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Statistical convergence through de la Vallée-Poussin mean in locally solid Riesz spaces
Advances in Difference Equations volume 2013, Article number: 66 (2013)
Abstract
The notion of statistical convergence was defined by Fast (Colloq. Math. 2:241-244, 1951) and over the years was further studied by many authors in different setups. In this paper, we define and study statistical τ-convergence, statistically τ-Cauchy and -convergence through de la Vallée-Poussin mean in a locally solid Riesz space.
MSC:40A35, 40G15, 46A40.
1 Introduction and preliminaries
Since 1951, when Steinhaus [1] and Fast [2] defined statistical convergence for sequences of real numbers, several generalizations and applications of this notion have been investigated. For more detail and related concepts, we refer to [3–29] and references therein. Quite recently, Di Maio and Kǒcinac [30] studied this notion in topological and uniform spaces and Albayrak and Pehlivan [31], and Mohiuddine and Alghamdi [32] for real and lacunary sequences, respectively, in locally solid Riesz spaces. Afterward, the idea was extended to double sequences by Mohiuddine et al.[33] in the framework of locally solid Riesz spaces.
Let K be a subset of ℕ, the set of natural numbers. Then the asymptotic density of K denoted by is defined as
where the vertical bars denote the cardinality of the enclosed set.
The number sequence is said to be statistically convergent to the number ℓ if for each ,
In this case, we write .
Remark 1.1 It is well known that every statistically convergent sequence is convergent, but the converse is not true. For example, suppose that the sequence is defined as
It is clear that the sequence is statistically convergent to 0, but it is not convergent.
Now we recall some definitions related to the notion of a locally solid Riesz space. Let X be a real vector space and ≤ be a partial order on this space. Then X is said to be an ordered vector space if it satisfies the following properties:
-
(i)
If and , then for each .
-
(ii)
If and , then for each .
If in addition X is a lattice with respect to the partial order ≤, then X is said to be a Riesz space (or a vector lattice) [34].
For an element x of a Riesz space X, the positive part of x is defined by , the negative part of x by and the absolute value of x by , where θ is the zero element of X.
A subset S of a Riesz space X is said to be solid if and imply .
A topological vector space is a vector space X which has a (linear) topology τ such that the algebraic operations of addition and scalar multiplication in X are continuous. The continuity of addition means that the function defined by is continuous on , and the continuity of scalar multiplication means that the function defined by is continuous on .
Every linear topology τ on a vector space X has a base for the neighborhoods of θ satisfying the following properties:
(C1) Each is a balanced set, that is, holds for all and every with .
(C2) Each is an absorbing set, that is, for every , there exists such that .
(C3) For each , there exists some with .
A linear topology τ on a Riesz space X is said to be locally solid (cf.[35, 36]) if τ has a base at zero consisting of solid sets. A locally solid Riesz space is a Riesz space equipped with a locally solid topology τ.
In this paper, we define and study statistical τ-convergence, statistically τ-Cauchy and -convergence through de la Vallée-Poussin mean in a locally solid Riesz space.
2 Generalized statistical τ-convergence
Throughout the text, we write for any base at zero consisting of solid sets and satisfying the conditions (C1), (C2) and (C3) in a locally solid topology. The following idea of λ-statistical convergence was introduced in [37] and further studied in [38–40].
Let be a non-decreasing sequence of positive numbers tending to ∞ such that
The generalized de la Vallée-Poussin mean is defined by
where .
A sequence is said to be -summable to a number ℓ if
A sequence is said to be strongly-summable to a number ℓ if
We denote it by as .
Let be a set of positive integers, then
is said to be the λ-density of K.
In case , the λ-density reduces to the natural density.
The number sequence is said to be λ-statistically convergent to the number ℓ if for each , , where , i.e.,
In this case, we write and we denote the set of all λ-statistically convergent sequences by . This notion was extended to double sequences in [41, 42].
Remark 2.1 As in Remark 1.1, we observe that if a sequence is -summable to a number ℓ, then it is also λ-statistically convergent to the same number ℓ, but the converse need not be true. For example, let the sequence be defined by
where denotes the integer part of . Then x is λ-statistically convergent to 0 but not -summable.
Definition 2.1 Let be a locally solid Riesz space. Then a sequence in X is said to be generalized statistically τ-convergent (or-convergent) to the number if for every τ-neighborhood U of zero,
In this case, we write or .
Definition 2.2 Let be a locally solid Riesz space. We say that a sequence in X is generalized statistically τ-bounded if for every τ-neighborhood U of zero, there exists some such that the set
has λ-density zero.
Theorem 2.1 Letbe a Hausdorff locally solid Riesz space andandbe two sequences in X. Then the following hold:
-
(i)
If and , then .
-
(ii)
If , then , .
-
(iii)
If and , then .
Proof (i) Suppose that and . Let U be any τ-neighborhood of zero. Then there exists such that . Choose any such that . We define the following sets:
Since and , we have . Thus and, in particular, . Now, let . Then
Hence, for every τ-neighborhood U of zero, we have . Since is Hausdorff, the intersection of all τ-neighborhoods U of zero is the singleton set . Thus, we get , i.e., .
-
(ii)
Let U be an arbitrary τ-neighborhood of zero and . Then there exists such that and also
Since Y is balanced, implies for every with . Hence, for every , we get
Thus, we obtain
for each τ-neighborhood U of zero. Now let and be the smallest integer greater than or equal to . There exists such that . Since , the set
has λ-density zero. Therefore, for all and , we have
Since the set Y is solid, we have . This implies that . Thus,
for each τ-neighborhood U of zero. Hence .
-
(iii)
Let U be an arbitrary τ-neighborhood of zero. Then there exists such that . Choose E in such that . Since and , we have , where
Let . Hence, we have . For all and , we get
Therefore,
Since U is arbitrary, we have . □
Theorem 2.2 Letbe a locally solid Riesz space. If a sequenceis generalized statistically τ-convergent, then it is generalized statistically τ-bounded.
Proof Suppose is generalized statistically τ-convergent to the point and let U be an arbitrary τ-neighborhood of zero. Then there exists such that . Let us choose such that . Since , the set
has λ-density zero. Since E is absorbing, there exists such that . Let . Since E is solid and , we have . Since E is balanced, implies . Then, for each and , we have
Thus
Hence, is generalized statistically τ-bounded. □
Theorem 2.3 Letbe a locally solid Riesz space. If, andare three sequences such that
-
(i)
for all ,
-
(ii)
,
then.
Proof Let U be an arbitrary τ-neighborhood of zero, there exists such that . Choose such that . From condition (ii), we have , where
Also, we get , and from (i) we have
for all . This implies that for all and , we get
Since Y is solid, we have . Thus,
for each τ-neighborhood U of zero. Hence . □
3 Generalized statistically τ-Cauchy and -convergence
Definition 3.1 Let be a locally solid Riesz space. A sequence in X is generalized statistically τ-Cauchy if for every τ-neighborhood U of zero there exists such that the set
has λ-density zero.
Theorem 3.1 Letbe a locally solid Riesz space. If a sequenceis generalized statistically τ-convergent, then it is generalized statistically τ-Cauchy.
Proof Suppose that . Let U be an arbitrary τ-neighborhood of zero, there exists such that . Choose such that . By generalized statistical τ-convergence to ξ, there is with and
Also, for all and , where
we have
and . Therefore the set
for all . For every τ-neighborhood U of zero there exists such that the set has λ-density zero. Hence is generalized statistically τ-Cauchy. □
Now we define another type of convergence in locally solid Riesz spaces.
Definition 3.2 A sequence in a locally solid Riesz space is said to be -convergent to if there exists an index set , , with such that . In this case, we write .
Theorem 3.2 A sequencein a locally solid Riesz spaceis generalized statistically τ-convergent to a number ξ if it is-convergent to ξ.
Proof Let U be an arbitrary τ-neighborhood of ξ. Since is -convergent to ξ, there is an index set , , with and , such that and imply . Then
Therefore . Hence x is generalized statistically τ-convergent to ξ. □
Note that the converse holds for a first countable space.
Recall that a topological space is first countable if each point has a countable (decreasing) local base.
Theorem 3.3 Letbe a first countable locally solid Riesz space. If a sequenceis generalized statistically τ-convergent to a number ξ, then it is-convergent to ξ.
Proof Let x be generalized statistically τ-convergent to a number ξ. Fix a countable local base at ξ. For each , put
By hypothesis, for each i. Since the ideal ℐ of all subsets of ℕ having λ-density zero is a P-ideal (see, for instance, [43]), then there exists a sequence of sets such that the symmetric difference is a finite set for any and .
Let , then . In order to prove the theorem, it is enough to check that .
Let . Since is finite, there is , without loss of generality, with , , such that
If and , then , and by (1), . Thus . So, we have proved that for all , there is , , with for every : without loss of generality, we can suppose for every . The assertion follows taking into account that the ’s form a countable local base at ξ. □
4 Conclusion
Recently, statistical convergence has been established as a better option than ordinary convergence. It is found very interesting that some results on sequences, series and summability can be proved by replacing the ordinary convergence by statistical convergence; and further, through some examples, where some efforts are required, we can show that the results for statistical convergence happen to be stronger than those proved for ordinary convergence (e.g., [44–49]). This notion has also been defined and studied in different setups. In this paper, we have studied this notion through de la Vallée-Poussin mean in a locally solid Riesz space to deal with the convergence problems in a broader sense.
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The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
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Mohiuddine, S.A., Alotaibi, A. & Mursaleen, M. Statistical convergence through de la Vallée-Poussin mean in locally solid Riesz spaces. Adv Differ Equ 2013, 66 (2013). https://doi.org/10.1186/1687-1847-2013-66
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DOI: https://doi.org/10.1186/1687-1847-2013-66