Research Article | Open Access

# Second Order Ideal-Ward Continuity

**Academic Editor:**Remi Léandre

#### Abstract

The main aim of the paper is to introduce a concept of second order ideal-ward continuity in the sense that a function is second order ideal-ward continuous if whenever and a concept of second order ideal-ward compactness in the sense that a subset of is second order ideal-ward compact if any sequence of points in has a subsequence of the sequence **x** such that where . We investigate the impact of changing the definition of convergence of sequences on the structure of ideal-ward continuity in the sense of second order ideal-ward continuity and compactness of sets in the sense of second order ideal-ward compactness and prove related theorems.

#### 1. Introduction

Let us start with basic definitions from the literature. Let , the set of all natural numbers, and . Then the *natural density* of is defined by if the limit exists, where the vertical bars indicate the number of elements in the enclosed set.

Fast [1] presented the following definition of statistical convergence for sequences of real numbers. The sequence is said to be *statistically convergent* to if for every , the set has natural density zero; that is, for each ,
In this case, we write or and denotes the set of all statistically convergent sequences. Note that every convergent sequence is statistically convergent but not conversely.

Some basic properties related to the concept of statistical convergence were studied in [2, 3]. In 1985, Fridy [4] presented the notion of statistically Cauchy sequence and determined that it is equivalent to statistical convergence. Caserta et al. [5] studied statistical convergence in function spaces, while Caserta and Koinac [6] investigated statistical exhaustiveness.

Kostyrko et al. [7] introduced the notion of ideal convergence. It is a generalization of statistical convergence. For details on ideal convergence we refer to [8–13].

Let be a nonempty set; then a family of sets (power sets of ) is called an *ideal* on if and only if(a),
(b)for each , we have ,(c)for each and each , we have .

A nonempty family of sets is a * filter* onif and only if(a),
(b)for each , we have ,(c)each and each , we have .

An ideal is called *nontrivial* ideal if and . Clearly is a nontrivial ideal if and only if is a filter on . A nontrivial ideal is called *admissible* if and only if . A nontrivial ideal is maximal if there cannot exist any nontrivial ideal containing as a subset.

*Definition 1 (see [7]). *A sequence of points in is said to be -convergent to the number if, for every , the set . One writes . One sees that a sequence being -convergent implies that .

Burton and Coleman [14] introduced the concept of quasi-Cauchy sequences as a sequence of points of is said to be a quasi-Cauchy sequence if is a null sequence where . Çakallı and Hazarika [15] introduced the concept of ideal quasi-Cauchy sequences. Recall from [15] that a sequence of points of is called ideal quasi-Cauchy if .

We say that a sequence is ward convergent to a number if where . Using the idea of continuity of a real function and the idea of compactness in terms of sequences, Çakallı [16] introduced the concept of ward continuity in the sense that a function is ward continuous if it transforms ward convergent to sequences to ward convergent to sequences; that is, is ward convergent to whenever is ward convergent to , and Çakallı [17] introduced the concept of ward compactness in the sense that a subset of is ward compact if any sequence of points in has a subsequence of the sequence such that where . Throughout the paper , , , and will denote the set of all convergent sequences, statistically convergent sequences, -convergent sequences, and the set of all -ward convergent to sequences of points in where a sequence is called -ward convergent to if .

Throughout the paper we assume is a nontrivial admissible ideal of .

#### 2. Second Order Ideal-Ward Continuity

We introduce the notion of second order ward convergent sequences as follows.

*Definition 2. *A sequence is said to be second order ward convergent to a number if where . For the special case , is called second order ward convergent to 0.

We note that any ward convergent to 0 sequence is also second order ward convergent to 0, but the opposite is not always true as it can be considering the sequence .

*Definition 3. *A sequence is said to be second order ideal-ward convergent to a number if where . For the special case , is called second order ideal-ward convergent to 0. One denotes by the set of all second order ideal-ward convergent sequences.

Now we give the definition of second order ideal-ward continuous function on a subset of .

*Definition 4. *A function is called ward continuous on if the sequence is ward convergent to whenever is a ward convergent to sequence of terms in .

*Definition 5. *A function is called second order ideal-ward continuous on if whenever , for a sequence of terms in .

Theorem 6. *If is second order ideal-ward continuous on a subset of , then it is an ideal-ward continuous on .*

*Proof. *Suppose that is a second order ideal-ward continuous function on a subset of . Let be a sequence with . Then we have the sequence
such that . Since is second order ideal-ward continuous, then we get the sequence
such that . It follows that
is second order ideal-ward convergent to 0. This completes the proof of the theorem.

The converse is not always true for this we consider the functionwhich is ideal-ward continuous but not second order ideal-ward continuous.

Corollary 7. *If is second order ward continuous, then it is an ideal continuous.*

Theorem 8. *If is second order ideal-ward continuous, then it is second order ward continuous.*

*Proof. *The proof is easy, so omitted.

*Definition 9. *A subset of is called second order ward compact if is a sequence of points in and there is a subsequence of such that .

Now we give the definition of second order ideal-ward compactness of a subset of .

*Definition 10. *A subset of is called second order ideal-ward compact if is a sequence of points in and there is a subsequence of such that .

Theorem 11. *A second order ideal-ward continuous image of any second order ideal-ward compact subset of is second order ideal-ward compact.*

*Proof. *Suppose that is a second order ideal-ward continuous function on a subset of and is a second order ideal-ward compact subset of . Let be a sequence of points in . Write where for each . A second order ideal-ward compactness of implies that there is a subsequence of with . Write . Since is second order ideal-ward continuous, so we have . Thus we have obtained a subsequence of the sequence with . Thus is second order ideal-ward compact. This completes the proof of the theorem.

Corollary 12. *A second order ideal-ward continuous image of any compact subset of is compact.*

*Proof. *The proof of this theorem follows from the preceding theorem.

Theorem 13. *If is a sequence of second order ideal-ward continuous functions defined on a subset of and is uniformly convergent to a function , then is second order ideal-ward continuous on .*

*Proof. *Let and be a sequence of points in such that . By the uniform convergence of there exists a positive integer such that for all whenever . Since is second order ideal-ward continuous on , then we have
But
Since is an admissible ideal, the right-hand side of relation (6) belongs to , and we have
This completes the proof of the theorem.

Theorem 14. *The set of all second order ideal-ward continuous functions on a subset of is a closed subset of the set of all continuous functions on ; that is, where is the set of all second order ideal-ward continuous functions on , denotes the set of all cluster points of .*

*Proof. *Let be an element in . Then there exists sequence of points in such that . To show that is second order ideal-ward continuous, consider a sequence of points in such that . Since converges to , there exists a positive integer such that, for all and for all , . Since is second order ideal-ward continuous on we have
But
Since is an admissible ideal, the right-hand side of the relation (9) belongs to , and we have
This completes the proof of the theorem.

Corollary 15. *The set of all second order ideal-ward continuous functions on a subset of is a complete subspace of the space of all continuous functions on .*

*Proof. *The proof follows from the preceding theorem.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### References

- H. Fast, “Sur la convergence statistique,”
*Colloquium Mathematicum*, vol. 2, pp. 241–244, 1951. View at: Google Scholar | Zentralblatt MATH | MathSciNet - T. Šalát, B. C. Tripathy, and M. Ziman, “On some properties of $I$-convergence,”
*Tatra Mountains Mathematical Publications*, vol. 28, pp. 279–286, 2004. View at: Google Scholar | MathSciNet - B. C. Tripathy and B. Hazarika, “$I$-monotonic and $I$-convergent sequences,”
*Kyungpook Mathematical Journal*, vol. 51, no. 2, pp. 233–239, 2011. View at: Publisher Site | Google Scholar | MathSciNet - J. A. Fridy, “On statistical convergence,”
*Analysis*, vol. 5, no. 4, pp. 301–313, 1985. View at: Google Scholar | Zentralblatt MATH | MathSciNet - A. Caserta, G. di Maio, and L. D. R. Kočinac, “Statistical convergence in function spaces,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 420419, 11 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Caserta and L. D. R. Kočinac, “On statistical exhaustiveness,”
*Applied Mathematics Letters*, vol. 25, no. 10, pp. 1447–1451, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. Kostyrko, T. Šalát, and W. Wilczyński, “$I$-convergence,”
*Real Analysis Exchange*, vol. 26, no. 2, pp. 669–685, 2000-2001. View at: Google Scholar | MathSciNet - B. Hazarika, “On generalized difference ideal convergence in random 2-normed spaces,”
*Filomat*, vol. 26, no. 6, pp. 1273–1282, 2012. View at: Google Scholar | Zentralblatt MATH | MathSciNet - B. Hazarika, “Ideal convergence in locally solid Riesz spaces,”
*Filomat*. In press. View at: Google Scholar - P. Kostyrko, M. Mačaj, T. Šalát, and M. Sleziak, “$I$-convergence and extremal $I$-limit points,”
*Mathematica Slovaca*, vol. 55, no. 4, pp. 443–464, 2005. View at: Google Scholar | MathSciNet - V. Kumar, “On
*I*and ${I}^{*}$-convergence of double sequences,”*Mathematical Communications*, vol. 12, no. 2, pp. 171–181, 2007. View at: Google Scholar | MathSciNet - T. Šalát, “On statistically convergent sequences of real numbers,”
*Mathematica Slovaca*, vol. 30, no. 2, pp. 139–150, 1980. View at: Google Scholar | Zentralblatt MATH | MathSciNet - I. J. Schoenberg, “The integrability of certain functions and related summability methods,”
*The American Mathematical Monthly*, vol. 66, pp. 361–375, 1959. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. Burton and J. Coleman, “Quasi-Cauchy sequences,”
*American Mathematical Monthly*, vol. 117, no. 4, pp. 328–333, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Çakallı and B. Hazarika, “Ideal-quasi-Cauchy sequences,”
*Journal of Inequalities and Applications*, vol. 2012, article 234, 11 pages, 2012. View at: Publisher Site | Google Scholar - H. Çakallı, “Forward continuity,”
*Journal of Computational Analysis and Applications*, vol. 13, no. 2, pp. 225–230, 2011. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. Çakallı, “Forward compactness,” in
*Proceedings of the Conference on Summability and Applications*, Shawnee State University, November 2009, http://webpages.math.luc.edu/~mgb/ShawneeConference/Articles/HuseyinCakalliOhio.pdf. View at: Google Scholar

#### Copyright

Copyright © 2014 Bipan Hazarika. 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.