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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 765060, 8 pages
On Some Further Generalizations of Strong Convergence in Probabilistic Metric Spaces Using Ideals
1Department of Mathematics, Jadavpur University, Kolkata 700032, India
2Department of Mathematical Engineering, Yıldız Technical University, Esenler, Istanbul 34210, Turkey
3Department of Mathematics, Kalyani Government Engineering College, Kalyani, Nadia 741235, India
Received 21 June 2013; Accepted 5 September 2013
Academic Editor: Sung Guen Kim
Copyright © 2013 Pratulananda Das 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.
Following the line of (Das et al., 2011, Savas and Das, 2011), we make a new approach in this paper to extend the notion of strong convergence and more general strong statistical convergence (Şençimen and Pehlivan, 2008) using ideals and introduce the notion of strong - and -statistical convergence and two related concepts, namely, strong -lacunary statistical convergence and strong --statistical convergence in a probabilistic metric space endowed with strong topology. We mainly investigate their interrelationship and study some of their important properties.
1. Introduction and Background
The usual idea of convergence is not enough to understand behaviours of those sequences which are not convergent. One of the approaches to include more sequences under purview is to consider those sequences which are convergent when restricted to some “big set of natural numbers.” To accomplish this, the idea of convergence of real sequences was extended to statistical convergence by Fast , and it was further developed by several authors [2–4]. Recall that “asymptotic density” of a set is defined as provided that the limit exists, where denotes the set of natural numbers and the vertical bar stands for cardinality of the enclosed set. The sequence of real is said to be statistically convergent to a real number if for each ,
In another direction, a new type of convergence called lacunary statistical convergence was introduced and studied in . More results related to this convergence can be found in . The concept of -statistical convergence was introduced by Mursaleen in  as a further extension of statistical convergence. Afterward, in , Karakaya et al. defined statistical convergence of sequences of functions in intuitionistic fuzzy normed spaces and also used the concepts of lacunary and -statistical convergence for sequences of functions in these spaces in [9, 10]. It must be mentioned in this context that some of the above mentioned convergence methods have applications in number theory, measure theory, fourier analysis, optimization, and many branches of mathematics.
The concepts of and -convergence were introduced and investigated by Kostyrko et al.  as further generalizations of statistical convergence. They were much more general than other approaches as they were based on the very general notion of ideals of . In recent years, a lot of investigations have been done on ideal convergence and in some particular new approaches were made in [12, 13] to generalize the above mentioned convergences (for works on ideal convergence, see e.g., the papers [12–17] where many more references can be found).
On the other hand, the idea of probabilistic metric space was first introduced by Menger  in the name of “statistical metric space.” In this theory, the concept of distance is probabilistic rather than deterministic. More precisely, the distance between two points , is defined as a distribution function instead of a nonnegative real number. For a positive number , is interpreted as the probability that the distance between the points and is less than . The theory of probabilistic metric spaces was brought to prominence by path breaking works of Schweizer et al. [19–22] and Tardiff  among others. Detailed theory of probabilistic metric space can be found in the famous book written by Schweizer and Sklar . Several topologies can be defined on this space, but the topology that was found to be most useful is the “strong topology.” Şençimen and Pehlivan  noted in their paper that as the strong topology is first countable and Hausdorff, it can be completely specified in terms of strong convergence of sequences. Since probabilistic metric spaces have many applications in applied mathematics, in order to provide a more general framework for applications, they have recently studied the statistical convergence and then strong ideal convergence in probabilistic metric spaces [25, 26] and also carried out further investigations in [27, 28].
In this paper, first of all, we introduce the notion of strong -statistical convergence in probabilistic metric spaces which happens to be more general than strong statistical convergence. We also introduce the concepts of strong -lacunary statistical convergence and strong --statistical convergence in probabilistic metric spaces and investigate some of their important properties, in particular their relations with strong -statistical convergence. As these concepts are more general than the concept of strong statistical convergence, we believe that they can extend the general framework introduced by  which in the future may enhance the applicability of strong convergence in probabilistic metric spaces.
First, we recall some basic concepts related to the probabilistic metric spaces (in short PM spaces) (see ).
Definition 1. A nondecreasing function defined on with and , where , is called a distribution function.
The set of all left continuous distribution functions over is denoted by .
We consider the relation “≤” on defined by if and only if for all . It can be easily verified that the relation “≤” is a partial order on .
Definition 2. For any , the unit step function at is denoted by and is defined to be a function in given by
Definition 3. A sequence of distribution functions converges weakly to a distribution function , and one writes if and only if the sequence converges to at each continuity point of .
Definition 4. The distance between and in is denoted by and is defined as the infimum of all numbers such that the inequalities hold for every .
Here, we are interested in the subset of consisting of those elements that satisfy .
Definition 5. A distance distribution function is a nondecreasing function defined on that satisfies and and is left continuous on .
The set of all distance distribution functions is denoted by .
The function is clearly a metric on . The metric space is compact and hence complete (see ).
Theorem 6 (see ). Let be given. Then, for any , if and only if . >
Note 7. Geometrically, is the abscissa of the point of intersection of the line and the graph of (if necessary we add vertical line segment at the point of discontinuity).
Definition 8. A triangle function is a binary operation on , which is commutative, associative, and nondecreasing in each place and has as identity.
Definition 9. A PM space is a triplet where is a nonempty set, is a function from into , and is a triangle function. The following conditions for a PM space are satisfied for all :(1),(2) if ,(3),(4).
In the sequel, we shall denote by and its value at by . Throughout this paper shall represent the PM space .
Definition 10. Let be a PM space. For and , the strong -neighbourhood of is defined as the set
The collection is called the strong neighbourhood system at , and the union is called the strong neighbourhood system for .
By Theorem 6, we can write
If is continuous then the strong neighbourhood system determines a Hausdorff topology for . This topology is called the strong topology for .
Definition 11. Let be a PM space. Then, for any the subset of is given by and it is called the strong -vicinity.
Theorem 12 (see ). Let be a PM space, and, be continuous. Then, for any , there exists a such that , where .
Note 13. Under the hypothesis of Theorem 12, we can say that for any there is a such that whenever and . Equivalently, it can be written as follows: for any there is an such that , whenever and .
If is continuous in a PM space , then the strong neighbourhood system determines a Kuratowski closure operation. It is termed as the strong closure. For any subset of , the strong closure of is denoted by and is defined as
Remark 14. Throughout the rest of the paper, we always assume that in a PM space , the triangle function is continuous and is endowed with strong topology.
Definition 15. Let be a PM space. A sequence in is said to be strongly convergent to a point if for any there exists a natural number such that whenever . One writes or .
Similarly, a sequence in is called a strong Cauchy sequence if for any there exists a natural number such that whenever .
Next, we recall some of the basic concepts related to the theory of -convergence, and we refer to  for more details.
Definition 16. Let be any nonempty set. Then, the family is called an ideal in if(1) imply ,(2)if and then .
Definition 17. Let be any nonempty set. The family is called a filter in if(1),(2) imply ,(3)if and then .
If is an ideal in , then is a filter in , which is called the filter associated with the ideal . An ideal in is called proper if and only if . is called nontrivial if . An ideal is called an admissible ideal if it is proper and contains for all . In other words, it is called an admissible ideal if it is proper and contains all of its finite subsets.
Definition 18. An admissible ideal is said to satisfy the condition (AP) if for every countable family of mutually disjoint sets belonging to there exists a countable family of sets such that is a finite set for every and .
Throughout the paper stands for a nontrivial admissible ideal of , and is the filter associated with the ideal of .
3. Strong - and -Statistical Convergence in PM Space
In this section, we extend the concept of strong statistical convergence in PM spaces  via ideals and prove some associated results.
Definition 19. A sequence in a PM space is said to be strong statistically convergent to in if for ,
Definition 20. A sequence of real numbers is said to be strong -statistically convergent to if for each and , In this case, we write .
We now introduce the definition of strong -statistical convergence in PM space. So, consider the following .
Definition 21. A sequence in a PM space is said to be strong -statistically convergent to if for each and , In this case, we write and the class of all strong -statistically convergent sequences is simply denoted by .
Theorem 22. Let be a PM space, and, let be continuous. Then, the strong -statistical limit of a sequence in is unique.
Proof. If possible, let the sequence converge to two different limits and in . Since , we have and so there exists a positive real number such that . Choose such that and imply that . Since and for this and , Clearly, . Let . Hence, and consequently, Then, there exists some such that and , that is, and . Combining these two, we have , which is a contradiction. Hence, the limit is unique.
Theorem 23. Let be a PM space. If and are two sequences in such that and , then one has .
Proof. For there exists a such that whenever and . Now, assume that and . Then, for any , we have Clearly, Since and , we have Therefore, union of these two sets also belongs to , and consequently . This completes the proof.
Definition 24. A sequence in a PM space is said to be -statistically convergent to if there exists a set such that for all , In this case, we write .
Theorem 25. If a sequence in a PM space is -statistically convergent, then it is strong -statistically convergent. Moreover, if is an admissible ideal satisfying the property (AP), then strong -statistical convergence implies -statistical convergence for any sequence in .
Proof. Let in be a sequence such that . Then, there exists a set such that
that is, for there exists a natural number such that for all . Now, we have
where . Since is admissible, the set on the right hand side belongs to and consequently . Hence, .
Next, suppose that . Clearly, the sequence is -convergent to zero. Since the ideal has the property (AP), the sequence is -convergent to zero. Therefore, .
Definition 26. A sequence in a PM space is said to be strong -statistically Cauchy if for every , there exists a positive integer such that for any ,
Theorem 27. In a PM space every strong -statistically convergent sequence is strong -statistically Cauchy.
Proof. Since , then for any and , It follows that Then, there exists a such that . Therefore, , and so Consequently, This completes the proof.
Definition 28. A sequence in is said to be -statistically Cauchy if for every there exists and a set such that
Theorem 29. In a PM space , every -statistically Cauchy sequence is strong -statistically Cauchy. Moreover, if is an admissible ideal satisfying the property (AP), then strong -statistically Cauchy sequence coincides with -statistically Cauchy sequence.
Proof. The proof is straightforward, and so it has been omitted.
4. Strong -Lacunary Statistical Convergence in PM Space
In this section, we discuss some of the results associated with the lacunary statistical convergence and extend certain summability methods using this notion.
By a lacunary sequence, we mean an increasing integer sequence such that and as . Throughout this paper, the interval determined by shall be denoted by , and the ratio shall be denoted by . The lacunary sequence is said to be the lacunary refinement of the lacunary sequence if .
Definition 30. Let be a lacunary sequence. A sequence in a PM space is said to be strong -lacunary statistically convergent to if for every and , In this case, we write . The class of all strong -lacunary statistically convergent sequences is simply denoted by .
Theorem 31. In a PM space , the strong -lacunary statistical limit of a sequence is unique.
Proof. It is similar to the proof of Theorem 22, and therefore it has been omitted.
Theorem 32. For a sequence in a PM space , the following conditions are equivalent:(1) is strong -lacunary statistically convergent to ;(2)for all ,
Proof. : First suppose that . Since for all then for , we have Consequently, we have : Next suppose that condition (2) holds. Now, for , we have Therefore, Then, for any , we have Therefore, . This completes the proof.
Theorem 33. For any lacunary sequence , strong -statistical convergence in a PM space implies strong -lacunary statistical convergence if and only if .
Proof. Suppose first that . Then, there exists a such that for sufficiently large . This implies that . Since , for every and for sufficiently large , we have Then, for any , we have This proves the result.
Let be an admissible ideal of , and let be any function. Define . We can easily show that is also an ideal of . First, we note that and so . If , then which implies that . Also if and , then and which in turn implies that . Therefore, is an ideal of .
Theorem 34. Let be a lacunary refinement of the lacunary sequence . Let and , . If there exists a such that then implies that .
Proof. Let . Therefore, For and for every , we can find such that . Thus, we obtain For , we have Since , there exists some such that Therefore, This completes the proof.
5. Strong --Statistical Convergence in PM Space
In this section, we introduce the concepts of strong --statistical convergence and -summability in a PM space.
Let be a nondecreasing sequence of positive numbers tending to such that , . The collection of all such sequences is denoted by . The generalised de la Vallée-Poussin mean is defined for the sequence of reals by where . A sequence of reals is said to be summable to if -lim , that is, for , [13, 31].
We now introduce the main definitions for this section.
Definition 35. Any sequence in a PM space is said to be strong --statistically convergent to if for every and , In this case, we write . The collection of all such sequences is simply denoted by .
Definition 36. Any sequence in a PM space is said to be -summable to in if for every , In this case, we write . The collection of all such sequences is simply denoted by .
Theorem 37. For any sequence in a PM space , the following conditions are equivalent.(1)The sequence is strong --statistically convergent to in .(2)The sequence is summable to in .
Proof. It is similar to the proof of Theorem 32, and therefore it has been omitted.
Example 38. Consider that with the usual metric and . Clearly, . We define by for all . Let us define for and . Clearly, is a PM space where is the continuous triangle function. Recall that for any , if and only if .
Let be an admissible ideal of and . Take a fixed and define a sequence in by where is a fixed element in with . Clearly, . Now, for every , there exists a such that for all . Therefore, for every , as and . Consequently, for every , for some . Since is admissible, the expression on the right hand side belongs to . It follows that .
Theorem 39. If , then .
Proof. For , we have If , then from definition is finite. Thus, for , Since is admissible, the set on the right hand side belongs to , and this completes the proof.
Theorem 40. Let be such that . Then, .
Proof. Let be given. Since , we can choose such that for all. Now, observe that for , for all . Hence, If , then the set on the right hand side belongs to and so the left hand side also belongs to . This shows that .
Remark 41. Consider the sequence where for and for all . Construct the sequence as in Example 38. For (the ideal of density zero sets in and , the sequence is an example of a sequence which is strong -statistically convergent but is not statistically convergent in a PM space.
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