#### Abstract

The aim of present work is to introduce and study lacunary statistical limit and lacunary statistical cluster points for generalized difference sequences of fuzzy numbers. Some inclusion relations among the sets of ordinary limit points, statistical limit points, statistical cluster points, lacunary statistical limit points, and lacunary statistical cluster points for these type of sequences are obtained.

#### 1. Introduction

The notion of statistical convergence of sequences of numbers was introduced by Fast  and Schoenberg  independently and latter discussed in , and so forth. In 1993, Fridy and Orhan  presented an interesting generalization of statistical convergence with the help of a lacunary sequence and called it lacunary statistical convergence or -convergence. Demirci  defined -limit and cluster points of number sequences and obtained some interesting results analogous to . In past years, statistical convergence has also become an interesting area of research for sequences of fuzzy numbers. The credit goes to Nuray and Savaş  who first introduced statistical convergence of sequences of fuzzy numbers. After their pioneer work, many authors have made their contribution to study different generalizations of statistical convergence for sequences of fuzzy numbers (see , etc.).

Quite recently, statistical convergence of sequences of fuzzy numbers is studied with the help of the difference operator . For instance, Bilgin  introduced strongly -summable and -statistical convergence of sequences of fuzzy numbers. Işik  studied some notions of generalized difference sequences of numbers. In 2006, Altin et al.  united lacunary sequences to introduce the concept of lacunary statistical convergence of generalized difference sequences of fuzzy numbers and obtained some interesting results. Some more work in this direction can be found in . In present work, we continue with this study and introduce the concepts of lacunary statistical limit and cluster points of generalized difference sequences of fuzzy numbers. We obtain some relations among the sets of ordinary limit, points, lacunary statistical limit, and cluster points for these type of sequences.

#### 2. Background and Preliminaries

We begin with the following terminology on fuzzy numbers. Given any interval , we shall denote its end points by and by the set of all closed bounded intervals on real line , that is, . For we define if and only if and . Moreover, the distance function defined by is a Hausdorff metric on and is a complete metric space. Also is a partial order on .

A fuzzy number is a function from to which is satisfying the following conditions: (i) is normal, that is, there exists such that ; (ii) is fuzzy convex, that is, for any and , ; (iii) is upper semicontinuous; and (iv) the closure of the set denoted by is compact.

Properties (i)–(iv) imply that for each , the -level set, , is a nonempty compact convex subset of . Let denote the set of all fuzzy numbers. The linear structure of induces an addition and a scalar multiplication in terms of -level sets by for each . Define a map by Puri and Ralescu  proved that is a complete metric space. Also the ordered structure on is defined as follows. For , we define if and only if and for each . We say that if and there exist such that or . The fuzzy numbers and are said to be incomparable if neither nor .

We next recall some definitions and results which form the base for present study. For any set , let denote the set and denote the number of elements in . The natural density of is defined by . The natural density may not exist for each set . But the upper density defined by always exists for each set . Moreover, different from zero means . Besides that, and if , then .

For any sequence of fuzzy numbers, we write to denote the range of . If is a subsequence of and , then we abbreviate by . If , is called a thin subsequence, otherwise if , is called nonthin subsequence of .

For , the set of all sequences of fuzzy numbers, the operator is defined by

Definition 1. A sequence of fuzzy numbers is said to be -statistically convergent to a fuzzy number , in symbol: , if for each , Let denote the set of all -statistically convergent sequences of fuzzy numbers.

Definition 2. Let be a sequence of fuzzy numbers. A fuzzy number is said to be a statistical limit point (s.l.p) of the generalized difference sequence of fuzzy numbers provided that there is a nonthin subsequence of that is -convergent to .
Let denote the set of all s.l.p. of the generalized difference sequence of fuzzy numbers.

Definition 3. Let be a sequence of fuzzy numbers. A fuzzy number is said to be a statistical cluster point (s.c.p) of the generalized difference sequence of fuzzy numbers provided that, for each , Let denote the set of all s.c.p of the generalized difference sequence of fuzzy numbers.

By a lacunary sequence we mean an increasing sequence of positive integers such that and as . The intervals determined by will be denoted by whereas the ratio is denoted by . Further, a lacunary sequence is called a lacunary refinement of the lacunary sequence if .

Definition 4 (see ). Let be a lacunary sequence. A sequence of fuzzy numbers is said to be lacunary statistical convergent to a fuzzy number provided that for each , Let denote the set of all lacunary statistically convergent sequences of fuzzy numbers.

Let be a lacunary sequence and a sequence of fuzzy numbers. If where is a subsequence of such that we call a -thin subsequence. On the other hand, is a -nonthin subsequence of provided that

Definition 5. Let be a lacunary sequence. A sequence of fuzzy numbers is said to be lacunary -statistically convergent to a fuzzy number , in symbol: , if for each , Let denote the set of all lacunary -statistically convergent sequences of fuzzy numbers.

We now consider the natural definitions of statistical limit and cluster points for generalized difference sequences of fuzzy numbers with respect to lacunary sequences.

#### 3. Main Results

Definition 6. Let be a lacunary sequence and a sequence of fuzzy numbers. A fuzzy number is said to be a lacunary statistical limit point (l.s.l.p) of the generalized difference sequence of fuzzy numbers provided that there is a -nonthin subsequence of that is -convergent to .

Let denote the set of all l.s.l.p. of the generalized difference sequence of fuzzy numbers.

Definition 7. Let be a lacunary sequence and a sequence of fuzzy numbers. A fuzzy number is said to be a lacunary statistical cluster point (l.s.c.p) of the generalized difference sequence of fuzzy numbers provided that, for each , Let denote the set of all l.s.c.p of the generalized difference sequence of fuzzy numbers.

Example 8. Let be a lacunary sequence. We define a sequence of fuzzy numbers as follows. For , define Then, we obtain Thus, for , it is clear that the sequence has two different subsequences which converge to and , respectively, where and . Hence, if denotes the set of ordinary limit points of , then ; however, .

Theorem 9. Let be a lacunary sequence and a sequence of fuzzy numbers. Then, one has .

Proof. Suppose . By definition, there is a -nonthin subsequence of which is -convergent to , and therefore we have Since, for every , so we have the containment Now, is -convergent to , which implies that, for every , is finite for which we have Thus from (15), we obtain using (13) and (16). This shows that and therefore the result is proved.

Theorem 10. Let be a lacunary sequence. Then, for any sequence of fuzzy numbers, one has .

Proof. Assume . By definition, for each we have We set a -nonthin subsequence of such that for . Since , it follows that is an infinite set. Thus we have a subsequence of that is -convergent to . This shows that . Hence .

Theorem 11. Let be a lacunary sequence. If and are two sequences of fuzzy numbers such that , then and .

Proof. We prove the theorem into two parts. In the first part we prove that ; however, in the second part we shall prove .
Part (i). Let . By definition, there is a -nonthin subsequence of that is -convergent to . Since , it follows that . Therefore, from the later set, we can yield a -nonthin subsequence of that is -convergent to . Hence, , and therefore we have . Also by symmetry one get . On combining we have .
Part (ii). Let . By definition, for each , Since for all most all , it follows that, for each , This shows that and therefore . By symmetry, we see that , whence .

Theorem 12. Let be a lacunary sequence. If is a sequence of fuzzy numbers such that , then .

Proof. We prove the theorem in two parts. In the first part, we prove that whereas in the second part we obtain .
Part (i). Suppose that , where , that is, is a l.s.l.p. of the generalized difference sequence different from . Choose such that . By definition there exist two -nonthin subsequences and of the sequence which are -convergent to and , respectively. Since is -convergent to , so for each , is a finite set for which Further, we can write for which we have Since is nonthin, so, by use of (21), we have Since , so for each and therefore we can write Furthermore for , which immediately gives the containment for which we have As left side of (29) cannot be negative, so we must have This contradicts (24). Hence, .
Part (ii). Let be a l.s.c.p. of the generalized difference sequence different from , that is, , where . Choose such that . Since is a l.s.c.p of , so for each we have Since for every , it follows that for which we have by (31), which is impossible as by (25) . In this way we obtained a contradiction. Hence, .

Theorem 13. Let be a lacunary sequence and a sequence of fuzzy numbers. Then one has the following:(i)if  , then ,(ii)if  , then ,(iii)if  , then .

Proof. (i) Suppose ; there exists a such that for sufficient large , which implies that . Assume that , then there is -nonthin subsequence of that is -convergent to and Since it follows by (33) that . Since is already -convergent to , so we have . Hence .
(ii) If , then there exists a real number such that for all . Without loss of generality, we can assume (as otherwise ). Now for all . Let , then there is a set with and . Let and . For any integer satisfying , we can write Suppose as . Since is a lacunary sequence and the first part on the right side of above expression is a regular weighted mean transform of the sequence , therefore it too tends to zero as . Since as , it follows that which is a contradiction as . Thus and therefore . Hence .
(iii) This is an immediate consequence of (i) and (ii).

Theorem 14. Let be a lacunary sequence and a sequence of fuzzy numbers. Then one has the following:(i)if  , then ,(ii)if  , then ,(iii)if  , then .

Proof. The proof of the theorem can be obtain on the similar lines as that of the above theorem and therefore is omitted here.

Theorem 15. For any lacunary refinement of a lacunary sequence , and .

Proof. Suppose each of contains the points of so that , where . Note that for all , . Let be the sequence of abutting intervals ordered by increasing right end points. Let , then for each , As before, write and . Now for each , we can write where is the characteristics function of the set and . Suppose . Then the right side of above expression is a regular weighted mean transform of and therefore tends to zero as which contradicts (36). Thus , which shows that . Hence .
Similarly, we can prove .

#### Acknowledgment

The authors are grateful to the referees for their valuable suggestions which improved the readability of the paper.