- Review
- Open access
- Published:
Statistical convergence in a paranormed space
Journal of Inequalities and Applications volume 2012, Article number: 39 (2012)
Abstract
In this article, we define the notion of statistical convergence, statistical Cauchy and strongly p-Cesàro summability in a paranormed space. We establish some relations between them.
AMS subject classification (2000): 41A10; 41A25; 41A36; 40A05; 40A30.
1 Introduction and preliminaries
The concept of statistical convergence for sequences of real numbers was introduced by Fast [1] and Steinhaus [2] independently in the same year 1951 and since then several generalizations and applications of this notion have been investigated by various authors, namely [3–11]. This notion was defined in normed spaces by Kolk [12] and in locally convex Hausdorff topological spaces by Maddox [13]. Çakalli [14] extended this notation to topological Hausdorff groups. Recently, in [15, 16], the concept of statistical convergence is studied in probabilistic normed space and in intuitionistic fuzzy normed spaces, respectively. In this article, we shall study the concept of statistical convergence, statistical Cauchy, and strongly p-Cesàro summability in a paranormed space.
Let K be a subset of the set of natural numbers ℕ. Then the asymptotic density of K denoted by δ(K), is defined as , where the vertical bars denote the cardinality of the enclosed set.
A number sequence x = (x k ) is said to be statistically convergent to the number L if for each ϵ > 0, the set K(ϵ) = {k ≤ n: |x k - L| > ϵ} has asymptotic density zero, i.e.,
In this case we write st-lim x = L.
A number sequence x = (x k ) is said to be statistically Cauchy sequence if for every ϵ > 0, there exists a number N = N(ϵ) such that
The concept of paranorm is a generalization of absolute value (see [17]).
A paranorm is a function g: X → ℝ defined on a linear space X such that for all x, y, z ∈ X
(P 1) g(x) = 0 if x = θ
(P 2) g(-x) = g(x)
(P 3) g(x + y) ≤ g(x) + g(y)
(P 4) If (α n ) is a sequence of scalars with α n → α0 (n → ∞) and x n , a ∈ X with x n → a (n → ∞) in the sense that g(x n - a) → 0 (n → ∞), then α n x n → α0a (n → ∞), in the sense that g(α n x n - α0a) → 0 (n → ∞).
A paranorm g for which g(x) = 0 implies x = θ is called a total paranorm on X, and the pair (X, g) is called a total paranormed space.
Note that each seminorm (norm) p on X is a paranorm (total) but converse need not be true.
In this article, we define and study the notion of convergence, statistical convergence, statistical Cauchy, and strong summability by a modulus function in a paranormed space.
Let (X, g) be a paranormed space.
Definition 1.1. A sequence x = (x k ) is said to be convergent (or g-convergent) to the number ξ in (X, g) if for every ε > 0, there exists a positive integer k0 such that g(x k - ξ) < ε whenever k ≥ k0. In this we write g-lim x = ξ, and ξ is called the g-limit of x.
Definition 1.2. A sequence x = (x k ) is said to be statistically convergent to the number ξ in (X, g) (or g(st)-convergent) if for each ϵ > 0,
In this case, we write g(st)-lim x = ξ. We denote the set of all g(st)-convergent sequences by S g .
Definition 1.3. A number sequence x = (x k ) is said to be statistically Cauchy sequence in (X, g) (or g(st)-Cauchy) if for every ϵ > 0 there exists a number N = N(ϵ) such that
2 Main results
Theorem 2.1. If a sequence x = (x k ) is statistically convergent in (X, g) then g(st)-limit is unique.
Proof. Suppose that g(st)-lim x = ξ1 and g(st)-lim x = ξ2. Given ε > 0, define the following sets as:
Since g(st)-lim x = ξ1, we have δ(K1(ε)) = 0. Similarly, g(st)-lim x = ξ2 implies that δ(K2(ε)) = 0. Now, let K(ε) = K1(ε)∪K2(ε). Then δ(K(ε)) = 0 and hence the compliment KC(ε) is a nonempty set and δ(KC(ε)) = 1. Now if k ∈ ℕ\K(ε), then we have g(ξ1-ξ2) ≤ g(x n -ξ1)+g(x n -ξ2) < ε/2+ε/2 = ε.
Since ε > 0 was arbitrary, we get g(ξ1 - ξ2) = 0 and hence ξ1 = ξ2.
Theorem 2.2. If g-lim x = ξ then g(st)-lim x = ξ but converse need not be true in general.
Proof. Let g-lim x = ξ. Then for every ε > 0, there is a positive integer N such that
for all n ≥ N. Since the set A(ϵ):= {k ∈ ℕ: g(x k - ξ) ≥ ε} ⊂ {1, 2, 3, ...}, δ(A(ϵ)) = 0. Hence g(st)-lim x = ξ.
The following examle shows that the converse need not be true.
Example 3.1. Let X = ℓ(1/k): = {x = (x k ): ∑ k |x k |1/k< ∞} with the paranorm g(x) = (∑ k |x k |1/k). Define a sequence x = (x k ) by
and write
We see that
and hence
Therefore g-lim x does not exist. On the other hand δ(K(ε)) = 0, that is, g(st)-lim x = 0.
Theorem 2.3. Let g(st)-lim x = ξ1 and g(st) - lim y = ξ2. Then
(i) g(st)-lim(x ± y) = ξ1 ± ξ2,
(ii) g(st)-lim αx = αξ1, α ∈ ℝ.
Proof. It is easy to prove.
Theorem 2.4. A sequence x = (x k ) in (X, g) is statistically convergent to ξ if and only if there exists a set K = {k1 < k2 < ⋯ < k n < ⋯ } ⊆ ℕ with δ(K) = 1 such that .
Proof. Suppose that g(st)-lim x = ξ. Now, write for r = 1, 2, ....
and
Then δ(K r ) = 0,
and
Now we have to show that for n ∈ M r , is g-convergent to ξ. On contrary suppose that is not g-convergent to ξ. Therefore there is ε > 0 such that for infinitely many terms. Let and . Then
and by (2.4.1), M r ⊂ M ε . Hence δ(M r ) = 0, which contradicts (2.4.2) and we get that is g-convergent to ξ.
Conversely, suppose that there exists a set K = {k1 < k2 < k3 < ⋯ < k n < ⋯} with δ(K) = 1 such that . Then there is a positive integer N such that g(x n - ξ) < ε for n > N. Put K ε (t): = {n ∈ ℕ: g(x n -ξ) ≥ ε} and K': = {kN+1, kN+2, ...}. Then δ(K') = 1 and K ε ⊆ ℕ-K' which implies that δ(K ε ) = 0. Hence g(st)-lim x = ξ.
Theorem 2.5. Let (X, g) be a complete paranormed space. Then a sequence x = (x k ) of points in (X, g) is statistically convergent if and only if it is statistically Cauchy.
Proof. Suppose that g(st)-lim x = ξ. Then, we get
where A(ε): = {n ∈ ℕ: g(x n - ξ) ≥ ε/2}. This implies that
Let m ∈ AC(ε). Then g(x m - ξ) < ε/2. Now, let B(ε): = {n ∈ ℕ: g(x m - x n ) ≥ ε}. We need to show that B(ε) ⊂ A(ε). Let n ∈ B(ε). Then g(x n - x m ) ≥ ε and hence g(x n - ξ) ≥ ε/2, i.e. n ∈ A(ε). Otherwise, if g(x n - ξ) < ε then
which is not possible. Hence B(ε) ⊂ A(ε), which implies that x = (x k ) is g(st)-convergent.
Conversely, suppose that x = (x k ) is g(st)-Cauchy but not g(st)-convergent. Then there exists M ∈ ℕ such that δ(G(ε) = 0,
where G(ε): = {n ∈ ℕ: g(x n - x M ) ≥ ε}, and δ(D(ε)) = 0, where D(ε): ={n ∈ ℕ: g(x n - ξ) < ε/2}, i.e., δ(DC(ε)) = 1. Since g(x n - x m ) ≤ 2g(x n - ξ) < ε,
if g(x n - ξ) < ε/2. Therefore δ(GC(ε)) = 0, i.e., δ(G(ε) = 1, which leads to a contradiction, since x = (x k ) was g(st)-Cauchy. Hence x = (x k ) must be g(st)-convergent.
3 Strong summability
In this section, we define the notion of strong summability by a modulus function and establish its relation with statistical convergence in a paranormed space.
Definition 3.1. A sequence x = (x k ) is said to be strongly p-Cesàro summable (0 < p < ∞) to the limit ξ in (X, g) if
and we write it as x k → ξ[C1, g] p . In this case ξ is called the [C1, g] p -limit of x.
Theorem 3.1. (a) If 0 < p < ∞ and x k → ξ[C1, g] p , then x = (x k ) is statistically convergent to ξ in (X, g).
(b) If x = (x k ) is bounded and statistically convergent to ξ in (X, g) then x k → ξ[C1, g] p .
Proof. (a) Let x k → ξ[C1, g] p , then
as n → ∞. That is, and so δ(K ε ) = 0, where K ε : = {k ≤ n: (g(x k - ξ))p≥ ε} . Hence x = (x k ) is statistically convergent to ξ in (X, g).
(b) Suppose that x = (x k ) is bounded and statistically convergent to ξ in (X, g). Then for ε > 0, we have δ(K ε ) = 0. Since x ∈ l∞, there exists M > 0 such that g(x k - ξ) ≤ M (k = 1, 2, ...). We have
where
Now if k ∉ K ε then S1(n) < εq. For k ∈ K ε , we have
as n → ∞, since δ(K ε ) = 0. Hence x k → ξ[C1, g] p .
This completes the proof of the theorem.
Recall that a modulus f is a function from [0, ∞) to [0, ∞) such that (i) f(x) = 0 if and only if x = 0, (ii) f(x + y) ≤ f(x) + f(y) for all x, y ≥ 0, (iii) f is increasing, and (iv) f is continuous from the right at 0.
Now we define the following:
Definition 3.2. Let f be a modulus. we say that a sequence x = (x k ) is strongly p-Cesàro summable with respect to f to the limit ξ in (X, g) if
(0 < p < ∞). In this case we write x k → ξ(w(f, g, p)).
As in [13], it is easy to prove the following:
Theorem 3.2. (a) Let f be any modulus and x k → ξ(w(f, g, p)). Then x = (x k ) is statistically convergent to ξ in (X, g).
(b) S g = w(f, g, p) If and only if f is bounded.
References
Fast H: Sur la convergence statistique. Colloq Math 1951, 2: 241–244.
Steinhaus H: Sur la convergence ordinaire et la convergence asymptotique. Colloq Math 1951, 2: 73–34.
Osama H, Edely H, Mursaleen M: On statistical A -summability. Math Comput Model 2009, 49: 672–680. 10.1016/j.mcm.2008.05.053
Fridy JA: On statistical convergence. Analysis 1985, 5: 301–313.
Fridy JA: Lacunary statistical summability. J Math Anal Appl 1993, 173: 497–504. 10.1006/jmaa.1993.1082
Moricz F: Statistical convergence of multiple sequences. Arch Math 2003, 81: 82–89. 10.1007/s00013-003-0506-9
Mursaleen M: λ -statistical convergence. Math Slovaca 2000, 50: 111–115.
Mursaleen M, Osama H, Edely H: Statistical convergence of double sequences. J Math Anal Appl 2003, 288: 223–231. 10.1016/j.jmaa.2003.08.004
Mursaleen M, Mohiuddine SA: Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 2009, 41: 2414–2421. 10.1016/j.chaos.2008.09.018
Mursaleen M, Mohiuddine SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J Comput Appl Math 2009, 233: 142–149. 10.1016/j.cam.2009.07.005
Šalát T: On statistically convergent sequences of real numbers. Math Slovaca 1980, 30: 139–150.
Kolk E: The statistical convergence in Banach spaces. Tartu Ul Toime 1991, 928: 41–52.
Maddox IJ: Statistical convergence in a locally convex space. Math Cambridge Phil Soc 1988, 104: 141–145. 10.1017/S0305004100065312
Çakalli H: On Statistical Convergence in topological groups. Pure Appl Math Sci 1996, 43: 27–31.
Karakus S: Statistical convergence on probabilistic normed spaces. Math Commun 2007, 12: 11–23.
Karakus S, Demirci K, Duman O: Statistical convergence on intuitionistic fuzzy normed spaces. Chaos Solitons & Fractals 2008, 35: 763–69. 10.1016/j.chaos.2006.05.046
Mursaleen M: Elements of Metric Spaces. Anamaya Publ., New Delhi; 2005.
Acknowledgements
The authors would like to thank the Deanship of Scientific Research at King Abdulaziz University for its financial support under a grant with number 156/130/1431.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors have equitably contributed in obtaining the new results presented in this article. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Alotaibi, A., Alroqi, A.M. Statistical convergence in a paranormed space. J Inequal Appl 2012, 39 (2012). https://doi.org/10.1186/1029-242X-2012-39
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2012-39