- Open Access
On generalized difference lacunary statistical convergence in a paranormed space
© Altundağ; licensee Springer. 2013
- Received: 14 December 2012
- Accepted: 6 May 2013
- Published: 20 May 2013
In this article, we introduce the concept of -lacunary statistical convergence and -lacunary strong convergence in a paranormed space. Also, we establish some connections between these concepts.
- Topological Space
- Normed Space
- Statistical Convergence
- Generalize Difference
- Modulus Function
Dedicated to Professor Hari M Srivastava.
In order to extend convergence of sequences, the notion of statistical convergence was introduced by Fast  and Steinhaus  and several generalizations and applications of this concept have been investigated by various authors [3, 4]. This notion was studied in normed spaces by Kolk , in locally convex Hausdorff topological spaces by Maddox , in topological Hausdorff groups by Çakallı  and in probabilistic normed space by Karakuş . Recently, Alotaibi and Alroqi  extended this notion in paranormed spaces.
In this article, we study the concept of statistical convergence from difference sequence spaces which are defined over paranormed space.
Let K be a subset of the set of natural numbers ℕ. Then the asymptotic density of K denoted by , where the vertical bars denote the cardinality of the enclosed set in .
By a lacunary ; , where , we shall mean an increasing sequence of non-negative integers with as . The intervals determined by θ will be denoted by and . The ratio will be denoted by .
for , c, , where for all .
for , c and , where , , .
The set of all -statistically convergent sequences was denoted by in .
A paranorm is a function defined on a linear space X such that for all ,
(i) if ;
(iv) If is a sequence of scalars with () and with () in the sense that (), then (), in the sense that ().
A paranorm g for which implies is called a total paranorm on X and the pair is called a total paranormed space.
Note that each seminorm (norm) p on X is a paranorm (total) but converse need not be true.
The concept of paranorm is a generalization of absolute value .
A modulus function f is a function from to such that
(i) if and only if ;
(ii) for all ;
(iii) f increasing;
(iv) f is continuous from at the right zero.
In , the notion of statistical convergence was defined in a paranormed space.
In this case, we write . We denote the set of all -convergent sequences by .
and we write it as . In this case, L is called the it of x .
In this article, we shall study the concept of -lacunary statistical convergence, -lacunary strong convergence and -lacunary strong convergence with respect to a modulus function in a paranormed space.
In this case, we write . We denote the set of all -statistically convergent sequences in by .
In this case, we write . We denote the set of all -lacunary statistically convergent sequences in by .
and we write it as . In this case L is called the of x.
and we write it as . In this case L is called the of x.
Theorem 3.1 Let θ be a lacunary sequence and be a paranormed space. Then
(i) If , then and the inclusion is strict;
(ii) If x is a -bounded sequence and , then ;
which yields the result.
In order to prove that the inclusion is proper, let θ be given and with the paranorm . Define to be at the first term in for every , between the second term and th term in , at the th term in and otherwise.
from which the result follows.
(iii) This is an immediate consequence of (i) and (ii). □
Corollary 3.1 If , then . If and if , then .
Theorem 3.2 Let θ be a lacunary sequence and be a paranormed space, then if and only if .
which proves the .
Conversely, suppose that . Since θ is lacunary, we can select a subsequence of θ satisfying and , where and with the paranorm .
and hence x is Δ-bounded in .
We can see that , but . Theorem 3.1(ii) implies that , but it follows from Corollary 3.1 that . Hence and implies that .
Then , but . By Theorem 3.1(i) we conclude that , but by Corollary 3.1 that . Hence, . This completes the proof. □
In this case, we write . If we take for all , we say .
Lemma 3.1 Let f be a modulus function and let . Then for each we have .
Theorem 3.3 Let f be a modulus function and be a paranormed space. Then .
Proof Let . Then we have as for some L.
from Lemma 3.1. Therefore . □
Theorem 3.4 Let . Then if and only if f is bounded.
Proof Following the technique applied for establishing Theorem 3.16 of , we can prove the theorem. □
The author would like to thank the referees for their careful reading of the manuscript and for their helpful suggestions.
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