On Cluster Points, Continuity, and Boundedness Associated with the Generalized Statistical Convergence in Probabilistic Normed Spaces

Das, Pratulananda; Dutta, Kaustubh; Karakaya, Vatan

doi:https://doi.org/10.1155/2014/909364

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2014 | Article ID 909364 | https://doi.org/10.1155/2014/909364

On Cluster Points, Continuity, and Boundedness Associated with the Generalized Statistical Convergence in Probabilistic Normed Spaces

Pratulananda Das,¹Kaustubh Dutta,¹and Vatan Karakaya²

Academic Editor: Csaba Varga

Received26 Oct 2013

Accepted06 Apr 2014

Published28 Apr 2014

Abstract

We consider the recently introduced notion of -statistical convergence (Das, Savas and Ghosal, Appl. Math. Lett., 24(9) (2011), 1509–1514, Savas and Das, Appl. Math. Lett. 24(6) (2011), 826–830) in probabilistic normed spaces and in the following (Şençimen and Pehlivan (2008 vol. 26, 2008 vol. 87, 2009)) we introduce the notions like strong -statistical cluster points and extremal limit points, and strong -statistical continuity and strong -statistical D-boundedness in probabilistic normed spaces and study some of their important properties.

1. Introduction

The idea of convergence of real sequences had been extended to statistical convergence by Fast [1] and basic ideas were further developed in [2–5]. 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 reals is said to be statistically convergent to a real number if, for each ,

The concepts of and -convergence, two important generalizations of statistical convergence, were introduced and investigated by Kostyrko et al. [6]. The ideas were based on the notion of ideal of . Subsequently, a lot of investigations have been done on ideal convergence (see [7–17] where many more references both on ideal as well as statistical convergence can be found). Very recently, ideals were used in a different way to generalize the notion of statistical convergence [18, 19] and certain new and summability methods were introduced and their basic properties were investigated. More recently these ideas were extended to double sequences in [20].

On the other hand, the idea of probabilistic metric space was first introduced by Menger [21] in the name of “statistical metric space.” Probabilistic normed space (briefly PN space) is a generalisation of an ordinary normed linear space. In a PN space, the norms of the vectors are represented by the distribution functions instead of nonnegative real numbers. Detailed theory of these spaces can be found in the famous book written by Schweizer and Sklar [22] and the monogram [23]. One can also see the papers [22, 24–35] where the basic ideas were established. 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 have very recently extended the notion of strong convergence to strong statistical convergence in probabilistic metric spaces [36] and carried out further investigations on statistical continuity and statistical -boundedness in PN spaces [37, 38]. These were followed by the studies of strong ideal convergence in PM and PN spaces in [10, 13, 39], studies of lacunary statistical convergence in PN spaces in [40]. As a natural extension, we had recently introduced the idea of strong -statistical convergence in PM spaces [41] and as a followup in this paper we investigate the notion of strong -statistical limit and cluster points in PN spaces. Further, we have introduced the concepts like strong -statistical continuity and strong -statistical -boundedness in such spaces and investigated some of their important properties.

2. Preliminaries

First, we recall some of the basic concepts related to the theory of probabilistic metric and normed spaces (see [22, 23, 31–39, 42] for more details).

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 . One considers the relation “” on defined by if and only if for all . It can be easily verified that the relation “” is a partially 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.

Theorem 6. Let be given. Then, for any , if and only if .

Note 2. 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 7. A triangular norm (briefly, a -norm) is a binary operation on the unit interval that is associative, commutative, nondecreasing in each place, and has 1 as identity. The operations defined by and are particular -norms. Given a -norm , its -conorm is defined as a mapping on by .

Definition 8. A triangle function is a binary operation on , , which is commutative, associative, and nondecreasing in each place, and has as identity. Triangle functions can be constructed through left-continuous -norms. If is such a -norm, then is a triangle function, where . If, moreover, is continuous, then is uniformly continuous on . If is a continuous -conorm, then is a triangle function which is uniformly continuous on .

Definition 9. A probabilistic metric space (briefly a PM space) is a triplet where is a nonempty set, is a function from into , and is a triangle function. The following conditions are satisfied for all :(PM1);(PM2) if ;(PM3);(PM4).
In the sequel, we will denote by and its value at by .

Definition 10. A probabilistic normed space (briefly a PN space) is a quadruple , where is a real linear space, and are continuous triangle functions with , and is a mapping (the probabilistic norm) from into the space of distribution functions such that, writing for for all , in , the following conditions hold:(N1) if and only if , the null vector in ;(N2);(N3);(N4) for all .
A Menger PN space under is a PN space in which and for some continuous -norm and its -conorm . It is denoted by .
Throughout the text, will represent the PN space .

Theorem 11. Let be a probabilistic normed space and let be the function from to defined by Then, is a probabilistic metric space (briefly PM space).

Definition 12. Let be a PN space. For and , the strong -neighbourhood of is defined as the set
Since is continuous, strong neighbourhood system that determines a Hausdorff and first countable topology for . This topology is called the strong topology for .

Remark 13. Throughout the rest of this paper, we always assume that in a PN space the triangle function is continuous and is endowed with strong topology.

Definition 14. A sequence in the PN space is said to be strongly convergent to a point in and one writes or if for any there exists a natural number such that whenever .

Definition 15. Given a nonempty set in the PN space , its probabilistic radius is defined by where denotes the left limit of the function at the point and .

Definition 16. A nonempty set in a PN space is said to be(1)certainly bounded if for some ;(2)perhaps bounded if for every and ;(3)perhaps unbounded if for some and ;(4)certainly unbounded if that is, if .
Moreover, is said to be distributionally bounded (-bounded) if either (1) or (2) holds; that is, if ; otherwise, if , then A is said to be -unbounded.

In the following, we now recall some of the basic concepts related to ideals.

Definition 17. Let be any nonempty set. A nonempty family is called an ideal in if(1) implies ;(2) and imply .

Definition 18. Let be any nonempty set. A nonempty family is called a filter in if(1);(2) implies ;(3) and imply .

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 .

Definition 19. 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 .

Definition 20 (see [18]). A sequence of real numbers is said to be -statistically convergent to if, for each and , In this case, we write .

Definition 21 (see [41]). 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 .

Definition 22 (see [41]). 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 ,

3. Strong -Statistical Limit Points and Strong -Statistical Cluster Points in Probabilistic Normed Spaces

In this section, we extend the notions of strong statistical limit points and strong statistical cluster points in PN spaces using ideals. Let be a PN space.

Definition 23 (see [36]). Let be a sequence in . We say that a point is a strong limit point of provided that there exists a subsequence of that strongly converges to . We denote the set of all strong limit points of by .

Definition 24. Let be a sequence in and let be a subsequence of . Denote. If, for all , then we say that is an -statistical thin subsequence of . If, for some , then is called an -statistical nonthin subsequence of .

In this sequel, we will abbreviate the subsequence of as , where .

Definition 25. Let be a sequence in . An element is a strong -statistical limit point of provided that there exists a set such that, for some , and the subsequence strongly converges to . We denote the set of all strong -statistical limit points of by .

Definition 26. Let be a sequence in . An element is a strong -statistical cluster point of provided that for every there is a such that We denote the set of all strong -statistical cluster points of by .

Theorem 27. For any sequence in , one has .

Proof. Assume that . Then, there exists a set such that, for some , say , and the subsequence of strongly converges to . Now, for every , . Since strongly, the set is finite for every . Let be given. If possible let, for every , . Then, we have where and . Clearly, . Now, choose small enough and in such a way that . Then, By our assumption, the set on the right-hand side belongs to and so , which is a contradiction. This implies that there is a such that . Therefore, .
Next, let . Then, for every , there is a such that . This means that there are infinitely many terms of the sequence in every strong -neighbourhood of ; that is,. This completes the proof.

Theorem 28. Let be a sequence in . If , then .

Proof. Let . Then, for all and . Clearly, . Therefore, . Now, assume that there exists at least one such that . Since the strong topology is Hausdorff, we can choose , such that Hence, we get, for any , Since , the set on the right-hand side belongs to . This implies that, for the chosen , for every . This contradicts the fact that . Therefore, we have .

Theorem 29. For any sequence in , the set of strong -statistical cluster points of is strongly closed.

Proof. Let , where denotes the strong closure of the set (see [22]). Choose . Then, . Let . Choose in such a way that . Since , there exists a such that . Since , it follows that As , consequently ≤ . Hence, ; that is,. This proves that is strongly closed.

Theorem 30. If and are two sequences in and there exists a set such that and , for all , then and .

Proof. Assume that . Let be an -statistical nonthin subsequence of that strongly converges to , where . Since , . But = + . Hence, ≤ ≤ . Since the set on the left-hand side does not belong to whereas the second set on the right-hand side belongs to , the first set on the right-hand side cannot belong to . This shows that is a -statistical nonthin subsequence of that strongly converges to . Hence, and so . Similarly, we can prove that . Hence, .
The second assertion can be similarly proved.

4. Strong -Statistical Continuity in Probabilistic Normed Spaces

In this section, we introduce the notion of the strong -statistical continuity and investigate the same for a probabilistic norm, vector addition operation, and scalar multiplication.

Definition 31. Let and be two probabilistic normed spaces. A function is said to be strongly -statistically continuous at a point if implies that .
If is strongly -statistically continuous at each point of a set , then is said to be strongly -statistically continuous on .

Theorem 32. Let be a PN space. Let be endowed with the strong topology and let be endowed with the -metric topology. Then, is a strongly -statistically continuous mapping from to .

Proof. It is known that the probabilistic norm is a uniformly continuous mapping from to ; that is, for any , there exists a such that whenever . Now, let be a sequence in such that . Then, we have, for each , . Clearly, for all and , Since , the set on the right-hand side belongs to . Consequently, for any , . Hence, by definition, we have . This means that is a strongly -statistically continuous mapping.

Theorem 33. Let be a PN space. Let be endowed with the strong topology and let be endowed with the -metric topology. Also assume that is endowed with the corresponding product topology. Then, vector addition is a strongly -statistically continuous mapping from to .

Proof. Let and be two sequences in such that and . As , for every . Again since continuity of implies its uniform continuity, it follows that for any there is a such that whenever and , where . Now, let . Then, we can find a such that whenever and . Hence, whenever and . Thus, we have, for all , Therefore, we have, for all and , Since and , each set on the right-hand side belongs to and so their union also belongs to . Therefore, we get, for each and , . This shows that which completes the proof.

Corollary 34. Let be a PN space. The mapping from to defined as for any is strongly -statistically continuous.

Proof. Proof of this result immediately follows from Theorems 32 and 33.

We now investigate the strong -statistical continuity properties of scalar multiplication given by for all and .

Lemma 35 (see [37]). For any , , and , there exists a such that whenever .

Theorem 36. The mapping is strongly -statistically continuous in its second place; that is, for a fixed , scalar multiplication is a strongly -statistically continuous mapping from to .

Proof. Let be fixed and let be a sequence in such that . Then, by Lemma 35, for any , we can find a such that . Therefore, for any , Since , the set on the right-hand side belongs to . Therefore, for any and , . Hence, .

However, in general, the mapping needs not to be strongly -statistically continuous in its first place.

Example 37. Let be the real line viewed as a one-dimensional linear space and let and , where and are the continuous triangle functions defined by For , define by setting and Clearly, is a PN space. Choose an infinite set . Now, consider the real sequence defined by It can be easily shown that but . This example shows that the mapping from into defined by is not strongly -statistically continuous for any fixed ; that is, the mapping is not strongly -statistically continuous in its first place.

A triangle function is called Archimedean if admits no idempotents other than and . More details on Archimedean triangle function can be found in the book [22]. If is Archimedean, then we can establish the following lemmas.

Lemma 38 (see [37]). If is Archimedean, then, for any such that and any , there exists a such that whenever .

Theorem 39. If is PN space such that is Archimedean and if for every , then for any fixed the mapping is strongly -statistically continuous in its first place.

Proof. Let be fixed and let be a real sequence such that . Let be given. By Lemma 38, we can find a such that whenever . In particular, implies that . Therefore, . It now readily follows that, for all , Since , the set on the right-hand side belongs to and, consequently, for any and , . Therefore, as desired.

The following lemmas will be needed to prove our next result.

Lemma 40 (see [37]). If , then for any .

Lemma 41 (see [37]). Let be a continuous triangle function and let be the set of all triplets in such that and . Then, for any , there exists a such that if is in and , then .

Theorem 42. Let be a PN space such that is Archimedean and for all . Then, the scalar multiplication is a jointly strong -statistically continuous mapping from endowed with the natural product topology onto . Furthermore, the mapping given by for any and any is also jointly strong -statistically continuous.

Proof. Let be a sequence in such that and let be a real sequence such that . Consider the set . Since , we have, for every , ; that is, . Now, if , then . From the properties of probabilistic norm, we have . Thus, if , then we have . Next, let . Since is uniformly continuous, we can find a such that whenever and . For such a , consider the sets If , then clearly . Hence, ≥ ≥ ≥ . Consequently, we can write ≥ ≥ ≤ ≥ ≤ . Then, for any , Clearly, all the three sets on the right-hand side of the expression belong to . Therefore, for any , which shows that . This completes the proof of the first part of the theorem.
Let us now show that the mapping is jointly strong -statistically continuous. Assume that and . Then, we have . Therefore, . Now, we can write, for every , and . By Lemma 41, we can say that for any there exists a such that whenever . Now, using the argument similar to that of the preceding proof, we obtain . Hence, it follows that is jointly strong -statistically continuous.

5. Strong -Statistically -Bounded Sequences in Probabilistic Normed Spaces

Definition 43 (see [38]). Let be a PN space. A sequence in is statistically -bounded provided that there exists a set with such that is -bounded.

In this section, we generalize the above definition for sequences in a PN space and introduce the concept of a strongly -statistically -bounded sequence.

Definition 44. Let be a PN space. A sequence in is strongly -statistically -bounded provided that there exists a set such that for any and is -bounded.

Clearly, in this case, . Note that a -bounded sequence is always strongly -statistically -bounded, but the converse is not generally true.

Theorem 45. A sequence in the PN space is strongly -statistically -bounded if and only if there exists a set with for any and a distribution function such that .

Proof. The proof of the theorem immediately follows from Theorem 2.1 of [29] and Definition 44.

Example 46. Let us consider the simple space , where denotes the usual norm on ; , , and the probabilistic norm is given by and, for , , This space is called Menger PN space under where is the -norm defined by . Now, assume that there is a such that and .
Next, assume that is a decomposition of (i.e., for ) where are infinite sets defined as . Denote by the class of all such that intersects only a finite number of . It can be easily verified that is an admissible ideal. Note that for all . Now, for every , there exists a such that . Therefore, (say). Clearly, . We define a sequence in the PN space by The subsequence , where stands for the complement, is certainly bounded and hence -bounded in . Moreover, the subsequence is certainly unbounded in . Therefore, the sequence is strongly -statistically -bounded, but it is not statistically -bounded as .

We now present certain results which are modifications of similar results proved for statistically -bounded sequences [38]. The proofs of these results are parallel to the corresponding results of [38] with necessary modifications.

Theorem 47. If is a strongly -statistically -bounded sequence in the PN space , then there exists a -bounded sequence and a strongly -statistically null sequence such that for all .

Proof. Let be a strongly -statistically -bounded sequence in the PN space. Then, there exists a set with , for any , such that is -bounded. Now, define We have, for each and , Since the set on the right-hand side belongs to , . Thus, . The sequence is clearly -bounded. It is easy to see that , where is -bounded and .

Theorem 48. Let be a PN space in which and is invariant under ; that is,. If is a sequence in such that , then is strongly -statistically -bounded.

Proof. Let . Then, for each , there exists a set with , for any , such that . Therefore, for all . By our assumption, we have and now the result follows from Theorem 45.

Theorem 49. Let be a PN space in which and is invariant under . If is strongly -statistically Cauchy sequence in , then it is strongly -statistically -bounded.

Proof. Let be strongly -statistically Cauchy. Then, for every , there exists a set for which for any and a , such that . Let be given, and then we can find a such that . By our assumption, we have and the rest follows from Theorem 45.

Theorem 50. Let be a strongly -statistically -bounded sequence in the PN space . If is invariant under , then the sequence is strongly -statistically -bounded for every fixed .

Proof. By axiom (N2), it is sufficient to consider the case . If or , then is strongly -statistically -bounded. Let . Since is strongly -statistically -bounded, there exists a set such that for any and . Thus, we have which means that , where . Therefore, is strongly -statistically -bounded. Now, let and set . Then, , where . Hence, which completes the proof.

Theorem 51. Let and be two strongly -statistically -bounded sequences in the PN space . If is invariant under , then the sequence is also strongly -statistically -bounded.

Proof. Let and be two strongly -statistically -bounded sequences in the PN space . Then, there exist sets and such that for any , and and . Now, consider the set . Obviously, . We also have . Since both sets on the right-hand side belong to , then . We observe that, for , and therefore we can write , where and . By our assumption . Thus, for all and for all . Thus, the sequence yields a -bounded subset of . This completes the proof.

Corollary 52. Let be a PN space. If is invariant under , that is,, then the set of all strongly -statistically -bounded sequences in forms a real linear space.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are thankful to the Council of Scientific and Industrial Research, HRDG, India, for granting the research Project no. 25(0186)/10/EMR-II during the tenure of which this work was done.

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Copyright © 2014 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.

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