Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 926193, 14 pages

http://dx.doi.org/10.1155/2012/926193

## -Statistical Convergence of Sequences of Functions in Intuitionistic Fuzzy Normed Spaces

^{1}Department of Mathematical Engineering, Yildiz Technical University, Davutpasa Campus, Esenler, 34210 Istanbul, Turkey^{2}Department of Mathematics, Istanbul Commerce University, Uskudar, 34672 Istanbul, Turkey^{3}Department of Mathematics, Yildiz Technical University, Davutpasa Campus, Esenler, 34220 Istanbul, Turkey

Received 18 July 2012; Accepted 17 September 2012

Academic Editor: Manuel Sanchis

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

We study -statistically convergent sequences of functions in intuitionistic fuzzy normed spaces. We define concept of -statistical pointwise convergence and -statistical uniform convergence in intuitionistic fuzzy normed spaces and we give some basic properties of these concepts.

#### 1. Introduction and Some Definitions

Fuzzy logic was introduced by Zadeh [1]. Since then, the importance of fuzzy logic has come increasingly to the present. There are many applications of fuzzy logic in the field of science and engineering, for example, population dynamics [2], chaos control [3, 4], computer programming [5], nonlinear dynamical systems [6], fuzzy topology [7], and so forth. The concept of intuitionistic fuzzy set, as a generalization of fuzzy logic, was introduced by Atanassov [8] in 1983.

In the literature, -norm and -conorm were defined by Schweizer and Sklar [9]. The norms on intuitionistic fuzzy sets are introduced firstly in [10]. Recently Park [11] has introduced the concept of intuitionistic fuzzy metric space and in [12], Saadati and Park introduced intuitionistic fuzzy normed spaces and concept of convergence of a sequence in intuitionistic fuzzy normed spaces. In light of these developments, intuitionistic fuzzy analogues of many concepts in classical analysis were studied by many authors [13–17], and so forth.

The concept of statistical convergence was introduced by Fast [18] and Steinhaus [19] independently. Mursaleen defined -statistical convergence in [20]. Also the concept of statistical convergence was studied in intuitionistic fuzzy normed space in [21]. Quite recently, Karakaya et al. [22] defined and studied statistical convergence of sequences of functions in intuitionistic fuzzy normed spaces. Mohiuddine and Lohani [23] defined and studied -statistical convergence in intuitionistic fuzzy normed spaces.

In this paper, we will study concept -statistical convergence for sequences of functions and investigate some basic properties related to the concept in intuitionistic fuzzy normed space.

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

*Definition 1.1 (see [12]). *Let be a continuous -norm, let be a continuous -conorm, and be a linear space over the field IF ( or ). If and are fuzzy sets on satisfying the following conditions, the five-tuple is said to be an intuitionistic fuzzy normed space and is called an intuitionistic fuzzy norm. 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 .

*Definition 1.2 (see [19, 24]). *Let and . Then, the natural density is defined by , where denotes the cardinality of . A sequence is said to be statistically convergent to the number if for every , the set has asymptotic density zero, where
This case is stated by .

*Definition 1.3 (see [22]). *Let and be two intuitionistic fuzzy normed linear spaces and let be sequences of functions. If for each and for all ,
then we say that the sequence is pointwise statistically convergent to with respect to intuitionistic fuzzy norm and we write it .

*Definition 1.4 (see [20]). *Let be a nondecreasing sequence of positive numbers tending to such that
Let . The number
is said to be -density of , where .

If , then -density is reduced to asymptotic density. A sequence is said to be -statistically convergent to the number if for every , the set has -density zero, where
This case is stated by .

*Definition 1.5 (see [23]). *Let be an intuitionistic fuzzy normed space. Then, a sequence is said to be -statistically convergent to with respect to intuitionistic fuzzy norm provided that for every and ,
or equivalently
This case is stated by .

#### 2. -Statistical Convergence of Sequence of Functions in Intuitionistic Fuzzy Normed Spaces

In this section, we define pointwise -statistical and uniformly -statistical 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. We investigate relations of these concepts with continuity. Let us start definition of pointwise -statistical convergence in intuitionistic fuzzy normed spaces.

*Definition 2.1. *Let and be two intuitionistic fuzzy normed linear spaces over the same field IF and let be sequences of functions. If for each and for all ,
or equivalently
then we say that the sequence is pointwise -statistically convergent with respect to intuitionistic fuzzy norm and we write it .

This means that for every , there is integer such that, for all and for every ,

*Remark 2.2. *Let be sequences of functions. If , since -density is reduced to asymptotic density, then is pointwise statistically convergent on with respect to , that is, .

Lemma 2.3. *Let be sequences of functions. Then for every and , the following statements are equivalent. *(i)*Consider .*(ii)*For each *(iii)*For each *(iv)*For each *(v)*For each *

*Example 2.4. *Let denote the space of real numbers with the usual norm, and let and for . For all and every , consider
In this case is intuitionistic fuzzy normed space. (Also, is intuitionistic fuzzy normed space.) Let be sequences of functions whose terms are given by
is pointwise -statistically convergent on with respect to intuitionistic fuzzy norm . It is fact that for each , and since
hence we have
Thus, for each , since
is -statistically convergent to with respect to intuitionistic fuzzy norm .

If we take , then we have Thus , then is -statistically convergent to with respect to intuitionistic fuzzy norm .

If we take , it can be seen easily that Thus, at , is -statistically convergent to with respect to intuitionistic fuzzy norm .

Consequently, since is -statistically convergent to different points with respect to intuitionistic fuzzy norm for each , is pointwise -statistically intuitionistic fuzzy convergent on .

Theorem 2.5. *Let be an intuitionistic fuzzy normed space and let be sequences of functions. If sequence is pointwise intuitionistic fuzzy convergent on to a function with respect to , then is pointwise -statistically convergent with respect to intuitionistic fuzzy norm .*

*Proof. *Let be pointwise intuitionistic fuzzy convergent in . In this case, is convergent with respect to for each . Then for every and , there is number such that
for all and for each . Hence for each , the set
has finite numbers of terms. Since finite subset of has -density , hence
That is, .

Theorem 2.6. *Let and be two sequences of functions from intuitionistic fuzzy normed space to . If and , then where IF .*

*Proof. *The proof is clear for and . Now let and . Since and , for each if we define
then and . 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 mentioned previously, we have
This implies that
Since and , hence
That is

*Definition 2.7. *Let be sequences of functions. is a pointwise -statistical Cauchy sequence in intuitionistic fuzzy normed space provided that for every and there exists a number such that

Theorem 2.8. *Let be a sequence of functions. If is a pointwise -statistical convergent sequence with respect to intuitionistic fuzzy norm , then is a pointwise -statistical Cauchy sequence with respect to intuitionistic fuzzy norm .*

*Proof. *Suppose that and let . For a given , 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 have. Let Then we have
in particular . In this case
which is not possible. On the other hand
in particular . In this case
which is not possible. Hence . Therefore, by . That is, is a pointwise -statistical Cauchy sequence with respect to intuitionistic fuzzy norm .

In the following, we introduce uniformly -statistical convergence in intuitionistic fuzzy normed spaces.

*Definition 2.9. *Let and be two intuitionistic fuzzy normed linear spaces over the same field IF and let be a sequence of functions. If for all and for all ,
or equivalently
then we say that the sequence is uniformly -statistically convergent with respect to intuitionistic fuzzy norm and we write it .

*Remark 2.10. *If , then .

Lemma 2.11. *Let be a sequence of functions. Then for every and , the following statements are equivalent:*(i)*Consider .*(ii)*For all *(iii)*For all *(iv)*For all *(v)*For all *

*Example 2.12. *Let be as Example 2.4. Let be a sequence of functions whose terms are given by
Since , hence we have
Thus, for all , since
is uniformly -statistically convergent to with respect to intuitionistic fuzzy norm .

*Definition 2.13. *Let be sequences of functions. The sequence is a uniformly -statistical Cauchy sequence in intuitionistic fuzzy normed space provided that for every and there exists a number such that

*Definition 2.14. *Let and be two intuitionistic fuzzy normed spaces, and let be a family of functions from to . The family is intuitionistic fuzzy equicontinuous at a point if for every and , there exists a such that and for all and all such that and . The family is intuitionistic fuzzy equicontinuous if it is equicontinuous at each point of . (For continuity, may depend on , and ; for equicontinuity, must be independent of .)

Theorem 2.15. *Let be intuitionistic fuzzy normed space. Assume that on where functions are intuitionistic fuzzy equicontinuous on and . Then is continuous on .*

*Proof. *Let be an arbitrary point. By the intuitionistic fuzzy equicontinuity of ’s, for every and there exist such that
for every and all such that and . Let ( stands for an open ball in with center and radius ) be fixed. Since on , for each , if we state, respectively, and by the sets
then and ; hence and is different from . Thus, there exists such that
Now, we will show that is intuitionistic fuzzy continuous at . Since and for every ’s are continuous, is also continuous for , we have
Thus, the proof is completed.

#### Acknowledgment

This work is supported by The Scientific and Technological Research Council of Turkey (TUBITAK) under the project no. 110T699.

#### References

- L. A. Zadeh, “Fuzzy sets,”
*Information and Computation*, vol. 8, pp. 338–353, 1965. View at Google Scholar · View at Zentralblatt MATH - L. C. Barros, R. C. Bassanezi, and P. A. Tonelli, “Fuzzy modelling in population dynamics,”
*Ecological Modelling*, vol. 128, pp. 27–33, 2000. View at Google Scholar - G. Feng and G. Chen, “Adaptive control of discrete-time chaotic systems: a fuzzy control approach,”
*Chaos, Solitons and Fractals*, vol. 23, no. 2, pp. 459–467, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. L. Fradkov and R. J. Evans, “Control of chaos: methods and applications in engineering,”
*Chaos, Solitons & Fractals*, vol. 29, pp. 33–56, 2005. View at Google Scholar - R. Giles, “A computer program for fuzzy reasoning,”
*Fuzzy Sets and Systems*, vol. 4, no. 3, pp. 221–234, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Hong and J.-Q. Sun, “Bifurcations of fuzzy nonlinear dynamical systems,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 11, no. 1, pp. 1–12, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Das, “Fuzzy topology on fuzzy sets: product fuzzy topology and fuzzy topological groups,”
*Fuzzy Sets and Systems*, vol. 100, no. 1–3, pp. 367–372, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Atanassov, “Intuitionistic fuzzy sets,” in
*7th ITKR'S Session*, V. Sgurev, Ed., Sofia, Bulgaria, 1983. - B. Schweizer and A. Sklar, “Statistical metric spaces,”
*Pacific Journal of Mathematics*, vol. 10, pp. 313–334, 1960. View at Google Scholar · View at Zentralblatt MATH - K. Atanassov, “Norms and metrics over intuitionistic fuzzy sets,” in
*Proceedings of the BUSEFAL*, vol. 55, pp. 11–20, 1993. - J. H. Park, “Intuitionistic fuzzy metric spaces,”
*Chaos, Solitons and Fractals*, vol. 22, no. 5, pp. 1039–1046, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Saadati and J. H. Park, “On the intuitionistic fuzzy topological spaces,”
*Chaos, Solitons and Fractals*, vol. 27, no. 2, pp. 331–344, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Mursaleen and Q. M. Danish Lohani, “Baire's and Cantor's theorems in intuitionistic fuzzy 2-metric spaces,”
*Chaos, Solitons & Fractals*, vol. 42, no. 4, pp. 2254–2259, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Mursaleen, Q. M. D. Lohani, and S. A. Mohiuddine, “Intuitionistic fuzzy 2-metric space and its completion,”
*Chaos, Solitons & Fractals*, vol. 42, no. 2, pp. 1258–1265, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Saadati and J. H. Park, “Intuitionistic fuzzy Euclidean normed spaces,”
*Communications in Mathematical Analysis*, vol. 1, no. 2, pp. 85–90, 2006. View at Google Scholar - T. K. Samanta and I. H. Jebril, “Finite dimensional intuitionistic fuzzy normed linear space,”
*International Journal of Open Problems in Computer Science and Mathematics*, vol. 2, no. 4, pp. 574–591, 2009. View at Google Scholar · View at Zentralblatt MATH - B. Dinda and T. K. Samanta, “Intuitionistic fuzzy continuity and uniform convergence,”
*International Journal of Open Problems in Computer Science and Mathematics*, vol. 3, no. 1, pp. 8–26, 2010. View at Google Scholar · View at Zentralblatt MATH - H. Fast, “Sur la convergence statistique,”
*Colloquium Mathematicum*, vol. 2, pp. 241–244, 1951. View at Google Scholar · View at Zentralblatt MATH - H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,”
*Colloquium Mathematicum*, vol. 2, pp. 73–74, 1951. View at Google Scholar - M. Mursaleen, “$\lambda $-statistical convergence,”
*Mathematica Slovaca*, vol. 50, no. 1, pp. 111–115, 2000. View at Google Scholar · View at Zentralblatt MATH - S. Karakus, K. Demirci, and O. Duman, “Statistical convergence on intuitionistic fuzzy normed spaces,”
*Chaos, Solitons and Fractals*, vol. 35, no. 4, pp. 763–769, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. Karakaya, N. Simsek, M. Ertürk, and F. Gürsoy, “Statistical convergence of sequences of functions in intuitionistic fuzzy normed spaces,”
*Abstract and Applied Analysis*. In press. - S. A. Mohiuddine and Q. M. D. Lohani, “On generalized statistical convergence in intuitionistic fuzzy normed space,”
*Chaos, Solitons & Fractals*, vol. 42, no. 3, pp. 1731–1737, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. A. Fridy, “On statistical convergence,”
*Analysis. International Journal of Analysis and its Application*, vol. 5, no. 4, pp. 301–313, 1985. View at Google Scholar · View at Zentralblatt MATH