Almost -Statistical and Strongly Almost -Convergence of Order of Sequences of Fuzzy Numbers

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.