#### Abstract

The main purpose of this article is to introduce the concepts of almost -statistical convergence and strongly almost -convergence of order of sequences of fuzzy numbers with respect to an Orlicz function. We give some relations between strongly almost -convergence and almost -statistical convergence of order of sequences of fuzzy numbers.

#### 1. Introduction

The concept of fuzzy set was introduced by Zadeh [1]. Matloka [2] introduced sequences of fuzzy numbers and provided that every convergent sequence of fuzzy numbers is bounded. Recently, sequences of fuzzy numbers have been discussed by Altınok et al. [3, 4], Aytar and Pehlivan [5], Çolak et al. [6, 7], Gökhan et al. [8], Nuray [9], and Talo and Başar [10].

The idea of statistical convergence was introduced by Fast [11] and the notion was linked with summability theory by Alotaibi et al. [12], Connor [13], Et et al. [14, 15], Fridy [16], Işık [17], Mohiuddine et al. [18, 19], and Tripathy [20]. Recently, the notion was generalized by Gadjiev and Orhan [21], Çolak [22], and Çolak and Bektaş [23].

In this paper, we study the concepts of and examine some properties of almost -statistical convergence and strongly almost -convergence of order of sequences of fuzzy numbers. The results which we obtained in this study are much more general than those obtained by Başarır et al. [24].

#### 2. Definitions and Preliminaries

A fuzzy set on is called a fuzzy number if it has the following properties:(i) is normal;(ii) is fuzzy convex;(iii) is upper semicontinuous;(iv) is compact, where cl denoted the closure of the enclosed set.

A sequence of fuzzy numbers is a function from the set of all positive integers into , where is fuzzy number space. A sequence is said to be bounded if the set is bounded. A sequence is said to be convergent if there exists a positive integer such that for , for every . By , , and we denote the set of , , and sequences of fuzzy numbers, respectively [2].

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

The generalized de la Vallée-Pousion mean is defined by , where is a nondecreasing sequence of positive numbers such that , , and as and .

The space was introduced by Lorentz [25] and Maddox [26] has defined to be strongly almost convergent to a number if , uniformly in .

*Definition 1. *Let be as above, let be an Orlicz function, let be any sequence of strictly positive real numbers, and let be given. A sequence is said to be almost -statistically convergent of order to , with respect to the Orlicz function , if for every where The set of all almost -statistically convergent sequences of order with respect to the Orlicz function will be denoted by . In this case we write . In the special case , for all we will write instead of and in the special case for all and we will write instead of . If we take , then we obtain the set of all almost -statistically convergent sequence . If for all then we will write instead of .

*Definition 2. *Let be as above, let be an Orlicz function, let be any sequence of strictly positive real numbers, and let be given. We define the following sets:

If , we say that is strongly almost -convergent of order with respect to the Orlicz function . In this case we write .(i)If we take for all , then we get , , and .(ii)If we take and for all , then we get , , and .(iii)If we take and for all , then we get , , and .

#### 3. Main Results

In this section we give the main results of this paper.

The proof of each of the following results is straightforward, so we state the following.

Theorem 3. *Let be bounded, let be as above, and let ; then .*

Theorem 4. *Let be bounded, let be as above, and let ; then , , , and are closed under the operations of addition and scalar multiplication.*

Theorem 5. *Let be bounded, let be as above, let , and let , be Orlicz functions; then*(i)*,*(ii)*,*(iii)*.*

Theorem 6. *Let and be bounded, let be as above, and let ; then .*

Theorem 7. *Let be bounded, let be as above, and let . If , then .*

*Proof. *For given we have and so Hence we get .

Theorem 8. *Let be bounded, let be as above, and let ; then .*

*Proof. *The following inequality yields the proof:

Theorem 9. *Let be bounded, let be as above, let , and let be an Orlicz function; then*(i)*,*(ii)*,*(iii)*.*

*Proof. *It is easy so it is omitted.

Theorem 10. *If and is strongly almost -convergent of order to , with respect to the Orlicz function , then is unique.*

*Proof. *Suppose that , , and ; then there exist and such that Let . Then we have where and . Thus Also, since clearly therefore . Hence .

Theorem 11. *Let be as above; then and , where *

*Proof. *Let . Then we can find a constant such that Hence .

Conversely, if we can find a constant such that for each and so Thus . Similar proof can be made for the other.

In Theorems 12 and 13, we will assume that the sequence is bounded and . From now on, will be denoting the sum over with and will be denoting the sum over with .

Theorem 12. *Let be as above and let such that ; then .*

*Proof. *One has ; then we have Hence .

Theorem 13. *Let be as above and let such that , , and ; then .*

*Proof. *Suppose that and . Since is bounded, there exists a constant such that . Let , and then we have Hence .

Let and be two sequences such that for all , where and . Suppose also that the parameters and are fixed real numbers such that . We will now give the following inclusion relations.

Theorem 14. *Let and be defined as above, let be an Orlicz function, and let such that . Then the following assertions hold true.*(i)*If* *then .*(ii)*If* *then .*

*Proof. *(i) It is easy so it is omitted.

(ii) Suppose that and (17) is satisfied. Since , we may write for all . Hence .

Theorem 15. *Let and be defined as above, let be an Orlicz function, and let such that . Then the following assertions hold true.*(i)*If (16) holds, then ,*(ii)*Let condition (17) be satisfied and ; then .*

*Proof. *
It is easy so it is omitted.

Theorem 16. *Let and be defined as above, let be an Orlicz function, and let such that . Then*(i)*if (16) holds, then ;*(ii)*let condition (17) be satisfied and ; then .*

*Proof. *(i) It is easy so it is omitted.

(ii) Suppose that and let be bounded; then there exists a constant such that . Then we may write for all . Hence we obtain .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.