Vol.5 No.5A(2013), Article ID:31862,5 pages DOI:10.4236/eng.2013.55A008

I-Pre-Cauchy Double Sequences and Orlicz Functions

Vakeel A. Khan1, Nazneen Khan1, Ayhan Esi2, Sabiha Tabassum3

1Department of Mathematics, Aligarh Muslim University, Aligarh, India

2Department of Mathematics, Science and Art Faculty, Adiyaman University, Adiyaman, Turkey

3Department of Applied Mathematics, Zakir Hussain College of Engineering and Technology, Aligarh Muslim University, Aligarh, India


Copyright © 2013 Vakeel A. Khan 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.

Received March 1, 2013; revised April 3, 2013; accepted April 12, 2013

Keywords: Ideal; Filter; Paranorm; I-Convergent; Invariant Mean; Monotone and Solid Space


Let be a double sequence and let M be a bounded Orlicz function. We prove that x is I-pre-Cauchy if and only if This implies a theorem due to Connor, Fridy and Klin [1], and Vakeel A. Khan and Q. M. Danish Lohani [2].

1. Introduction

The concept of statistical convergence was first defined by Steinhaus [3] at a conference held at Wroclaw University, Poland in 1949 and also independently by Fast [4], Buck [5] and Schoenberg [6] for real and complex sequences. Further this concept was studied by Salat [7], Fridy [8], Connor [9] and many others. Statistical convergence is a generalization of the usual notation of convergence that parallels the usual theory of convergence.

A sequence is said to be statistically convergent to if for a given

A sequence is said to be statistically precauchy if

Connor, Fridy and Klin [1] proved that statistically convergent sequences are statistically pre-cauchy and any bounded statistically pre-cauchy sequence with a nowhere dense set of limit points is statistically convergent. They also gave an example showing statistically pre-cauchy sequences are not necessarily statistically convergent (see [10]).

Throughout a double sequence is denoted by A double sequence is a double infinite array of elements for all

The initial works on double sequences is found in Bromwich [11], Tripathy [12], Basarir and Solancan [13] and many others.

Definition 1.1. A double sequence is called statistically convergent to if

where the vertical bars indicate the number of elements in the set.

Definition 1.2. A double sequence is called statistically pre-cauchy if for every there exist and such that

Definition 1.3. An Orlicz Function is a function which is continuous, nondecreasing and convex with for and, as.

If convexity of is replaced by , then it is called a Modulus function (see Maddox [14]). An Orlicz function may be bounded or unbounded. For example,

is unbounded and

is bounded (see Maddox [14]).

Lindenstrauss and Tzafriri [15] used the idea of Orlicz functions to construct the sequence space,

The space is a Banach space with the norm

The space is closely related to the space which is an Orlicz sequence space with for.

An Orlicz function M is said to satisfy condition for all values of if there exists a constant such that for all values of

The study of Orlicz sequence spaces have been made recently by various authors [1,2,16-20]).

In [1], Connor,Fridy and Klin proved that a bounded sequence is statistically pre-cauchy if and only if

The notion of I-convergence is a generalization of statistical convergence. At the initial stage it was studied by Kostyrko, Salat, Wilezynski [21]. Later on it was studied by Salat, Tripathy, Ziman [22] and Demirci [23], Tripathy and Hazarika [24-26]. Here we give some preliminaries about the notion of I-convergence.

Definition 1.4. [20,27] Let X be a non empty set. Then a family of sets (denoting the power set of X) is said to be an ideal in X if


(ii) I is additive i.e.

(iii) I is hereditary i.e.

An Ideal is called non-trivial if. A non-trivial ideal is called admissible if .

A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal containing I as a subset.

For each ideal I, there is a filter corresponding to I. i.e.

Definition 1.5. [10,21,28] A double sequence is said to be I-convergent to a number L if for every,

In this case we write

Definition 1.6. [21] A non-empty family of sets is said to be filter on X if and only if

(i)(ii) For we have

(iii) For each and implies.

2. Main Results

In this article we establish the criterion for any arbitrary double sequence to be I-pre-cauchy.

Theorem 2.1. Let be a double sequence and let M be a bounded Orlicz function then is I-preCauchy if and only if

Proof: Suppose that

For each and we have that



Now by (1) and (2) we have

thus is I-pre-Cauchy.

Now conversely suppose that is I-pre-Cauchy, and that has been given.

Then we have


Let be such that Since M is a bounded Orlicz function there exists an integer B such that for all. Therefore, for each


Since is I-pre-Cauchy, there is an such that the right hand side of (3) is less than for all. Hence

Theorem 2.2. Let be a double sequence and let M be a bounded Orlicz function then x is I-convergent to L if and only if

Proof: Suppose that

with an Orlicz function M, then is I-convergent to L (See [1])

Conversely suppose that is I-convergent to L. We can prove this in similar manner as in Theorem 2.1 assuming that

and M being a bounded Orlicz function.

Corollary 2.3. A sequence is I-convergent if and only if

Proof: Let Then





Therefore from (4) and (5) we have,


if and only if

By an immediate application of Theorem 2.1 we get the desired result.

Corollary 2.4. A sequence is I-convergent to L if and only if

Proof: Let

We can prove this in the similar manner as in the proof of Corollary 2.3.

3. Acknowledgements

The authors would like to record their gratitude to the reviewer for his careful reading and making some useful corrections which improved the presentation of the paper.


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