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Ideal convergence of double sequences in random 2-normed spaces
Advances in Difference Equations volume 2012, Article number: 149 (2012)
Quite recently, Alotaibi and Mohiuddine (Adv. Differ. Equ. 2012:39, 2012) studied the idea of a random 2-normed space to determine some stability results concerning the cubic functional equation. In this paper, we define and study the concepts of -convergence and -convergence for double sequences in random 2-normed spaces and establish the relationship between these types of convergence, i.e., we show that -convergence implies -convergence in random 2-normed spaces. Furthermore, we have also demonstrated through an example that, in general, -convergence does not imply -convergence in random 2-normed spaces.
Menger  generalized the metric axioms by associating a distribution function with each pair of points of a set. This system, called a probabilistic metric space, originally a statistical metric space, has been developed extensively by Schweizer and Sklar . The idea of Menger was to use distribution functions instead of nonnegative real numbers. An important family of probabilistic metric spaces are probabilistic normed spaces. Probabilistic normed spaces are real linear spaces in which the norm of each vector is an appropriate probability distribution function rather than a number. Such spaces were first introduced by S̆erstnev in 1963 . In , Alsina et al. gave a new definition of PN spaces that includes S̆erstnev’s and leads naturally to the identification of the principal class of PN spaces, the Menger spaces. Recently, the concept of probabilistic normed spaces was extended to random/probabilistic 2-normed spaces by Golet  using the concept of 2-norm of Gähler .
The concept of statistical convergence for sequences of real numbers was introduced by Fast  in 1951, and since then several generalizations and applications of this notion have been investigated by various authors (see [8–21]). The notion of statistical convergence is a very useful functional tool for studying the convergence problems of numerical sequences/matrices (double sequences) through the concept of density. The concept of I-convergence, which is a generalization of statistical convergence, was introduced by Kastyrko, Salat and Wilczynski  by using the ideal I of subsets of the set of natural numbers and further studied in [23–27] and references therein. Recently, Şahiner et al., and Gürdal and Acik  studied ideal convergence and I-Cauchy sequences respectively in 2-normed spaces, and also Gürdal and Pehlivan  studied statistical convergence in a 2-Banach space. Most recently, Alotaibi and Mohiuddine  determined the stability of a cubic functional equation in a random 2-normed space.
In this paper, we study the concept of -convergence and -convergence in a more general setting, i.e., in a random 2-normed space. We discuss the relationship between -convergence and -convergence and prove some interesting results.
2 Definitions, notations and preliminary results
In this section, we recall some basic definitions and notations which form the background of the present work.
A distribution function is an element of , where , and the subset is the set . Here denotes the left limit of the function f at the point x. The space is partially ordered by the usual pointwise ordering of functions, i.e., if and only if for all . For any , is a distribution function defined by
The set Δ, as well as its subsets, can be partially ordered by the usual pointwise order: in this order, is the maximal element in .
A triangle function is a binary operation on , namely a function , that is associative, commutative, nondecreasing, and has as unit; that is, for all , we have:
A t-norm is a binary operation such that for all we have:
∗ is associative and commutative, (ii) ∗ is continuous, (iii) , (iv) whenever and .
The concept of a 2-normed space was first introduced by Gähler .
Let X is a linear space of a dimension d, where . A 2-norm on X is a function satisfying the following conditions: for every , (i) if and only if x and y are linearly dependent; (ii) ; (iii) , ; (iv) . In this case is called a 2-normed space.
Example 2.1 Take being equipped with the 2-norm = the area of the parallelogram spanned by the vectors x and y, which may be given explicitly by the formula
Recently, Goleţ  introduced the notion of a random 2-normed space as follows.
Let X be a linear space of a dimension greater than one, τ a triangle function, and . Then is called a probabilistic 2-norm on X 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 every x, y in X,
for every , and ,
If (v) is replaced by
(v′) , for all and ,
then triple is called a random 2-normed space (for short, RTN-space).
Example 2.2 Let be a 2-normed space with , , and for . For all , and nonzero , consider
Then is a random 2-normed space.
Remark 2.3 Note that every 2-normed space can be made a random 2-normed space in a natural way, by setting , for every , and , .
In 1900, Pringsheim  introduced the notion of convergence of double sequences as follows: A double sequence is said to converge to the limit L in Pringsheim’s sense (shortly, P-convergent to L) if for every there exists an integer N such that whenever . In this case L is called the P-limit of x.
Let be a two-dimensional set of positive integers and let . Then the two-dimensional analogue of natural density can be defined as follows.
In case the sequence has a limit in Pringsheim’s sense, then we say that K has a double natural density and is defined as
For example, let . Then
i.e., the set K has double natural density zero, while the set has double natural density .
A real double sequence is said to be statistically convergent to the number ℓ if for each the set
has double natural density zero. In this case we write and denote the set of all statistically convergent double sequences.
If X is a non-empty set, then a family of subsets of X is called an ideal in X if and only if
for each and we have ,
I is called a nontrivial ideal if and is the power set of X.
Let X be a non-empty set. A non-empty family of sets is called a filter on X if and only if
for each and we have .
A nontrivial ideal I in X is called an admissible ideal if it is different from and contains all singletons, i.e., for each .
Let be a nontrivial ideal. Then a class is a filter on X, called the filter associated with the ideal I.
An admissible ideal is said to satisfy the condition (AP) if for every sequence of pairwise disjoint sets from I there are sets , such that the symmetric difference is a finite set for every n and .
Let I be a nontrivial ideal of . A sequence is said to be I-convergent to if, for each , the set . In this case we write .
3 Main results
In this section, we study the concept of -convergence and -convergence of double sequences in a random 2-normed space. We shall assume throughout this paper that as a nontrivial ideal in and is a random 2-normed space.
Definition 3.1 A double sequence is convergent in or simply -convergent to ℓ if, for every , , there exists a positive integer N such that whenever and nonzero . In this case we write and ℓ is called the -limit of .
Definition 3.2 Let a be nontrivial ideal of . A double sequence is said to be -convergent in or simply -convergent to ℓ if for every , and nonzero , we have , where
In this case we write .
Definition 3.3 We say that a double sequence is said to be -convergent in or simply -convergent to ℓ if there exists a subset of such that (i.e., ) and . In this case we write and ℓ is called the -limit of the double sequence .
Theorem 3.4 If a double sequenceis-convergent, then-limit is unique.
Proof Suppose that and . Given , choose such that . Then, for any and nonzero , define the following sets as:
Since , we have . Similarly implies that . Now let . Then from the definition of , , and hence its compliment is a non-empty set which belongs to . Now, if , then we have
Since was arbitrary, we get for all and nonzero . Hence .
This completes the proof of the theorem. □
Theorem 3.5 Letbe an admissible ideal. If a double sequenceis-convergent to ℓ, then it is-convergent to the same limit. But the converse need not be true.
Proof Let . Then for every , and nonzero , there is a positive integer N such that
for all . Since the set
is contained in , where and the ideal is admissible, . Hence .
For the converse, we construct the following example.
Example 3.6 Let , with the 2-norm , , , and for all . Let for all , . Now we define a double sequence by
We see that
Taking limit , we get
Hence, a double sequence is not convergent in . But if we take , then since , , that is, .
This completes the proof of the theorem. □
Theorem 3.7 Letbe an admissible ideal andbe a double sequence. Ifthen.
Proof Suppose that . Then there exists a subset of such that (i.e., ) and . But, for each and there exists such that for all . Since is contained in and the ideal is admissible, we have
for all and . Therefore, we conclude that .
This completes the proof of the theorem. □
The following example shows that the converse of the above theorem need not be true.
Example 3.8 Let , with the 2-norm , , , and for all . Define for all and . Then is a RTN-space. Let be a decomposition of such that, for any each contains infinitely many ’s, where , and for . Now we define a double sequence by if . Then as . Hence .
On the other hand, suppose that . Then there exists a subset of such that and . Since , there is a set such that . Now, from the definition of , there exists say such that
But , therefore for infinitely many ’s from K which contradicts . Therefore, the assumption leads to the contradiction. Hence the converse of the Theorem 3.7 need not be true.
Now the question arises under what condition the converse may hold. The following theorem shows that the converse holds if the ideal satisfies the condition (AP).
An admissible ideal is said to satisfy the condition (AP) if, for every sequence of pairwise disjoint sets from , there are sets , such that the symmetric difference is a finite set for every n and .
Theorem 3.9 Suppose the idealsatisfies the condition (AP). Ifis a double sequence in X such that, then.
Proof Suppose satisfies the condition (AP) and . Then, for each , and nonzero , we have
We define the set for , and as
Obviously, is countable and belongs to , and for . By the condition (AP), there is a countable family of sets such that the symmetric difference is a finite set for each and . From the definition of the associated filter there is a set such that . To prove the theorem it is sufficient to show that the subsequence is convergent to ℓ in . Let and . Choose such that . Then
Since , are finite, there exists such that
If , and , then . Therefore by (3.1), we have . Hence, for every , and , we have . Since s was arbitrary, we have .
This completes the proof of the theorem. □
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The authors declare that they have no competing interests.
The authors contributed equally and significantly in writing this paper. Both the authors read and approved the final manuscript.
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Mohiuddine, S.A., Alotaibi, A. & Alsulami, S.M. Ideal convergence of double sequences in random 2-normed spaces. Adv Differ Equ 2012, 149 (2012). https://doi.org/10.1186/1687-1847-2012-149
- double sequences
- random 2-normed spaces
- ideal convergence