through the concept of density. Later on it was further investigated by various authors in different frameworks (see [7–18]). Mursaleen and Edely [19] extended these concepts from single to double sequences by using two dimensional analogue of natural density of subsets of and established relationship between statistical convergence and strongly Cesáro summable double sequences. Mohiuddine et al. [20] and Mursaleen and Mohiuddine [21] defined these notions for double sequences in fuzzy normed spaces and intuitionistic fuzzy normed spaces, respectively. Recently, Mohiuddine et al. [22] introduced these notions for double sequences in locally solid Riesz spaces and proved some interesting results. Fridy and Orhan [23] presented an interesting generalization of statistical convergence with the help of lacunary sequence and called it lacunary statistical convergence. Savaş and Patterson [24, 25] extended the notion of lacunary statistical convergence from single sequences to double sequences with the help of double lacunary density and proved some interesting results related to this concept. For more details related to the concept of lacunary statistical convergence for single and double sequences and applications to approximation theorems, we refer to [26–42].

On the other hand, the concept of Riesz space was introduced by Riesz [43]. Since then, with a view to utilize this concept in topology and analysis, many authors have extensively developed the theory of Riesz spaces along with their applications (e.g., [7, 22, 44, 45]).

#### 2. Definitions and Notations

In this section, we recall some of the basic concepts related to the notions of statistical convergence and lacunary sequence which we will use throughout the paper.

Let . Then the natural density of is denoted by and is defined by where the vertical bar denotes the cardinality of the respective set.

*Definition 1 (see [14]). *A sequence in a topological space is said to be *statistically convergent* to if for every neighborhood of
In this case, we write .

By a lacunary sequence , where , we will mean an increasing sequence of nonnegative integers with as . The intervals determined by will be denoted by and the ratio will be defined by (see [46]).

*Definition 2. *Let be a lacunary sequence and let . Let . The number is called the *lacunary density* or -*density* of if

The generalized lacunary mean is defined by

*Definition 3. *A sequence is said to be -*summable* to number if as . In this case we write that is the -limit of . If , then -summable reduces to -summable (see [46]).

By the convergence of a double sequence we mean the convergence in the Pringsheim sense [47]. A double sequence has a *Pringsheim* limit (denoted by ) provided that given an there exists an such that whenever . We will describe such an more briefly as “*convergent*.”

Let , and let denote the number of in such that and (see [19]). Then the lower natural density of is defined by . In case that the sequence has a limit in Pringsheim’s sense, then we say that has a *double natural density* and is defined by .

For example, let . Then that is, the set has double natural density zero, while the set has double natural density .

The double sequence is called *double lacunary sequence* if there exist two increasing sequences of integers such that (see [25])

*Notations*. , , and is determined by

*Definition 4 (see [26]). *Let be a double lacunary sequence. Let . The number
is said to be *double lacunary density*, that is, -*density* of , provided the limit exists.

We define the generalized double lacunary mean by

#### 3. Locally Solid Riesz Spaces

Let be a real vector space and let be a partial order on this space. Then is said to be an *ordered vector space* if it satisfies the following properties:(i)if and , then for each ;(ii)if and , then for each .

If, in addition, is a lattice with respect to the partial order, then is said to be a *Riesz space* (or a *vector lattice*) (see [45]).

For an element of a Riesz space , the *positive part* of is defined by , the *negative part* of by , and the *absolute value* of by , where is the zero element of .

A subset of a Riesz space is said to be *solid* if and implies .

A *topological vector space * is a vector space which has a topology (linear) , such that the algebraic operations of addition and scalar multiplication in are continuous. Continuity of addition means that the function defined by is continuous on , and continuity of scalar multiplication means that the function defined by is continuous on .

Every linear topology on a vector space has a base for the neighborhoods of satisfying the following properties.(1)Each is a *balanced set*; that is, holds for all and for every with .(2)Each is an *absorbing set*; that is, for every , there exists such that .(3)For each there exists some with .

A linear topology on a Riesz space is said to be *locally solid* [48] if has a base at zero consisting of solid sets. A *locally solid Riesz space * is a Riesz space equipped with a locally solid topology .

Recall [49] that a topological space is first countable if each point has a countable (decreasing) local base.

The purpose of this paper is to give certain characterizations of lacunary statistically convergent double sequences in locally solid Riesz spaces and obtain extensions of a decomposition theorem and some inclusion results related to the notions statistically convergence and lacunary statistically convergence in locally solid Riesz spaces.

Throughout the paper, the symbol will denote any base at zero consisting of solid sets and satisfying the conditions , , and in a locally solid topology.

#### 4. Double Lacunary Statistical Convergence in Locally Solid Riesz Spaces

Throughout the paper will denote the Hausdorff locally solid Riesz space which is first countable.

The idea of lacunary statistical convergence for single sequences in locally solid Riesz spaces has been recently studied by Mohiuddine and Alghamdi [50] as follows.

*Definition 5 (see [50]). *Let be a locally solid Riesz space. A sequence of points in is said to be -*convergent* to an element of if for each -neighborhood of zero,
that is,
In this case, we write .

Albayrak and Pehlivan [7] introduced the notion of statistical convergence in locally solid Riesz spaces. Afterward, Mohiuddine et al. [22] defined and studied the concept of statistical convergence in this setup as follows.

*Definition 6 (see [22]). *Let be a locally solid Riesz space. Then, a double sequence in is said to be *statistically **-convergent* to the number if for every -neighborhood of zero,
In this case we write or .

Now we recall the definition of lacunary statistical convergence of double sequences in the framework of locally solid Riesz spaces as follows.

*Definition 7. *Let be a locally solid Riesz space. A double sequence of points in is said to be *double lacunary statistical* *-convergent* or -*convergent* to an element of if for each -neighborhood of zero,
that is
In this case, we write or .

Now we prove our results.

Theorem 8. *Let be a locally solid Riesz space. If a double sequence of points in is -convergent to in , then there are double sequences and such that and , for all and and is a -null sequence.*

*Proof. *Let be a nested base of -neighborhoods of zero. Take and choose an increasing sequence of positive integers such that
Let us define the sequences and as follows:
and suppose , for ,

To show that, (i) and (ii) is a -null sequence.(i) Let be an arbitrary -neighborhood of zero. Since is first countable, we may choose a positive integer such that . Then , for . If , then . Hence .(ii) It is enough to show that . For any -neighborhood of zero, we have
If , then
If and , then
This implies that . Hence is a -null sequence.

Theorem 9. *Let be a locally solid Riesz space and let be a double sequence of points in . If there is a -convergent sequence in such that , then is also -convergent.*

*Proof. *Suppose that and . Then for an arbitrary -neighborhood of zero, we have
Now,
Therefore, we have

This completes the proof of the theorem.

#### 5. Some Inclusions Relations in Locally Solid Riesz Spaces

Here, we prove some inclusion type results. We begin with the following interesting result.

Theorem 10. *Let be a locally solid Riesz space and let be a double sequence of points in . For any double lacunary sequence if and only if .*

*Proof. *Suppose first that , and (say). Write . Then there exists an integer such that for . Hence for ,
Suppose that . We prove that . Let be an arbitrary -neighborhood of zero. Then for all , we have

Since . Therefore this inequality implies that . Hence .

Next, we suppose that . We can select a subsequence of the double lacunary sequence such that
where , . Take . Now we define a sequence by
Then . To see this, let be an arbitrary -neighborhood of zero. We choose such that and . On the other hand, for each we can find a positive number such that , . Then
Therefore . Now let us see that . Let be a -neighborhood of zero such that . Thus
and for ,
Hence neither nor can be double lacunary statistical limit of . No other point of can be double lacunary statistical limit of the sequence as well. Thus . This completes the proof of the theorem.

Theorem 11. *Let be a locally solid Riesz space and let be sequence in . For any double lacunary sequence if and only if .*

*Proof. *Suppose that . Then there exists an such that for all . Let . Let be an arbitrary -neighborhood of zero. Let . We write
By the definition of double lacunary statistical convergence, there are positive numbers , such that
Let and let be two integers satisfying ; then we can write
Since , there exist positive integers such that
Hence for
It follows that .

Next we suppose that . Take an element . Let be a subsequence of the double lacunary sequence such that . Define a sequence by
Let be a -neighborhood of zero such that . Then for
Hence . But , because
This completes the proof of the theorem.

Corollary 12. *Let be a locally solid Riesz space and let be a double sequence in . For any double lacunary sequence if and only if .*

Theorem 13. *Let be a locally solid Riesz space, let be a double sequence in . For any double lacunary sequence , if , then .*

*Proof. *Let and , and . Suppose that . Since is a Hausdorff, then there exists a -neighborhood of zero value such that . We choose such that . Then, we have

It follows from this inequality that
We write
where
Since , we have . Therefore the regular weighted mean transform of also tends to 0; that is,
Also since , we have
From (39), (43), and (44), we have
This contradiction completes the proof of the theorem.

#### 6. Double Statistical Lacunary Summable in Locally Solid Riesz Spaces

In this section, we introduce some new concepts by using the notions of statistical lacunary summable for double sequences.

*Definition 14. *Let be a locally solid Riesz space. A sequence is said to be *double lacunary summable* (or shortly, -*summable*) in or simply to an element if for each -neighborhood of zero value that , where . In this case, we write .

*Definition 15. *Let be a locally solid Riesz space. A sequence of points in is said to be *double statistical lacunary* *-summable* or simply -*summable* to an element of if for each -neighborhood of zero value, the set has double natural density zero; that is,

That is
In this case, we write .

Theorem 16. *Let be a locally solid Riesz space. A double sequence in is -summable to if and only if there exists a set , , such that and .*

*Proof. *Let be an arbitrary -neighborhood of zero. Suppose that ; there exists a set , , with and , such that for and . Write and . Then and which implies that . Hence is -summable to .

Conversely suppose that is -convergent to . Fix a countable local base at . For each , put
By hypothesis for each . Since the ideal of all subsets of having double density zero is a -ideal (see e.g., [51]), then there exists a sequence of sets such that the symmetric difference is a finite set for any and .

Let , then . In order to prove the theorem, it is enough to check that .

Let . Since is a finite, there is , without loss of generality with , , such that
If and then , and by (48) . Thus . So we have proved that for all there is , with for every : without loss of generality, we can suppose and for every . The assertion follows taking into account that the s form a countable local base at .

#### Acknowledgment

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.

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