Abstract and Applied Analysis

Volume 2012 (2012), Article ID 157467, 19 pages

doi:10.1155/2012/157467

Research Article

Statistical Convergence of Sequences of Functions in Intuitionistic Fuzzy Normed Spaces

Vatan Karakaya,¹ Necip Şimşek,² Müzeyyen Ertürk,³ and Faik Gürsoy³

¹Department of Mathematical Engineering, Yildiz Technical University, Davutpasa Campus, Esenler, 34210 Istanbul, Turkey

²Department of Mathematics, Istanbul Ticaret University, Uskudar, 34672 Istanbul, Turkey

³Department of Mathematics, Yildiz Technical University, Davutpasa Campus, Esenler, 34220 Istanbul, Turkey

Received 29 June 2012; Revised 16 September 2012; Accepted 27 October 2012

Academic Editor: Ljubisa Kocinac

Copyright © 2012 Vatan Karakaya 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.

Abstract.

The purpose of this work is to investigate types of convergence of sequences of functions in intuitionistic fuzzy normed spaces and some properties related with these concepts.

1. Introduction and Preliminaries

The fuzzy theory has emerged as one of the most active area of research in many branches of mathematics and engineering. This new theory was introduced by Zadeh [1] in 1965 and since then a large number of research papers have appeared by using the concept of fuzzy set/numbers and fuzzification of many classical theories have also been made. It has also very useful application in various fields, for example, population dynamics [2], chaos control [3], computer programming [4], nonlinear dynamical systems [5], fuzzy physics [6], fuzzy topology [7, 8], and so forth. The notion of intuitionistic fuzzy sets, a generalization of fuzzy sets, was introduced by Atanassov [9] in 1986 and later there has been much progress in the study of intuitionistic fuzzy sets by many authors including Mursaleen et al. [10], Mursaleen et al. [11], and Yılmaz [12]. Using the idea of intuitionistic fuzzy sets, Park [13] defined the notion of intuitionistic fuzzy metric spaces with the help of the continuous t-norms and the continuous t-conorms as a generalization of fuzzy metric spaces due to George and Veeramani [14]. Samanta and Jebril [15] introduced the definitions of intuitionistic fuzzy continuity and sequential intuitionistic fuzzy continuity and proved that they are equivalent. A few of the algebraic and topological properties of intuitionistic fuzzy continuity and uniformly intuitionistic fuzzy continuity were investigated by Dinda and Samanta [16]. On the other hand, the Fast [17] introduced the concept of statistical convergence for real number sequences. Different types of statistical convergence of sequences of real functions and related notions were first studied in [18], and some important results and references on statistical convergence and function sequences can be found in [19–23].

In this paper, primarily following the line of [18], we define statistical convergence of sequences of functions in intuitionistic fuzzy normed space (IFNS for short), and we investigate some properties related with these concepts. To explain main problems of this work, we have to give some definitions and literature acknowledgment. In [24], Schweizer and Sklar introduced a continuous t-norm and a continuous t-conorm. Afterward Saadati and Park [8] introduced the following definitions by using concepts mentioned above.

We now first recall some basic notions of intuitionistic fuzzy normed spaces.

Definition 1.1 (see [8]). Let be a continuous t-norm, a continuous t-conorm, and a linear space over the intuitionistic fuzzy field ( or ). If and are fuzzy sets on satisfying the following conditions, the five-tuple is said to be an IFNS and is called an intuitionistic fuzzy norm (IFN for short). For every and ,

(i),(ii),(iii),(iv) for each ,(v) ,(vi) is continuous,(vii) and ,(viii), (ix),(x) for each ,(xi) ,(xii) is continuous,(xiii) and .  For an IFNS, we further assume that satisfy the following axiom:(see [16])(xiv)

Definition 1.2 (see [8]). Let be a intuitionistic fuzzy normed space. A subset of is said to be IF-bounded if there exist and such that and for each .

Definition 1.3 (see [25]). Let be a intuitionistic fuzzy metric space. Let be any subset of . Define

(i) is said to be q-bounded if and , (ii) is said to be semibounded if and (iii) is said to be unbounded if and .

Theorem 1.4 (see [25]). Let be a intuitionistic fuzzy metric space. subset of X is IF bounded if and only if is q-bounded or semibounded.

Definition 1.5 (see [8]). Let be an IFNS and be a sequence in . The sequence is said to be convergent to with respect to IFN if for every and , there exists a positive integer such that and whenever . In this case, we write as .

Definition 1.6 (see [16]). Let and be two IFNS. A mapping from to is said to be intuitionistic fuzzy continuous at if for any given , there exist such that for all and for all ,

Definition 1.7 (see [16]). Let be a sequence of functions. The sequence is said to be pointwise intuitionistic fuzzy convergent on to a function with respect to if for each , the sequence is convergent to with respect to .

Definition 1.8 (see [16]). Let be a sequence of functions. The sequence is said to be uniformly intuitionistic fuzzy convergent on to a function with respect to , if given , there exist a positive integer such that and ,

Now, we recall the notion of the statistical convergence of sequences in intuitionistic fuzzy normed spaces.

Definition 1.9 (see [26]). Let and . Then the asymptotic density is defined by , where denotes the cardinality of .

Definition 1.10. Let be subset of . If a property holds for all with , we say that holds for almost all .

Definition 1.11 (see [26]). A sequence is said to be statistically convergent to the number , or in short , if for every , the set has asymptotic density zero, where

that is,

Definition 1.12 (see [23]). Let be an IFNS. Then, sequence is said to be statistically convergent to with respect to IFN provided that for every and ,

or equivalently

In this case, we write .

Definition 1.13 (see [23]). Let be an IFNS. Sequence is said to be statistically Cauchy with respect to IFN provided that for every and , there exists a number satisfying

2. Statistical Convergence of Sequences of Functions in Intuitionistic Fuzzy Normed Spaces

In this section, we define pointwise statistically and uniformly statistically convergent sequences of functions in intuitionistic fuzzy normed spaces. Also, we give the statistical analog of the Cauchy convergence criterion for pointwise and uniformly statistical convergent in intuitionistic fuzzy normed space. Finally, we prove that uniformly statistical convergence preserves continuity.

Definition 2.1. Let and be two IFNS and : a sequence of functions. We say that a sequence pointwise statistically converges to with respect to intuitionistic fuzzy norm if statistically converges to for each with respect to intuitionistic fuzzy norm and we write .

Theorem 2.2. Let be a sequence of functions. If is pointwise intuitionistic fuzzy convergent on with respect to , then . But the converse of this is not true.

Proof. Let be pointwise intuitionistic fuzzy convergent on . In this case the sequence is convergent with respect to for each . Then for each and , there is number such that

for all and for each . Hence for each the set

has finite numbers of terms. Since density of finite subset of is , hence

That is, .

Example 2.3. Let denote the space of real numbers with the usual norm, and let and for . For all and every , we consider

In this case, is an IFNS (also, is intuitionistic fuzzy normed space.). Let be a sequence of functions whose terms are given by

Then sequence is pointwise statistically intuitionistic convergent on with respect to . Indeed, for , since

we have

which yields . Thus, for each , sequence is statistically convergent to with respect to intuitionistic fuzzy norm .

If we take , then we have

Therefore, density of is and for each , sequence is statistically convergent to with respect to IFN . If we take , it can be seen easily that is intuitionistic fuzzy convergent to . Hence is intuitionistic fuzzy statistically convergent to . That is

Since is statistically convergent to different points with respect to intuitionistic fuzzy norm for each , it can be seen that is pointwise statistically intuitionistic fuzzy convergent on .

Lemma 2.4. Let be sequence of functions. Then the following statements are equivalent:

(i),(ii) for each , for each and ,(iii) and for each , for each and ,(iv) and for each and .

Theorem 2.5. Let and be two sequences of functions from to . If and , then where or .

Proof. The proof is clear for and . Now let and . Since and , for each , if we define

then

Since and , if we state by then

Hence, and there exists such that

Let

We will show that for each

Let . In this case,

Using those above, we have

This implies that

Since and , hence

that is

which means

Definition 2.6. Let be a sequence of functions. The sequence is a pointwise statistically Cauchy sequence in IFNS provided that for each and there exists such that

That is, there exists a number for each such that

Theorem 2.7. Let : be a sequence of functions. If is a pointwise statistically convergent sequence with respect to intuitionistic fuzzy norm , then is a pointwise statistically Cauchy sequence with respect to intuitionistic fuzzy norm .

Proof. Suppose that and let , . For given each , choose such that and . If we state, respectively, and by

for each . Then, we have

which implies that

Let . Then

We want to show that there exists a number such that

Therefore, define for each ,

We have to show that

Suppose that

In this case, has at least one different element which does not has. Let . Then we have

in particularly . In this case,

which is not possible. On the other hand

in particularly . In this case,

which is not possible. Hence . Therefore, by . That is, is a pointwise statistical Cauchy sequence with respect to intuitionistic fuzzy norm .

Afterward this step, we introduce a uniformly statistical convergence of sequences of function in an IFNS. To do this, we need the following definition

Definition 2.8. Let and be two intuitionistic fuzzy normed linear space over the same field IF and be a sequence of functions. converges uniform statistically to f with respect to , and   and and ,

Lemma 2.9. Let be a sequence of functions. Then the following statements are equivalent:

(i).(ii) for all , for every and .(iii) and for all , for every and .(iv) and for all and .

Proposition 2.10. Let the sequence and be bounded functions from to . is intuitionistic fuzzy uniformly statistically convergent to if and only if

where the supremum and infimum are taken over all .

Proof. Suppose that on . Since and are bounded in for each , by using Definition 1.3 and Theorem 1.4, we have and for each and for each . By using of Lemma 2.9, we get

where the supremum and infimum are taken over all .

Conversely, suppose that

where the supremum and infimum are taken over all . Since

from definition statistical convergence, for , for every and

For all

Therefore, and for all and . From Lemma 2.9, we get .

Example 2.11. Let be as Example 2.3. Consider be sequence of functions whose terms are given by

Then, for every and for every , we define

For all , we have

Since

for all , the sequence is uniformly statistically intuitionistic fuzzy convergent to on .

Example 2.12. Let be as Example 2.3. Consider be sequence of functions whose terms are given by

Since for all , is uniformly statistically intuitionistic fuzzy convergent to on . We can show this using Proposition 2.10 as the following, since is bounded functions sequence on .

We want to find

Firstly, we need to find and for over all and for over all , respectively. In case of , we have . Since

we get

On the other hand, we have

and so

In case of we have . Since

we get

On the other hand, we have

and so

That is, the sequence is uniformly statistically intuitionistic fuzzy convergent to on .

Remark 2.13. If , then . But the converse of this is not true.

We prove this with the following example.

Example 2.14. Let us define the sequence of functions

on . This sequence of functions is pointwise statistically intuitionistic fuzzy convergent to (indeed, for and for ). But, it is not uniformly statistical intuitionistic fuzzy convergent. Since the sequence of functions is bounded on , we can use Proposition 2.10 to prove our claim. Let us take infumum for and supremum for , over all . Firstly, we try to find and . For this, we have

Since and , we get and . Then,

so, we get

Therefore, we conclude that does not intuitionistic fuzzy uniformly statistical convergent to .

Theorem 2.15. Let  : be a sequence of functions. If is uniformly intuitionistic fuzzy convergent on to a function with respect to , then . But the converse of this is not true.

Proof. Let be uniformly intuitionistic fuzzy convergent on to a function . In this case, given , there exist a positive integer such that and ,

That is, for

is satisfied and these ’s are finite. Since finite set has -density, density of complement of finite set is . If complement of this finite set is stated by , for every , there exist , and such that and and ,

This shows that .

Definition 2.16. Let be a sequence of functions.The sequence is a uniformly statistically Cauchy sequence in intuitionistic fuzzy normed space provided that for every and , there exists a number such that

Theorem 2.17. Let  : be sequence of functions. If is a uniformly statistically convergent sequence with respect to intuitionistic fuzzy norm , then is uniformly statistically Cauchy sequence with respect to intuitionistic fuzzy norm .

Proof. Suppose that . In this case, , there exists , and such that and ,

Choose . So, and . We investigate such that

or

For every , we have

Since is a uniformly statistically Cauchy sequence in intuitionistic fuzzy normed space.

Theorem 2.18. Let and be two IFNS and the mapping be the intuitionistic fuzzy continuous on of sequence of functions. If , the mapping is the intuitionistic fuzzy continuous on .

Proof. Let be an arbitrary point. By the intuitionistic fuzzy continuity of ’s, for every and there exists such that

for every and all such that and . Let be fixed stands for an open ball in with center and radius ). Since on , for all , if we state, respectively, and by these sets

then, and , hence and is different from . Thus, there exists such that

Now, we will show that is intuitionistic fuzzy contunious at . Using Definition 1.1, we have

Thus, the proof is completed.

Acknowledgments

This work is supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK) under the Project number 110T699. We have benefited a lot from the report of the anonymous referee. So, the authors are thankful for his/her valuable comments and careful corrections on the first draft of this paper which improved the presentation and readability.

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