Lacunary statistical convergence of double sequences in topological groups
© Savaş; licensee Springer. 2014
Received: 20 August 2014
Accepted: 7 November 2014
Published: 2 December 2014
Recently, Patterson and Savaş (Math. Commun. 10:55-61, 2005), defined the lacunary statistical analog for double sequences as follows: A real double sequences is said to be P-lacunary statistically convergent to L provided that for each , . In this case write or .
In this paper we introduce and study lacunary statistical convergence for double sequences in topological groups and we shall also present some inclusion theorems.
The notion of statistical convergence, which is an extension of the usual idea of convergence, was introduced by Fast  and also independently by Schoenberg  for real and complex sequences, but rapid developments were started after the papers of Šalát  and Fridy . Nowadays it has become one of the most active area of research in the field of summability. Di Maio and Kočinac  introduced the concept of statistical convergence in topological spaces and statistical Cauchy condition in uniform spaces and established the topological nature of this convergence. Statistical convergence has several applications in different fields of mathematics: summability theory, number theory, trigonometric series, probability theory, measure theory, optimization, and approximation theory. Recently a lot of interesting developments have occurred in double statistical convergence and related topics (see [6–9] and ).
Before continuing with this paper we present some definitions and preliminaries.
The set of all statistically convergent sequences in X is denoted by and the set of all statistically Cauchy sequences in X is denoted by . It is well known that when X is complete.
By a lacunary sequence, we mean an increasing sequence of positive integers such that and as . Throughout this paper, the intervals determined by θ will be denoted by , and the ratio will be abbreviated by .
where denotes the cardinality of . In  the relation between lacunary statistical convergence and statistical convergence was established among other things. In , Mursaleen and Mohiuddine extended the idea of lacunary statistical convergence with respect to the intuitionistic fuzzy normed space.
2 Definitions and notations
By the convergence of a double sequence we mean the convergence in Pringsheim’s sense (see ). A double sequence is said to be convergent in Pringsheim’s sense if for every there exists such that whenever . L is called the Pringsheim limit of x. We shall describe such an x more briefly as ‘P-convergent’.
A double sequence is said to be Cauchy sequence if for every there exists , where N is the set of natural numbers such that for all and .
(i.e., the set K has double natural density zero), while the set has double natural density .
Recently the studies of double sequences have seen rapid growth. The concept of double statistical convergence, for the complex case, was introduced by Mursaleen and Edely  and others, while the idea of statistical convergence of single sequences was first studied by Fast . Also the double lacunary statistical convergence was introduced by Patterson and Savaş .
Mursaleen and Edely have given the main definition.
Definition 2.1 ()
In this case we write and we denote the set of all statistical convergent double sequences by .
is -convergent. Nevertheless it neither is convergent nor bounded.
It should be noted that in , the authors proved the following important theorem.
x is statistically convergent to L;
x is statistically Cauchy;
- (c)there exists a subsequence y of x such that
Notations: , , θ is determined by , , , , , and . We will denote the set of all double lacunary sequences by .
Example 1 Let and . Then . But it is obvious that .
In 2005, Patterson and Savaş  studied double lacunary statistically convergence by giving the definition for complex sequences.
In this case we write or .
In this presentation, our goal is to extend a few results known in the literature from ordinary (single) sequences to double sequences in topological groups and to give some important inclusion theorems.
Quite recently, Çakalli and Savaş  defined the statistical convergence of double sequences of points in a topological group as follows.
has double natural density zero. In this case we write and we denote the set of all statistically convergent double sequences by .
Now we are ready to give the definition of double lacunary statistical convergence in topological groups.
It is obvious that every double lacunary statistically convergent sequence has only one limit, that is, if a sequence is double lacunary statistically convergent to and then .
3 Inclusion theorems
In this section, we prove some analogs for double sequences. For single sequences such results have been proved by Çakalli .
Theorem 3.1 For any double lacunary sequence , if and only if and .
Therefore neither x nor 0 can be a double lacunary statistical limit of . No other point of X can be a double lacunary statistical limit of the sequence as well. Thus . This completes the proof of this theorem. □
Theorem 3.2 For any lacunary sequence , if and only if and .
Finally it follows that .
which implies that cannot be double statistically convergent. This completes the proof of the theorem. □
In Section 2 we mentioned that the -limit is unique. However, it is possible for a sequence to have different -limits for different θ’s. The following theorem shows that this situation cannot occur if .
This contradiction completes the proof. □
Before presenting the next theorem, let us consider the following definition.
Finally we conclude this paper by presenting the multidimensional analog of Çakalli .
Theorem 3.6 A sequence is -convergent if and only if it is an -Cauchy sequence.
Since , in Pringsheim sense, and , by the assumption, it follows from the last inequality that .
in Pringsheim’s sense. Thus the theorem is proven. □
The author wishes to thank the referees for their valuable suggestions, which have improved the presentation of the paper. This paper was presented during the International Congress in Honour of Professor Ravi P Agarwal at The Auditorium at the Campus of Uludag University, Bursa-Turkey, 23-26 June 2014.
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