Research Article | Open Access
On Lacunary Ideal Convergence in Random -Normed Space
We introduce the notions of lacunary -convergence and lacunary -Cauchy in the topology induced by random -normed spaces and prove some important results.
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 [2, 3]. The idea of Menger was to use distribution function instead of nonnegative real numbers as values of the metric, which was further developed by several other authors. 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 . Using this concept, Šerstnev  introduced the concept of probabilistic normed spaces. It provides an important area into which many deterministic results of linear normed spaces can be generalized. The studies of continuity properties, linear operators, statistical convergence, and ideal convergence in probabilistic normed spaces have gained many attractions, and such studies have diverse applications into various fields [5–13].
Since the introduction of the notion of statistical convergence for sequences of real numbers by Steinhaus  and Fast  independently, the theory has been investigated and developed by several authors. There has been an effort to introduce several generalizations and variants of statistical convergence in different spaces. One very important generalization of this notion was introduced by Kostyrko et al.  by using an ideal of subsets of the set of natural numbers, which they called -convergence.
Another important variant of statistical convergence is the notion of lacunary statistical convergence introduced by Fridy and Orhan . Recently, Mohiuddine and Aiyub  studied lacunary statistical convergence by introducing the concept -statistical convergence in random -normed space. Their work can be considered as a particular generalization of the statistical convergence. In , Mursaleen and Mohiuddine extended the idea of lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, and Debnath  investigated lacunary ideal convergence in intuitionistic fuzzy normed linear spaces. Also, lacunary statistically convergent double sequences in probabilistic normed space were studied by Mohiuddine and Savaş in .
The notion of lacunary ideal convergence has not been studied previously in the setting of -normed linear spaces. Motivated by this fact, in this paper, as a variant of -convergence, the notion of lacunary ideal convergence is introduced in a random -normed space, and some important results are established. Finally, the notions of lacunary -Cauchy and lacunary -Cauchy sequences are introduced and studied.
Throughout the paper, will denote the set of all natural numbers. First, we recall some of the basic concepts which will be used in this paper.
Definition 1 (see ). Let , and let be a real vector space of dimension , where . A real-valued function on , satisfying the following properties:(i) if and only if are linearly dependent,(ii) is invariant under any permutation of ,(iii), where (set of real numbers),(iv), is called an -norm on , and the pair is called an -normed space.
Given an -normed space , one can derive a topology for it via the following definition of the limit of a sequence: a sequence in is said to be convergent to in if for every .
All the concepts listed later are studied in depth in the fundamental book by Schweizer and Sklar .
Definition 2. Let denote the set of real numbers, , and the closed unit interval. 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 .
Definition 3. 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 and associative, and for every .
Definition 4 (see ). Let be a linear space of dimension greater than one, a triangle function, and a mapping from into . Then, is called a probabilistic -norm and a probabilistic -normed space if the following conditions are satisfied:(i) if and only if are linearly dependent, where denotes the value of at ;(ii) if and only if are linearly independent;(iii) is invariant under any permutation ;(iv) for every , and ;(v) whenever , and .If (v) is replaced by (vi) for all and ,then is called a random -normed space (for short, RnN space).
Remark 5. Note that every -norm space can be made a random -normed space in a natural way, by setting(i), for every , and , ;(ii) for every , and for .
Let be a RnN 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 that and vice versa.
A sequence in is said to have -convergence to if for every , and for each nonzero there exists a positive integer such that or equivalently, In this case, we write -.
Lemma 6. Let be a real -normed space, and let be RnN space induced by the random norm , where and . Then, for every sequence and nonzero in ,
Proof. Suppose that . Then, for every and for every , there exists a positive integer such that We observe that for any given , which is equivalent to Therefore, by letting , we have This implies that for each as desired.
Definition 7 (see [21, 29]). A subset of is said to have density if exists. A number sequence is said to be statistically convergent to if for every ,. If is statistically convergent to , we write st-, which is necessarily unique.
Definition 8 (see ). A family of subsets of a nonempty set is said to be an ideal in if (i) ; (ii) imply ; (iii) , imply . A nontrivial ideal in is called an admissible ideal if it is different from and it contains all singletons, that is, for each .
Let be a nontrivial ideal. A class , called the filter associated with the ideal , is a filter on .
Definition 9 (see ). Let be a nontrivial ideal in . Then, a sequence in is said to be -convergent to , if for each the set belongs to .
Definition 10. 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 . Let . The number is said to be the -density of , provided the limit exists (see ).
Definition 11 (see ). Let be a nontrivial ideal in . A sequence of numbers is said to be lacunary -convergent to a number if, for every ,
In this case, we write -.
2. Lacunaryand-Convergence for Sequences in RnN Spaces
In this section, we study the concepts lacunary and -convergence of a sequence in and prove some important results.
Definition 12. Let be RnN space, and let be a proper ideal in . A sequence in is said to be -convergent to (-convergent to with respect to the random -norm -topology) if for each , and each nonzero ,
In this case, the vector is called the -limit of the sequence , and we write -.
Definition 13. Let be RnN space, and let be a proper ideal in . A sequence in is said to be lacunary -convergent to with respect to the random -norm -topology if for each , and each nonzero ,
In this case, the vector is called the -limit of the sequence , and we write -.
Lemma 14. Let be RnN space. If a sequence is -convergent with respect to the random -norm , then -limit is unique.
Proof. Let us assume that where . Since , select , and each nonzero such that and are disjoint neighborhoods of and . Since and both are -limit of the sequence , we have both belonging to . This implies that the sets and belong to . In this way, we obtain a contradiction to the fact that the neighborhoods and of and are disjoints. Hence, we have . This completes the proof.
Lemma 15. Let be RnN space. Then one has(i)-, then -;(ii)if - and -, then -;(iii)if - and , then -;(iv)if -, and -, then -.
Proof. (i) Suppose that -. Let , and nonzero . Then, there exists positive integer such that for each . Since the set
and the ideal is admissible, we have . This shows that -.
(ii) Let , and nonzero . Choose such that . Since - and -, the sets belong to . Let . Since is an ideal, it is sufficient to show that . This is equivalent to show that where and belong to . Let , that is, and , and we have Since , we have .
(iii) It is trivial for . Now, let , , , and nonzero . Since -, we have This implies that Let . Then, we have So, Hence, -.
(iv) The result follows from (ii) and (iii).
We introduce the concept of -convergence closely related to -convergence of sequence in random -normed space and show that -convergence implies -convergence but not conversely.
Definition 16. Let be RnN space, and let be an admissible ideal. We say that a sequence in is said to be lacunary -convergent to with respect to the random -norm if there is a set such that and - for each nonzero .
In this case, we write -, and is called the -limit of the sequence .
Theorem 17. Let be RnN space, and let be an admissible ideal. If -, then -.
Proof. Suppose that -. Then, by definition, there exists such that and - for each nonzero . Let , and nonzero be given. Since -, there exists such that for every . Since is contained in and the ideal is admissible, we have . Hence, for , and nonzero . Therefore, we conclude that -.
The following example shows that the converse of Theorem 17 needs not to be true.
Example 18. Consider with
where for each , and let for all . For all and , consider
Then, is RnN space. Consider a decomposition of as such that for any , each contains infinitely many ’s, where and for . Let be the class of all subsets of which intersect almost a finite number of ’s. Then, is an admissible ideal. We define a sequence as follows: , , if . Then, for nonzero , we have
as . Hence, -.
Now, we show that -. Suppose that Then, by definition, there exists a subset such that and -. Since , there exists such that . Then, there exist positive integers such that Thus, , and so for infinitely many values ’s in . This contradicts the assumption that -. Hence, -.
Hence, the converse of the theorem needs not to be true.
The following theorem shows that the converse holds if the ideal satisfies condition .
Definition 19 (see ). An admissible ideal is said to satisfy the condition () if for every sequence of pairwise disjoint sets from there are sets , , such that the symmetric difference is a finite set for every and .
Theorem 20. Let be RnN space, and let the ideal satisfy the condition (). If is a sequence in such that -, then -.
Proof. Since -, so for every , and nonzero , the set We define the set for as Then, it is clear that is a countable family of mutually disjoint sets belonging to , and so by the condition (), there is a countable family of sets such that the symmetric difference is a finite set for each and . Since , there is a set such that . Now, we prove that the subsequence is convergent to with respect to the random -norm . Let , and nonzero . Choose a positive such that . Then, Since is a finite set for each , there exists such that If and , then and . Hence, for every and , we have Since this holds for every , and nonzero , so we have -. This completes the proof of the theorem.
3. and -Cauchy Sequences in RnN Spaces
In this section, we study the concepts -Cauchy and -Cauchy of a sequence in . Also, we will study the relations between these concepts.
Definition 21. Let be RnN space, and let be an admissible ideal of . Then, a sequence of elements in is called lacunary -Cauchy sequence in if for every , and nonzero , there exists satisfying
Definition 22. Let be RnN space, and let be an admissible ideal of . We say that a sequence of elements in is called lacunary -Cauchy sequence in if for every , and nonzero , there exists a set such that , and is a lacunary Cauchy with respect to the random -norm .
The next theorem gives that -Cauchy sequence implies -Cauchy sequence.
Theorem 23. Let be RnN space, and let be a nontrivial ideal of . If is a -Cauchy sequence, then is a -Cauchy sequence too.
Proof. Let be a -Cauchy sequence. Then, for , and nonzero , there exist and a number such that for every . Now, fix . Then, for every , and nonzero , we have Let . It is obvious that and Therefore, for every , and nonzero , we can find such that , that is, is a -Cauchy sequence.
Now, we will prove that -convergence implies -Cauchy condition in random -normed space.
Theorem 24. Let be RnN space, and let be an admissible ideal of . If a sequence is -convergent, then it is a -Cauchy sequence.
Proof. By assumption, there exists a set such that and is a lacunary Cauchy with respect to the random -norm . Choose such that . Since for every , , each nonzero in and ,, we have for every and each nonzero , that is, in is an -Cauchy sequence in . Then, by Theorem 23 is a -Cauchy sequence in RnN space.
Theorem 25. Let be RnN space, and let be an admissible ideal of . If a sequence of elements in is -convergent, then it is -Cauchy sequence.
Proof. Suppose that is -convergent to . Let , and nonzero be given. Then, we have This implies that Choose such that . Then, for every , Hence. This implies that that is, is a -Cauchy sequence.
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