On generalized double statistical convergence in a random 2-normed space
© Savas; licensee Springer 2012
Received: 12 March 2012
Accepted: 22 August 2012
Published: 25 September 2012
Recently, the concept of statistical convergence has been studied in 2-normed and random 2-normed spaces by various authors. In this paper, we shall introduce the concept of λ-double statistical convergence and λ-double statistical Cauchy in a random 2-normed space. We also shall prove some new results.
MSC:40A05, 40B50, 46A19, 46A45.
The probabilistic metric space was introduced by Menger  which is an interesting and an important generalization of the notion of a metric space. The theory of probabilistic normed (or metric) space was initiated and developed in [2–6]; further it was extended to random/probabilistic 2-normed spaces by Goleţ  using the concept of 2-norm which is defined by Gähler (see [8, 9]); and Gürdal and Pehlivan  studied statistical convergence in 2-normed spaces. Also statistical convergence in 2-Banach spaces was studied by Gürdal and Pehlivan in . Moreover, recently some new sequence spaces have been studied by Savas [12–14] by using 2-normed spaces.
In order to extend the notion of convergence of sequences, statistical convergence of sequences was introduced by Fast  and Schoenberg  independently. A lot of developments have been made in this areas after the works of S̆alát  and Fridy . Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory. Recently, Mursaleen  studied λ-statistical convergence as a generalization of the statistical convergence, and in  he considered the concept of statistical convergence of sequences in random 2-normed spaces. Quite recently, Bipan and Savas  defined lacunary statistical convergence in a random 2-normed space, and also Savas  studied λ-statistical convergence in a random 2-normed space.
where the vertical bars denote the cardinality of the enclosed set.
2 Definitions and preliminaries
We begin by recalling some notations and definitions which will be used in this paper.
It is obvious that for all .
A t-norm is a continuous mapping such that is an Abelian monoid with unit one and if and for all . A triangle function τ is a binary operation on , which is commutative, associative and for every .
In , Gähler introduced the following concept of a 2-normed space.
if and only if , are linearly dependent,
is invariant under permutation,
, for any ,
is called a 2-norm on X and the pair is called a 2-normed space.
where for each .
Recently, Goleţ  used the idea of a 2-normed space to define a random 2-normed space.
Definition 3 Let X be a linear space of dimension (d may be infinite), τ a triangle, and . Then is called a probabilistic 2-norm and a probabilistic 2-normed space if the following conditions are satisfied:
() if x and y are linearly dependent, where denotes the value of at ,
() if x and y are linearly independent,
() , for all ,
() , for every , and ,
() , whenever .
If () is replaced by
() , for all and ;
then is called a random 2-normed space (for short, R2NS).
Remark 1 Every 2-normed space can be made a random 2-normed space in a natural way by setting for every , and , .
Then is a random 2-normed space.
Definition 4 A sequence in a random 2-normed space is said to be double convergent (or -convergent) to with respect to if for each , , there exists a positive integer such that , whenever and for nonzero . In this case we write , and ℓ is called the -limit of .
Definition 5 A sequence in a random 2-normed space is said to be double Cauchy with respect to if for each , there exist and such that , whenever and and for nonzero .
In this case we write , and ℓ is called the -limit of x. Let denote the set of all double statistically convergent sequences in a random 2-normed space .
In this article, we study λ-double statistical convergence in a random 2-normed space which is a new and interesting idea. We show that some properties of λ-double statistical convergence of real numbers also hold for sequences in random 2-normed spaces. We establish some relations related to double statistically convergent and λ-double statistically convergent sequences in random 2-normed spaces.
3 λ-double statistical convergence in a random 2-normed space
Recently, the concept of λ-double statistical convergence has been introduced and studied in  and . In this section, we define λ-double statistically convergent sequence in a random 2-normed space . Also we get some basic properties of this notion in a random 2-normed space.
where , and , is said to be the λ-double density of K, provided the limit exists.
In this case, we write .
Now we define λ-double statistical convergence in a random 2-normed space (see ).
Let denote the set of all λ-double statistically convergent sequences in a random 2-normed space .
If for every n, m then λ-double statistically convergent sequences in a random 2-normed space reduce to double statistically convergent sequences in a random 2-normed space .
Definition 9 immediately implies the following lemma.
Theorem 1 Letbe a random 2-normed space. Ifis a sequence in X such thatexists, then it is unique.
Proof Suppose that , where .
Let be given. Choose such that .
Since and , we have Lemma 1 and for all .
Now, let , then it is easy to observe that . But we have .
Since was arbitrary, we get for all and nonzero . Hence .
This completes the proof. □
Next theorem gives the algebraic characterization of λ-statistical convergence on random 2-normed spaces. We give it without proof.
If and , then .
If and , then .
Theorem 3 Letbe a random 2-normed space. Ifis a sequence in X such that, then.
has at most finitely many terms. Since every finite subset of has -density zero, finally we have . This shows that . □
Remark 2 The converse of the above theorem is not true in general. It follows from the following example.
This shows that , while it is obvious that .
Theorem 4 Letbe a random 2-normed space. Ifis a sequence in X, thenif and only if there exists a subsetsuch thatand.
Furthermore, implies that , which contradicts (3.1) as . Hence .
Thus . This completes the proof. □
We now have
Theorem 5 Letbe a random 2-normed space. Then a sequencein X is λ-double statistically convergent if and only if it is λ-double statistically Cauchy in random 2-normed space X.
Since , it follows that , and finally, .
then to prove the result it is sufficient to prove that .
which is not possible. Thus . Since , it follows that . This shows that is λ-double statistically Cauchy.
If , then .
i.e., , which contradicts (3.5) as . Hence is λ-double statistically convergent.
This completes the proof. □
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