Available online at www.sciencedirect.com Chaos, Solitons and Fractals 40 (2009) 1106–1117 www.elsevier.com/locate/chaos Generalized difference sequences of fuzzy numbers Rifat Çolak *, Hıfsı Altınok, Mikail Et Department of Mathematics, Firat University, 23119 Elazığ, Turkey Accepted 29 August 2007 Abstract The idea of difference sequences of real (or complex) numbers was generalized by Et and Çolak [Et M, Çolak R. On some generalized difference sequence spaces. Soochow J Math 1995; 21(4): 377–86; Çolak R, Et M. On some generalized difference sequence spaces and related matrix transformations. Hokkaido Math J 1997; 26(3): 483–92]. In this paper, using the difference operator Dm and an Orlicz function, we introduce and examine some sequence spaces of fuzzy numbers. We study some of their properties like completeness, solidity, symmetricity, etc. We also give some relations related to these spaces.  2007 Elsevier Ltd. All rights reserved. 1. Introduction The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh [35] and subsequently several authors have discussed various aspects of the theory and applications of fuzzy sets such as fuzzy topological spaces, similarity relations and fuzzy orderings, fuzzy measures of fuzzy events, fuzzy mathematical programming. Especially, the concept of fuzzy topology has very important applications in quantum particle physics, particularly in connections with both string and e(1) theory which were given and studied by El Naschie ([13,14]). Recently, Saadati and Park [30] has introduced the notion of intuitionistic fuzzy normed space. Matloka [26] introduced bounded and convergent sequences of fuzzy numbers, studied some of their properties and showed that every convergent sequence of fuzzy numbers is bounded. In addition, sequences of fuzzy numbers have been discussed by Altin et al. [1], Altinok et al. [2], Aytar and Pehlivan [5], Basßarır and Mursaleen ([6,27]), Et et al. [17], Nuray [28], Savas [32] and many others. The notion of statistical convergence was introduced by Fast [18] and Schoenberg [33], independently. Over the years and under different names statistical convergence has been discussed in the theory of fourier analysis, ergodic theory and number theory. Later on it was further investigated from the sequence space point of view and linked with summability theory by Connor [8], Fridy [19], Šalát [31], Tripathy [34] and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations are also connected with subsets of the Stone-Čech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability ([9,10]). * Corresponding author. E-mail addresses: rcolak@firat.edu.tr (R. Çolak), hifsialtinok@yahoo.com (H. Altınok), mikailet@yahoo.com (M. Et). 0960-0779/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.065 R. Çolak et al. / Chaos, Solitons and Fractals 40 (2009) 1106–1117 1107 The study of Orlicz sequence spaces was initiated with a certain specific purpose in Banach space theory. Indeed, Lindberg [24] got interested in Orlicz spaces in connection with finding Banach spaces with symmetric Schauder bases having complementary subspaces isomorphic to c0 or ‘p (1 6 p < 1). Subsequently Lindenstrauss and Tzafriri [25] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space ‘M contains a subspace isomorphic to ‘p (1 6 p < 1) [20]. The existing literature on statistical convergence appears to have been restricted to real or complex sequences, but in Basßarır and Mursaleen ([6,27]), Bilgin [7], Et et al.[17], Nuray [28] extended the idea to apply to sequences of fuzzy numbers. In the present paper, using an Orlicz function and the generalized difference operator Dm, we introduce and examine some sequence spaces of fuzzy numbers. In Section 2, we give a brief overview about statistical convergence, fuzzy numbers, Orlicz function and the operator Dm. In Section 3, we establish some relations about the sequence spaces c(Dm,F,M,p), c0(Dm,F,M,p), ‘1(Dm,F,M,p) and S(Dm,F,M,p). 2. Definitions and preliminaries The definitions of statistical convergence and strong p–Cesàro convergence of a sequence of real numbers were introduced in the literature independently of one another and have followed different lines of development since their first appearance. It turns out, however, that the two definitions can be simply related to one another in general and are equivalent for bounded sequences. The idea of statistical convergence depends on the density of subsets of the set N of natural numbers. The density of a subset E of N is defined by dðEÞ ¼ lim n!1 n 1X v ðkÞ n k¼1 E provided the limit exists, where vE is the characteristic function of E. It is clear that any finite subset of N has zero natural density and d(Ec) = 1  d(E). A sequence (xk) is said to be statistically convergent to ‘if for every e > 0, dðfk 2 N : jxk  ‘j P egÞ ¼ 0. In this case we write S  lim xk = ‘. Fuzzy sets are considered with respect to a nonempty base set X of elements of interest. The essential idea is that each element x 2 X is assigned a membership grade u(x) taking values in [0,1], with u(x) = 0 corresponding to nonmembership, 0 < u(x) < 1 to partial membership, and u(x) = 1 to full membership. According to Zadeh a fuzzy subset of X is a nonempty subset {(x,u(x)):x 2 X} of X · [0,1] for some function u:X ! [0,1]. The function u itself is often used for the fuzzy set. Let CðRn Þ denote the family of all nonempty, compact, convex subsets of Rn . If a; b 2 R and A; B 2 CðRn Þ, then aðA þ BÞ ¼ aA þ aB; ðabÞA ¼ aðbAÞ; 1A ¼ A and if a,b P 0, then (a + b)A = aA + bA. The distance between A and B is defined by the Haussdorff metric d1 ðA; BÞ ¼ maxfsup inf ka  bk; sup inf ka  bkg; a2A b2B b2B a2A where k  k denotes the usual Euclidean norm in Rn . It is well known that ðCðRn Þ; d1 Þ is a complete metric space. Denote LðRn Þ ¼ fu : Rn ! ½0; 1ju satisfies ðiÞ  ðivÞ belowg; where (i) (ii) (iii) (iv) u is normal, that is, there exists an x0 2 Rn such that u(x0) = 1; u is fuzzy convex, that is, for x; y 2 Rn and 0 6 k 6 1,u(kx + (1  k)y) P min[u(x),u(y)]; u is upper semicontinuous; and the closure of fx 2 Rn : uðxÞ > 0g; denoted by [u]0, is compact. If u 2 LðRn Þ, then u is called a fuzzy number, and LðRn Þ is said to be a fuzzy number space. For 0 < a 6 1, the a-level set [u]a is defined by ½ua ¼ fx 2 Rn : uðxÞ P ag: Then from (i)–(iv), it follows that the a-level sets ½ua 2 CðRn Þ. For the addition and scalar multiplication in LðRn Þ; we have 1108 R. Çolak et al. / Chaos, Solitons and Fractals 40 (2009) 1106–1117 ½u þ va ¼ ½ua þ ½va ; ½kua ¼ k½ua where u; v 2 LðRn Þ; k 2 R. Define, for each 1 6 q < 1, d q ðu; vÞ ¼ Z 0 1 1=q ½d1 ð½u ; ½v Þ da a a q and d 1 ðu; vÞ ¼ sup d1 ð½ua ; ½va Þ; where d1 is the Haussdorff metric. Clearly d 1 ðu; vÞ ¼ lim d q ðu; vÞ with dq 6 ds if q 6 s q!1 06a61 ([12,23]). A sequence X = (Xk) of fuzzy numbers is a function X from the set N of all positive integers into LðRn Þ. Thus, a sequence of fuzzy numbers (Xk) is a correspondence from the set of positive integers to a set of fuzzy numbers, i.e., to each positive integer k there corresponds a fuzzy number X(k). It is more common to write Xk rather than X(k) and to denote the sequence by (Xk) rather than X. The fuzzy number Xk is called the kth term of the sequence. Let X = (Xk) be a sequence of fuzzy numbers. The sequence X = (Xk) of fuzzy numbers is said to be bounded if the set fX k : k 2 Ng of fuzzy numbers is bounded and convergent to the fuzzy number X0, written as limkXk =X0, if for every e > 0 there exists a positive integer k0 such that d(Xk,X0) < e for k > k0. Let ‘1(F) and c(F) denote the set of all bounded sequences and all convergent sequences of fuzzy numbers, respectively [26]. Recall ([20,21,29]) that an Orlicz function is a function M:[0,1) ! [0,1), which is continuous, non-decreasing and convex with M(0) = 0, M(x) > 0 for x > 0 and M(x) ! 1 as x ! 1. Lindenstrauss and Tzafriri [25] used the idea of Orlicz function to construct the sequence space ( )   1 X jxk j ‘M ¼ x 2 w : M < 1; for some q > 0 : q k¼1 The space ‘M is a Banach space with the norm ( )   1 X jxk j M kxk ¼ inf q > 0 : 61 q k¼1 and this space is called an Orlicz sequence space. For M(t) = tp, 1 6 p < 1, the space ‘M coincides with the classical sequence space ‘p. The difference spaces ‘1(D), c(D) and c0(D), consisting of all real valued sequences x = (xk) such that Dx = D1x = (xk  xk+1) in the sequence spaces ‘1, c and c0, were defined by Kızmaz [22]. Continuing on this way, Basßar and Altay [4] have recently introduced the difference space bvp of real sequences whose D-transforms are in the space ‘p, where Dx = (xk  xk1) and 1 6 p 6 1. Moreover some spectral properties of the operator D are given by Altay and Basßar [3]. The idea of difference sequences was generalized by Çolak and Et ([11,16]) Et and Basßarır [15]. Let w(F) be the set of all sequences of fuzzy numbers. The operator Dm:w(F) ! w(F) is defined by ðD0 X Þk ¼ X k ; ðD1 X Þk ¼ D1 X k ¼ X k  X kþ1 ; ðDm X Þk ¼ D1 ðDm1 X Þk ; ðm P 2Þ; for all k 2 N: Definition 2.1. [17] Let X = (Xk) be a sequence of fuzzy numbers. Then the sequence X = (Xk) is said to be Dm-bounded if the set fDm X k : k 2 Ng of fuzzy numbers is bounded, and Dm-convergent to the fuzzy number X0, written as limkDmXk = X0, if for every e > 0 there exists a positive integer k0 such that d(DmXk,X0) < e for all k > k0. By ‘1(Dm,F) and c(Dm,F) denote the sets of all Dm-bounded sequences and all Dm-convergent sequences of fuzzy numbers, respectively. Definition 2.2. Let X = (Xk) be a sequence of fuzzy numbers, p = (pk) be any sequence of strictly positive real numbers and M be an Orlicz function. We define the following sequence spaces    pk dðDm X k ; X 0 Þ ¼ 0; X ¼ ðX k Þ : lim M k!1 q (
pk   d Dm X k ;  0 ¼ 0; c0 ðDm ; F ; M; pÞ ¼ X ¼ ðX k Þ : lim M k!1 q cðDm ; F ; M; pÞ ¼  for some q > 0 ; ) for some q > 0 ; 1109 R. Çolak et al. / Chaos, Solitons and Fractals 40 (2009) 1106–1117     pk dðDm X k ; 0Þ < 1; for some q > 0 ; ‘1 ðDm ; F ; M; pÞ ¼ X ¼ ðX k Þ : sup M q kP0     pk dðDm X k ; X 0 Þ s m ¼ 0 for some q > 0 ; SðD ; F ; M; pÞ ¼ X ¼ ðX k Þ : lim M k!1 q ( )
pk   m d D X k ; 0 s ¼ 0 for some q > 0 ; S 0 ðDm ; F ; M; pÞ ¼ X ¼ ðX k Þ : lim M k!1 q where  1; t ¼ ð0; 0; 0; . . . ; 0Þ and s in the brackets means that the limit is statistical in the definition of 0; otherwise m m S(D ,F,M,p) and S0(D ,F,M,p). By using these spaces, we can construct the sequence spaces  0ðtÞ ¼ mðDm ; F ; M; pÞ ¼ SðDm ; F ; M; pÞ \ ‘1 ðDm ; F ; M; pÞ m0 ðDm ; F ; M; pÞ ¼ S 0 ðDm ; F ; M; pÞ \ ‘1 ðDm ; F ; M; pÞ: Throughout the paper Z will denote any one of the notation c, c0, ‘1, S, S0, m and m0. If we take M(x) = x and pk ¼ 1ðfor all k 2 NÞ in the spaces Z(Dm,F,M,p), then we get the sequence spaces Z(Dm,F). In the case pk = 1 for all k 2 N we shall write Z(Dm,F,M) instead of Z(Dm,F,M,p). The sequence spaces Z(Dm,F,M,p) contain the sequences of fuzzy numbers which are unbounded and divergent, too. Example 1. Let M(x) = x and 8 > < x  2k þ 1; X k ðxÞ ¼ x þ 2k þ 1; > : 0; pk = 1 for allk 2 N. Consider the sequence if x 2 ½2k  1; 2k if x 2 ½2k; 2k þ 1 : otherwise Then the sequence X = (Xk) is Dm-bounded and Dm-convergent, but it is neither bounded nor convergent (Fig. 1). Really, for a 2 (0,1], a-level sets of Xk and DmXk are ½X k a ¼ ½2k  1 þ a; 2k þ 1  a and ½Dm X k a ¼  ½4 þ 2a; 2a; if m ¼ 1 m m ½2 ð1  aÞ; 2 ð1  aÞ; if m P 2 for m = 1,2, . . . , respectively. It is clear that (Dm Xk) is convergent to fuzzy number X0 for m P 2, where [X0]a = [  2m(1  a),2m(1  a)]. s s For the classical sets, (xk) converges to ‘ implies (Dmxk) converges to 0 (or xk ! ‘ implies Dm xk ! 0Þ. The following example shows that this is not valid for the sequences of fuzzy numbers. Example 2. Define the sequence (Xk) as follows: 8 9 x  k þ 1; if k  1 6 x 6 k = > > > > > x þ k þ 1; if k < x 6 k þ 1 ; > > ; < 0; otherwise 9 X k ðxÞ ¼ x  1; if 1 6 x 6 2 = > > > > > x þ 3; if 2 < x 6 3 ; > > ; : 0; otherwise if k ¼ n2 ðn ¼ 1; 2; 3; . . .Þ : otherwise ∆X k 1 2 3 m X1 X2 Xk ∆ Xk ∆ Xk ∆ Xk 2k-1 —2 m -8 -4 —2 0 2 4 8 2k+1 2k m 2 Fig. 1. A fuzzy sequence which is Dm-bounded and Dm-convergent, but un bounded and divergent. 1110 R. Çolak et al. / Chaos, Solitons and Fractals 40 (2009) 1106–1117 Then we obtain  ½k  1 þ a; k þ 1  a; ½X k a ¼ ½1 þ a; 3  a; if k ¼ n2 otherwise and 8 > < ½k  4 þ 2a; k  2a; ½DX k a ¼ ½k  1 þ 2a; k þ 3  2a; > : ½2 þ 2a; 2  2a; if k ¼ n2 if k þ 1 ¼ n2 ; ðn > 1Þ : otherwise s s Thus we have X k ! ‘1 , where [‘1]a = [1 + a,3  a] and DX k ! ‘2 , where [‘2] a = [  2 + 2a,2  2a]. Moreover, taking difs s ference order of m, we have Dm X k ! ‘3 and ‘3 –0, where [‘3]a = [  2m(1  a), 2m(1  a)] in case X k ! ‘1 . m m m It can be shown that c0(D ,F,M)  c(D ,F,M)  ‘1(D ,F,M) and the inclusions are strict. This is clear from the following example. Example 3. Let M(x) = x 8 x  1; > > > > > > x þ 3; > > > < 0; X k ðxÞ ¼ > x  3; > > > > > x þ 5; > > > : 0; and consider the sequence X = (Xk) where 9 if x 2 ½1; 2 > = if k is odd if x 2 ½2; 3 > ; otherwise 9 : if x 2 ½3; 4 > = if k is even if x 2 ½4; 5 > ; otherwise Then the sequence X = (Xk) is Dm- bounded, but not Dm-convergent. Indeed, for a 2 (0,1], a-level sets of Xk and DmXk are  ½1 þ a; 3  a; if k is odd ½X k a ¼ ½3 þ a; 5  a; if k is even and ½Dm X k a ¼  ½2m ð2  aÞ; 2m a; ½2m a; 2m ð2  aÞ; if k is odd if k is even for m = 1,2, . . . , respectively. Thus (DmXk) is bounded, but not convergent (Fig. 2). A sequence space E(F) is said to be normal (or solid) if (Xk) 2 E(F) and (Yk) is such that dðY k ;  0Þ 6 dðX k ;  0Þ implies (Yk) 2 E(F). A sequence space E(F) is said to be monotone if E(F) contains the canonical pre-images of all its step spaces. Let K ¼ fk n : k n < k nþ1 ; n 2 Ng # N and E(F) be a sequence space. A K-step space of E(F) is a sequence space EðF Þ kK ¼ fðX kn Þ 2 wðF Þ : ðX n Þ 2 EðF Þg. EðF Þ A canonical pre-image of a sequence ðX kn Þ 2 kK is a sequence (Yn) 2 w(F) defined as  X n ; if n 2 K; Yn ¼  : 0; otherwise ∆mXk ( k odd) ∆2Xk (k odd) ∆Xk (k even) ∆2Xk (k even) ∆Xk (k odd) ∆mXk (k even) 1 -2m+1 -2 m -8 -4 -2 0 2 4 Xk (k odd) 8 2m Xk (k even) Fig. 2. A fuzzy sequence which is Dm-bounded, but not Dm-convergent. 2 m+1 1111 R. Çolak et al. / Chaos, Solitons and Fractals 40 (2009) 1106–1117 EðF Þ EðF Þ A canonical pre-image of a step space kK is a set of canonical pre-images of all elements in kK , i.e., Y is in canonEðF Þ EðF Þ ical pre-image kK if and only if Y is canonical pre-image of some X 2 kK . Remark. If a sequence space E(F) is solid, then E(F) is monotone. A sequence space E(F) is said to be symmetric if (Xp(n)) 2 E(F), whenever (Xk) 2 E(F), where p is a permutation of N. A sequence space E(F) is said to be a sequence algebra if (Xk  Yk) 2 E(F), whenever (Xk), (Yk) 2 E(F). A sequence space E(F) is said to be convergence free if (Yk) 2 E(F), whenever (Xk) 2 E(F) and X k ¼  0 implies Y k ¼  0. 3. Main results In this section, we establish some relations about the sequence spaces c(Dm,F,M,p), c0(Dm,F,M,p), ‘1(Dm,F,M,p) and S(Dm,F,M,p). The proofs of the following results are easy and thus omitted. Theorem 3.1. The sequence spaces Z(Dm,F,M,p) are closed under the operations of addition and scalar multiplication. Theorem 3.2. The sequence spaces c(Dm,F), c0(Dm,F), m(Dm,F) and m0(Dm,F) are closed subspaces of the complete metric space ‘1(Dm,F) with the metric qD ðX ; Y Þ ¼ m X dðX k ; Y k Þ þ sup½dðDm X k ; Dm Y k Þ: ð1Þ kP0 k¼1 Theorem 3.3. Let M1 and M2 be two Orlicz functions. Then (i) Z(Dm, F, M1, p) \ Z(Dm, F, M2, p)  Z(Dm, F, M1 + M2, p), (ii) Z(Dm, F, M1)  Z(Dm, F, M1oM2). Theorem 3.4. The spaces c(Dm,F,p) and ‘1(Dm,F,p) are complete metric spaces with the metric dD ðX ; Y Þ ¼ m X pk dðX k ; Y k Þ þ sup½dðDm X k ; Dm Y k Þ K ; ð2Þ kP0 k¼1 where K = max(1,supkP0pk). Proof. We shall prove only for the space ‘1(Dm,F,p). The other can be treated, similarly. It can be shown that dD is a metric on ‘1(Dm,F,p). Let (Xs) be a Cauchy sequence in ‘1(Dm,F,p), where X s ¼ ðX si Þi ¼ ðX s1 ; X s2 ; . . .Þ 2 ‘1 ðDm ; F ; pÞ for each s 2 N. Then dD ðX s ; X t Þ ! 0; and, therefore, we get m X dðX sk ; X tk Þ ! 0 as s; t ! 1 and so that dðDm X sk ; Dm X tk Þ ! 0 ð3Þ k¼1 as s,t ! 1 for each k 2 N. Since dðX skþm ; X tkþm Þ 6 dðDm X sk ; Dm X tk Þ þ    
m m d X skþm1 ; X tkþm1 dðX sk ; X tk Þ þ . . . þ m1 0 we write dðX sk ; X tk Þ ! 0 for all k 2 N as s,t ! 1. Hence ðX sk Þs ¼ ðX 1k ; X 2k ; . . .Þ is a Cauchy sequence in LðRn Þ. Since LðRn Þ is complete, ðX sk Þs is a convergent sequence for each k 2 N. Take lims X sk ¼ X k . Now, we will show the sequence (Xs) is convergent and its limit is X = (Xk) = (X1,X2, . . . ). Since (Xs) is a Cauchy sequence in ‘1(Dm,F,p), for every e > 0 there is a number n0 = n0(e) such that dD(Xs,Xt) < e for all s,t P n0. From here we get m m X

X d X sk ; X k < e d X sk ; X tk ¼ lim t k¼1 k¼1 1112 R. Çolak et al. / Chaos, Solitons and Fractals 40 (2009) 1106–1117 and
pk 
pk  lim d Dm X sk ; Dm X tk K ¼ d Dm X sk ; Dm X k K < e t for s P n0 and so that dD ðX s ; X Þ ¼ m X k¼1 
pk dðX sk ; X k Þ þ supk d Dm X sk ; Dm X k K < 2e; for s P n0, that is, Xs ! X for s ! 1. On the other hand, since

d Dm X k ; 0 6 dðDm X nk 0 ; Dm X k Þ þ d Dm X nk 0 ; 0 ; we obtain 
p 
p  p d Dm X k ; 0 k 6 D ½dðDm X nk 0 ; Dm X k Þ k þ d Dm X nk 0 ;  0 k ; where 0 < pk 6 supkpk = G and D = max(1,2G1). Hence X 2 ‘1(Dm,F,p). h Theorem 3.5. The inclusions Z(Dm1,F,M)  Z(Dm,F,M) are strict, for m P 1. In general Z(Di,F,M)  Z(Dm,F,M) for i = 1,2, . . . ,m  1 and the inclusion is strict. Proof. We give the proof for Z = ‘1 only. Choose q = 2q1. Then we observe that (Xk) 2 ‘1 (Dm1,F,M) implies (Xk) 2 ‘1(Dm,F,M) from the inequality ("
 
!# "
!#)   d Dm X k ; 0 d Dm1 X k ; 0 d Dm1 X kþ1 ;  0 1 þ M : M M 6 2 q q1 q1 We get ‘1 ðDi ; F ; MÞ # ‘1 ðDm ; F ; MÞ for i = 0,1, . . . ,m  1 by applying induction. The inclusion is strict, for this consider the following example. Example 4. Let M(x) = x, m = 2. Consider the sequence (Xk) of fuzzy numbers as follows: 8 2 x > <  k 2 1 þ 1; for x 2 ½k  1; 0 X k ðxÞ ¼  2x þ 1; for x 2 ½0; k 2 þ 1 : > : k þ1 0; otherwise For a 2 (0,1], a-level sets of Xk, D Xk and D2Xk are ½X k a ¼ ½ðk 2  1Þð1  aÞ; ðk 2 þ 1Þð1  aÞ; ½DX k a ¼ ½ð2k  3Þð1  aÞ; ð2k þ 1Þð1  aÞ; ½D2 X k a ¼ ½2ð1  aÞ; 6ð1  aÞ; respectively. It is easy to see that the sequence (DXk) is not bounded although (D2Xk) is bounded. Theorem 3.6. Let 0 < pk 6 qk < 1 for each k. Then cðDm ; F ; M; pÞ # cðDm ; F ; M; qÞ and c0 ðDm ; F ; M; pÞ # c0 ðDm ; F ; M; qÞ: Proof. Let (Xk) 2 c0(Dm,F,M,p). Then, there exists a number q > 0 such that
pk   d Dm X k ; 0 lim M ¼ 0: k!1 q h R. Çolak et al. / Chaos, Solitons and Fractals 40 (2009) 1106–1117 1113 This requires
  d Dm X k ; 0 M 61 q for sufficiently large k. Therefore, since pk 6 qk for each k, we get
qk
pk     d Dm X k ; 0 d Dm X k ; 0 6 lim M ¼ 0; lim M k!1 k!1 q q i.e. (Xk) 2 c0(Dm,F,M,q). This completes the proof. h m Similarly, it can be shown that c(D ,F,M,p)  c(Dm,F,M,q). If we take pk = 1 and qk = 1 for all k 2 N, in previous theorem, we get the following results. Corollary 3.7 (a) Let 0 < inf pk 6 pk 6 1. Then (i) c(Dm,F,M,p)  c(Dm,F,M) (ii) c0(Dm,F,M,p)  c0(Dm,F,M). (b) Let 1 6 pk 6 suppk < 1. Then (i) c(Dm,F,M)  c(Dm,F,M,p) (ii) c0(Dm,F,M)  c0(Dm,F,M,p). Theorem 3.8. The sequence spaces c0(F,M) and ‘1(F,M) are solid and hence monotone, but the sequence spaces Z(Dm,F,M,p) are neither monotone nor solid. Proof. We give the proof for ‘1(F,M). Let (Xk) 2 ‘1(F,M) and (Yk) be a sequence of fuzzy numbers such that dðY k ; 0Þ 6 dðX k ; 0Þ for all k 2 N. Since M is non-decreasing, we have

   d Y k ; 0 d X k ; 0 6 sup M : sup M q q Hence ‘1(F,M) is solid. h It follows from the following example that the spaces Z(Dm,F,M,p) are not monotone and are not solid. Example 5. Let M(x) = x, m = 1 and pk = 1 for all k 2 N. Consider the sequence X = (Xk) defined by 8 k 9 k x  2k3 ; for 1 6 x 6 3k3 > > k > > > 2k3 > > = > if k ¼ 3n 1; for 3k3 6 x 6 3kþ3 > k k > > ; > k 5k 3kþ3 > ðn ¼ 1; 2; 3; . . .Þ > > < 32k x  32k ; for k 6 x 6 5 > > ; otherwise X k ðxÞ ¼ 0; : > 9 > 1 > > kx  8k þ 1; for 8  k 6 x 6 8 = > > > > 1 > > ; otherwise kx þ 8k þ 1; for 8 6 x 6 8 þ > k> > : ; 0; otherwise Then a ½X k  ¼ and ( ½1 þ a; 5  a;   8  1k ð1  aÞ; 8 þ 1k ð1  aÞ ; if k ¼ 3n otherwise i 8h a1 1a > ; ; 3  a þ 7 þ a þ > kþ1 kþ1 > <   a a1 1a 3 þ k þ a; 7 þ k  a ; ½DX k  ¼ > h i > > : a1 þ a1 ; 1a þ 1a ; k kþ1 k kþ1 Thus (Xk) 2 Z(D,F), for Z = S, m, S0 and m0. if k ¼ 3n if k þ 1 ¼ 3n ; ðn > 1Þ : otherwise 1114 R. Çolak et al. / Chaos, Solitons and Fractals 40 (2009) 1106–1117 ^ Let mðD; F ÞJ be the canonical pre-image of the J-step space m(D,F)J of m(D,F) defined as follows, where J ¼ fk 2 N : k ¼ 2i þ 1; for i 2 Ng is a subset of N. ^ F ÞJ is the canonical pre-image of (Xk) 2 m(D,F), then If ðY nk Þ 2 mðD;  X k ; for k 2 J Yk ¼  : 0; for k R J Now, and 8 þ a; 5  a; > < ½1   a 8  1k ð1  aÞ; 8 þ 1k ð1  aÞ ; ½Y k  ¼ > : ½0; 0; for k 2 J ; for k 2 J ; for k R J 8 ½1 þ a; 5  a; > >  > > < 8  1k ð1  aÞ; 8 þ 1k ð1  aÞ ; a ½DY k  ¼ ½5 þ a; 1  a; > > h i > > : 8  1 ð1  aÞ; 8 þ 1 ð1  aÞ ; kþ1 kþ1 k ¼ 3n ; k–3n ; for k 2 J ; k ¼ 3n ; for k 2 J ; for k R J ; k–3n ; k þ 1 ¼ 3n ; : otherwise Hence we have (Yk) R Z(D,F), for Z = S, m, S0 and m0. Now, it follows that the spaces S(D,F), m(D,F), S0(D,F) and m0(D,F) are not monotone and thus, these are not solid, either. Theorem 3.9. The sequence spaces c0(F,M), c(F,M) and ‘1(F,M) are symmetric, but the sequence spaces Z(Dm,F,M,p) are not symmetric. Proof. It can be shown that c0(F,M), c(F,M) and ‘1(F,M) are symmetric, following the technique used for establishing the classical set cases. The spaces Z(Dm,F,M,p) are not symmetric which follow from the following example. h Example 6. Let M(x) = x, m = 1 and pk = 1 for all k 2 N. Consider the sequence (Xk) defined as follows: 8 k 9 k x  2k3 ; for 1 6 x 6 3k3 > > 2k3 k > > > > > 3k3 3kþ3 = > for k ¼ 3n 1; for 6 x 6 > < k k ; k 5k 3kþ3 X k ðxÞ ¼ 32k x  32k ; for k 6 x 6 5 > > ðn ¼ 1; 2; 3; . . .Þ : > > ; > > > 0; otherwise > > : 0; otherwise Then we have (Xk) 2 Z(D,F), where Z = S, m, S0 and m0. Now let (Yk) be rearrangement of the sequence (Xk) defined as follows: ðY k Þ ¼ ðX 3 ; X 1 ; X 9 ; X 2 ; X 27 ; X 4 ; X 81 ; X 5 ; . . .Þ: Then ½Y k a ¼  ½1 þ a; 5  a; ½0; 0; for k odd for k even and a ½DY k  ¼  ½1 þ a; 5  a; for k odd : ½5 þ a; 1  a; for k even Then (Yk) R Z(D,F) and thus the spaces S(D,F), m(D,F), S0(D,F) and m0(D,F) are not symmetric. Theorem 3.10. The sequence spaces c(Dm,F,M,p), c0(Dm,F,M,p), ‘1(Dm,F,M,p) and S(Dm,F,M,p) are not sequence algebra. Proof. These spaces are not sequence algebra which follow from the following example. h 1115 R. Çolak et al. / Chaos, Solitons and Fractals 40 (2009) 1106–1117 Example 7. Let M(x) = x, m = 1 and pk = 1 for all k 2 N. Define the sequences (Xk), (Yk) as 8 k 9 k x  2k3 ; for 1 6 x 6 3k3 > > 2k3 k > > > > > 3k3 3kþ3 = > for k ¼ 3n 6 x 6 1; for > k k > > ; > k 5k 3kþ3 > ðn ¼ 1; 2; 3; . . .Þ > > < 32k x  32k ; for k 6 x 6 5 > > ; X k ðxÞ ¼ 0; otherwise > 9 > > > kx  8k þ 1; for 8  1k 6 x 6 8 = > > > > > > ; otherwise x þ 9; for 8 6 x 6 9 > > > : ; 0; otherwise and Now 8 9 x  k  1; for k þ 1 6 x 6 k þ 2 > > > = > > > > x þ k þ 3; for k þ 2 6 x 6 k þ 3 ; > > > > ; < 0; otherwise 9 Y k ðxÞ ¼ > x  k; for k 6 x 6 k þ 1 > > = > > > 3 > ; 2x þ 2k þ 3; for k þ 1 6 x 6 k þ > 2> > > ; : 0; otherwise ½X k  ¼ ( ½1 þ a; 5  a; for k ¼ 3n   1 8  k ð1  aÞ; 9  a ; otherwise ½Y k a ¼ ( ½k þ 1 þ a; k þ 3  a; for k ¼ 3n   ; k þ a; k þ 32  a2 ; otherwise a and so we get and i 8h 1a > ; 8 þ 2a; 3  a þ > kþ1 > <   a 1a 3  k þ a; 8  2a ; ½DX k  ¼ > h i > > : 1  1a þ a; 1  a þ 1a ; k kþ1 8  3 3a  > <  2 þ 2 ; 2  2a ; 4 þ 2a;  12  3a2 ; ½DY k a ¼ > :  5 3a 1 3a 2 þ 2 ;2  2 ; for k ¼ 3n ðn ¼ 1; 2; 3; . . .Þ : otherwise for k ¼ 3n for k þ 1 ¼ 3n ; ðn > 1Þ otherwise for k ¼ 3n for k þ 1 ¼ 3n ; ðn > 1Þ : otherwise For the sequences (Xk) and (Yk) we have (Xk), (Yk) 2 m(D,F)  S(D,F). Moreover, we get ½DðX k  Y k Þa ¼ ½X k  Y k  X kþ1  Y kþ1 a i 8h að1aÞ n a2 43 2 > > 8k þ 2ak þ 9a þ 2 ; 3k  ak  17a þ 8 þ a þ kþ1 ; for k ¼ 3 > > <h i 2 Þ ; for k þ 1 ¼ 3n ; ðn > 1Þ: ¼ 3k þ ak þ 18a  a2  21  að1a ; 8k  2ka  9a  a2  2 þ a þ27 k 2 > > h i > > : k þ ak þ 16a  1  að1aÞ  a2 þ45 ; k  ak  15a  7 þ a2 þ27 þ að1aÞ ; otherwise 2 2 k kþ1 Hence we have (Xk  Yk) R S(D,F) (m(D,F)). Theorem 3.11. The sequence spaces c(Dm,F,M,p), c0(Dm,F,M,p), ‘1(Dm,F,M,p) and S(Dm,F,M,p) are not convergence free. Proof. It follows from the following example that these spaces are not convergence free. h 1116 R. Çolak et al. / Chaos, Solitons and Fractals 40 (2009) 1106–1117 Example 8. Let M(x) = x, m = 1 8 0; > > > < k ðx  3Þ þ 1; X k ðxÞ ¼ 2 k >  ðx  3Þ þ 1; > > : 2 0 Then a ½X k  ¼ and ( and pk = 1 for all k 2 N. Define the sequence (Xk) as for k ¼ 3n 9 for 3k2 6 x 6 3> = k ; for 3 6 x 6 3kþ2 k > ; otherwise ½0; 0;   3  2k ð1  aÞ; 3 þ 2k ð1  aÞ ; otherwise : for k ¼ 3n otherwise  8 3  2k ð1  aÞ; 3 þ 2k ð1  aÞ ; for k ¼ 3n > >  < 2 þ 2k ð1  aÞ ; for k þ 1 ¼ 3n ; ðn > 1Þ : ½DX k a ¼ h3  k ð1 aÞ; 3   i > > : ða  1Þ 2 þ 2 ; ð1  aÞ 2 þ 2 ; otherwise k kþ1 k kþ1 Thus we have (Xk) 2 Z(D,F), for Z = S, m, S0 and m0. For the sequence (Yk), defined as 8 n 0 > > 9 for k ¼ 3 > < x  k; for k 6 x 6 k þ 1 > = : Y k ðxÞ ¼ 2 1 k 2 þ2 ; otherwise x  ; for k þ 1 6 x 6 k þ 2 > 2 2 > kk kk 1 1 > > : ; 0; otherwise  n ½0; 0; for k ¼ 3 ½Y k a ¼ 2 2 ½k þ a; k þ 2 þ aðk  k  1Þ; otherwise and 8 2 2 > < ½ðk þ 1Þ þ aðk þ k þ 1Þ  2; ðk þ 1 þ aÞ; a 2 ½DY k  ¼ ½k þ a; k þ 2 þ aðk  k 2  1Þ; > : ½k  ðk þ 1Þ2  2 þ aðk 2 þ k þ 2Þ; k 2  k þ 1 þ aðk  k 2  2Þ; for k ¼ 3n for k þ 1 ¼ 3n ; ðn > 1Þ : otherwise This implies that (Yk) R Z(D,F), for Z = S, m, S0 and m0. 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