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ISRN Mathematical Analysis
Volume 2013 (2013), Article ID 189721, 5 pages
Lacunary Statistical Limit Points in Random 2-Normed Spaces
Department of Mathematics, Suleyman Demirel University, 32260 Isparta, Turkey
Received 29 April 2013; Accepted 13 June 2013
Academic Editors: F. Colombini and J.-L. Wu
Copyright © 2013 A. Güncan 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.
We introduce the notion -cluster points, investigate the relation between -cluster points and limit points of sequences in the topology induced by random 2-normed spaces, and prove some important results.
1. Introduction and Background
An interesting and important generalization of the notion of metric space was introduced by Menger  under the name of statistical metric space, which is now called probabilistic metric space. In this theory, the notion of distance has a probabilistic nature. Namely, the distance between two points and is represented by a distribution function ; and for , the value is interpreted as the probability that the distance from to is less than . In fact the probabilistic theory has become an area of active research for the last forty years. An important family of probabilistic metric spaces are probabilistic normed spaces. The notion of probabilistic normed spaces was introduced in  and further it was extended to random/probabilistic 2-normed spaces by Goleţ  using the concept of 2-norm of Gähler . Applications of this concept have been investigated by various authors, for example, [5–7].
The concept of statistical convergence for sequences of real number was introduced by Fast in  and Steinhaus in  independently in the same year 1951. A lot of developments have been made in this area after the works of Salat  and Fridy . Recently, Mohiuddine and Aiyub  studied lacunary statistical convergence as generalization of the statistical convergence and introduced the concept -statistical convergence in random 2-normed space. In , Mursaleen and Mohiuddine extended the idea of lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. Also lacunary statistically convergent double sequences in probabilistic normed space was studied by Mohiuddine and Savaş in .
The aim of this work is to introduce and investigate the relation between -statistical cluster points, -statistical limit points, and ordinary limit points of sequence in random 2-normed spaces.
First, we recall some of the basic concepts that will be used in this paper. All the concepts listed below are studied in depth in the fundamental book by Schweizer and Sklar .
Let denote the set of real numbers and . A mapping is called a distribution function if it is nondecreasing and left continuous with and .
We denote the set of all distribution functions by such that . If , then , where It is obvious that for all .
A triangular norm (-norm) is a continuous mapping such that is an abelian monoid with unit one and if and for all . A triangle function is a binary operation on which is commutative, associative and for every .
Let be a real vector space of dimension , where . A 2-norm on is a function which satisfies (i) if and only if and are linearly dependent; (ii) ; (iii) , ; (iv) . The pair is then called a 2-normed space.
As an example of a 2-normed space we may take being equipped with the 2-norm the area of the parallelogram spanned by the vectors and , which may be given explicitly by the formula
In 2006, Goleţ  introduced the notion of random 2-normed space.
Let be a linear space of dimension greater than one, a triangle, and . Then is called a probabilistic 2-norm and a probabilistic -normed space if the following conditions are satisfied: (i) if and are linearly dependent, where denotes the value of at ,(ii) if and are linearly independent,(iii) for all ,(iv) for every , and ,(v) whenever .
If is replaced by(v)′ for all and , then is called a random -normed space (for short, RTN space).
Remark 1. Note that every 2-normed space can be made a random 2-normed space in a natural way, by setting for every , and , .
Let be a RTN space. Since is a continuous -norm, the system of -neighborhoods of (the null vector in ) where determines a first countable Hausdorff topology on , called the -topology. Thus, the -topology can be completely specified by means of -convergence of sequences. It is clear that means and vice versa.
A sequence in is said to be -convergence to if for every , and for each nonzero there exists a positive integer such that or equivalently, In this case we write , .
2. The Main Results
It is known (see ) that statistical cluster and statistical limit points set of a given sequence are not altered by changing the values of a subsequence, the index set of which has density zero. Moreover, there is a strong connection between -statistical cluster points and ordinary limit points of a given sequence. We will prove that these facts are satisfied for -statistical cluster points and -statistical limit point sets of a given sequence in the topology induced by random 2-normed spaces.
The notion of statistical convergence depends on the density of subsets of , the set of natural numbers.
Definition 2 (see [8, 11]). Let be a subset of . Then the asymptotic density of denoted by , where the vertical bars denote the cardinality of the enclosed set. A number sequence is said to be statistically convergent to if for every , . If is statistically convergent to , we write -.
By a lacunary sequence we mean an increasing integer sequence such that and as . Throughout this paper the intervals determined by will be denoted by , and the ratio will be abbreviated by . Let . The number is said to be the -density of , provided the limit exists (see ).
Definition 3 (see ). Let be a lacunary sequence. Then a sequence is said to be -convergent to the number if for every the set has -density zero, where In this case we write or .
Definition 4 (see ). Let be a RTN space and let be a lacunary sequence. A sequence in a random 2-normed spaces is said to be -statistically convergent or -convergent to with respect to if for every , and nonzero such that or equivalently In this case we write -, or .
Now we define some concepts in RTN-space.
Definition 5. Let be a RTN space and let be a lacunary sequence. Let be a subsequence of and then one denotes by . If then is called a -thin sequence. On the other hand, is a -nonthin subsequence of provided that
Definition 6. Let be a RTN space and be a lacunary sequence. is called a -statistical limit point of a sequence provided that there is a -nonthin subsequence of that converges to . Let denotes the set of all -limit points.
Definition 7. Let be a RTN space and let be a lacunary sequence. is called a -statistical cluster point of a sequence provided that, for every , and nonzero Let denote the set of -statistical cluster point of the sequence .
Theorem 8. Let be a RTN space and let be a lacunary sequence. If and are sequences in such that then and .
Proof. Assume that and ; say is a -nonthin sequence of that converges to . Since it follows that Therefore, the latter set yields a -nonthin subsequence of that converges to . Hence and . By symmetry we see that ; hence . Now let and let . Since , we can write for every , , and nonzero . Since for almost all , for every , , and nonzero . Hence, and . By symmetry we see that ; hence .
Theorem 9. Let be a RTN space and let be a lacunary sequence. For any sequence , one has .
Proof. Suppose ; then there is a -nonthin subsequence of that converges to , that is, Since for every , , and nonzero , we have Since converges to , the set is finite for any , , and nonzero . Therefore, Hence, which means that .
Theorem 10. Let be a RTN space and let be a lacunary sequence. Let be the set of ordinary limit points of and for any sequence , .
Proof. Assume that ; then
for every , , and nonzero . We set a -nonthin subsequence of such that
for every , , and nonzero , and . Since there are infinitely many elements in , .
The converse of the theorem does not hold.
Theorem 11. Let be a RTN space and let be a lacunary sequence. If for sequence , -, then .
Proof. First, we show that . Fix , , and nonzero . Assume that such that . In this case, there exist and -nonthin subsequences of that converge to and , respectively. Since converges to , we have
which is a finite set. Consider that
Therefore, we can write
For every ,
This contradicts (31). Hence, .
Now we assume that such that for some , , and nonzero . Then Since for every , Therefore From (37), the right side of (40) is greater than zero and from (32), the left side of (40) equals to zero. This is a contradiction. Hence, .
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