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λ -Difference sequence spaces of fuzzy numbers
Fuzzy Sets and Systems 160 (2009) 3128 – 3139 www.elsevier.com/locate/fss
-Difference sequence spaces of fuzzy numbers Hıfsı Altınok∗ , Rifat Çolak, Mikail Et Department of Mathematics, Firat University, 23119 Elazı˘g, Turkey Received 8 December 2007; received in revised form 3 June 2009; accepted 4 June 2009 Available online 12 June 2009
Abstract In this paper, using the difference operator of order m and an Orlicz function, we introduce and examine some classes of sequences of fuzzy numbers. We give the relations between the strongly Cesàro type convergence and statistical convergence in these spaces. Furthermore, we study some of their properties like completeness, solidity, symmetricity, etc. We also give some inclusion relations related to these classes. © 2009 Elsevier B.V. All rights reserved. MSC: 40A05; 40C05; 46A45; 03E72 Keywords: Orlicz function; Fuzzy number; Statistical convergence; Difference sequence
1. Introduction The notion of statistical convergence was introduced by Fast [12] and Schoenberg [25], 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 [7], Fridy [13], Šalát [23] 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 ˇ of the Stone–Cech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability [8]. The existing literature on statistical convergence appears to have been restricted to real or complex sequences, but Nuray and Sava¸s [22], Nuray [21] and Bilgin [4] extended the idea to apply to sequences of fuzzy numbers and also Matloka [19], Nanda [20], Ba¸sarır and Mursaleen [3], Fang and Huang [11], Burgin [5] and Sava¸s [24] studied the sequences of fuzzy numbers. The study of Orlicz sequence spaces was initiated with a certain specific purpose in Banach space theory. Indeed, Lindberg [17] got interested in Orlicz spaces in connection with finding Banach spaces with symmetric Schauder bases having complementary subspaces isomorphic to c0 or p (1 ⱕ p < ∞). Subsequently Lindenstrauss and Tzafriri [18] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space M contains a subspace isomorphic to p (1 ⱕ p < ∞) [14]. ∗ Corresponding author.
E-mail addresses: [email protected] (H. Altınok), [email protected] (R. Çolak), [email protected] (M. Et). 0165-0114/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2009.06.002
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In the present paper we study some generalized sequence spaces of fuzzy numbers. The main purpose of this paper is to study the difference sequences of order m of fuzzy numbers for some sequence spaces defined by an Orlicz function so as to fill up the existing gaps in the literature. In Section 2 we give a brief overview about statistical convergence, fuzzy numbers and sequences of fuzzy numbers. In Section 3, using the generalized difference operator m , an Orlicz function and a nondecreasing sequence = (n ) of positive real numbers such that n+1 ⱕ n + 1, 1 = 1, n → ∞ (n → ∞), we introduce the concepts -statistical convergence and strong -Cesàro convergence of generalized difference F (m , M, p, ) and sequences of fuzzy numbers, define the sequence spaces w0F (m , M, p, ), w F (m , M, p, ), w∞ m F S ( , M, p, ) and establish some relations about these spaces. In addition, we study the solidity, monotonicity and symmetricity of the spaces. It is particularly interesting to use an Orlicz function for introducing a sequence space of fuzzy numbers. The spaces we define by using Orlicz functions are much more general then those we already have in the literature and these spaces contain them in the special choices of Orlicz functions. 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 (E) = lim
n→∞
n 1 E (k) provided the limit exists, n k=1
where E is the characteristic function of E. It is clear that any finite subset of N has zero natural density and (E c ) = 1 − (E). A sequence (xk ) of complex numbers is said to be statistically convergent to if for every > 0, ({k ∈ N : |xk − | ⱖ }) = 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 ∈ 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 [26] a fuzzy subset of X is a nonempty subset {(x, u(x)) : x ∈ X } of X × [0, 1] for some function u : X −→ [0, 1]. The function u itself is often used for the fuzzy set. Let C(Rn ) denote the family of all nonempty, compact, convex subsets of Rn . If , ∈ R and A, B ∈ C(Rn ), then (A + B) = A + B, ()A = (A), 1A = A and if , ⱖ 0, then ( + )A = A + A. The distance between A and B is defined by the Hausdorff metric ∞ (A, B) = max sup inf a − b , sup inf a − b , a∈A b∈B
b∈B a∈A
where · denotes the usual Euclidean norm in Rn . It is well known that (C(Rn ), ∞ ) is a complete metric space. Denote L(Rn ) = {u : Rn −→ [0, 1] : u satisfies (i)–(iv) below}, where (i) (ii) (iii) (iv)
u is normal, that is, there exists an x0 ∈ Rn such that u(x0 ) = 1; u is fuzzy convex, that is, for x, y ∈ Rn and 0 ⱕ ⱕ 1, u(x + (1 − )y) ⱖ min[u(x), u(y)]; u is upper semicontinuous; the closure of {x ∈ Rn : u(x) > 0}, denoted by [u]0 , is compact.
If u ∈ L(Rn ), then u is called a fuzzy number, and L(Rn ) is said to be a fuzzy number space.
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For 0 < ⱕ 1, the -level set [u] is defined by [u] = {x ∈ Rn : u(x) ⱖ }. Then from (i) to (iv), it follows that the -level sets [u] ∈ C(Rn ). Some arithmetic operations for -level sets are defined as follows: Let u, v ∈ L(Rn ) and the -level sets be [u]a = [u 1 , u 2 ], [v]a = [v1 , v2 ], ∈ [0, 1]. Then we have [u + v]a = [u 1 + v1 , u 2 + v2 ], [u − v]a = [u 1 − v2 , u 2 − v1 ], [ku 1 , ku 2 ] if k ⱖ 0, [ku] = [ku 2 , ku 1 ] otherwise. Define, for each 1 ⱕ q < ∞, dq (u, v) = 0
1
1/q [∞ ([u] , [v] )] d
q
and d∞ (u, v) = sup0 ⱕ ⱕ 1 ∞ ([u] , [v] ), where ∞ is the Hausdorff metric. Clearly d∞ (u, v) = limq→∞ dq (u, v) with dq ⱕ ds if q ⱕ s [9,16]. For simplicity in notation, throughout the paper d will denote the notation dq with 1 ⱕ q ⱕ ∞. A sequence X = (X k ) of fuzzy numbers is a function X from the set N of all positive integers into L(Rn ). Thus, a sequence of fuzzy numbers (X k ) 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 X k rather than X (k) and to denote the sequence by (X k ) rather than X. The fuzzy number X k is called the k-th term of the sequence. Let X = (X k ) be a sequence of fuzzy numbers. The sequence X = (X k ) of fuzzy numbers is said to be bounded if the set {X k : k ∈ N} of fuzzy numbers is bounded and convergent to the fuzzy number X 0 , written as limk X k = X 0 , F if for every > 0 there exists a positive integer k0 such that d(X k , X 0 ) < for k > k0 . Let F ∞ and c denote the set of all bounded sequences and all convergent sequences of fuzzy numbers, respectively [19,20]. Recall that an Orlicz function is a function M : [0, ∞) → [0, ∞), which is continuous, nondecreasing and convex with M(0) = 0, M(x) > 0 for x > 0 and M(x) → ∞ as x → ∞. Let w be the space of all sequences (xk ) of complex numbers. Lindenstrauss and Tzafriri [18] used the idea of Orlicz function to construct the sequence space ∞ |xk | < ∞ for some > 0 . M M = x = (xk ) ∈ w : k=1
The space M is a Banach space with the norm ∞ |xk | ⱕ1 M x = inf > 0 : k=1
and this space is called an Orlicz sequence space. For M(t) = t p , 1 ⱕ p < ∞, the space M coincides with the classical sequence space p . The difference sequence spaces ∞ (), c() and c0 (), consisting of all real valued sequences x = (xk ) such that 1 x = (xk − xk+1 ) in the sequence spaces ∞ , c and c0 , were defined by Kızmaz [15]. Continuing on this way, Ba¸sar and Altay [2] have recently introduced the difference sequence space bv p of real sequences whose -transforms are in the space p , where x = (xk − xk−1 ) and 1 ⱕ p ⱕ ∞. The idea of difference sequences is generalized by Et and Çolak [10], Altin et al. [1], and Çolak et al. [6]. Let w F be the set of all sequences of fuzzy numbers. The operator m : w F → w F is defined by (0 X )k = X k , (1 X )k = 1 X k = X k − X k+1 , (m X )k = 1 (m−1 X )k (m ⱖ 2) for all k ∈ N.
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¯ ⱕ d(X k , 0) ¯ for all k ∈ N, implies (Yk ) ∈ E F , A sequence space E F ⊂ w F is said to be normal (or solid), if d(Yk , 0) F whenever (X k ) ∈ E , where 1, t = (0, 0, . . . , 0), ¯ = 0(t) 0 otherwise. 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 = {kn : kn < kn+1 , n ∈ N} ⊆ N and E F be a sequence space. A K -step space of E F is a sequence space E F = {(X ) ∈ w F : (X ) ∈ E F }.
K kn n E F is a sequence (Y ) ∈ w F defined as A canonical pre-image of a sequence (X kn ) ∈ K n X n if n ∈ K , Yn = 0¯ otherwise. F
F
E is a set of canonical pre-images of all elements in E , i.e., Y is in A canonical pre-image of a step space K K F E EF . canonical pre-image K if and only if Y is canonical pre-image of some X ∈ K
Remark. If a sequence space E F is solid, then E F is monotone [14]. A sequence space E F is said to be symmetric if (X (n) ) ∈ E F , whenever (X k ) ∈ E F , where is a permutation of N. 3. Main results In this section using an Orlicz function M, a sequence p = ( pk ) of positive real numbers, a nondecreasing sequence = (n ) of positive real numbers such that n+1 ⱕ n + 1, 1 = 1, n → ∞ (n → ∞) and the operator m , we F (m , M, p, ) and S F (m , M, p, ). Then we define the sequence spaces w0F (m , M, p, ), w F (m , M, p, ), w∞ establish some inclusion relations about these spaces and show that the space w F (m , po , ) is complete metric space. m Furthermore we give the relation between the strongly m ( p) -Cesàro convergence and -statistical convergence of a sequence X = (X k ) of fuzzy numbers with respect to the Orlicz function M. Definition 3.1 (Çolak et al. [6]). Let M be an Orlicz function, p = ( pk ) be any sequence of strictly positive real m F m numbers and m be a natural number. Then the sequence spaces cF (m , M, p), c0F (m , M, p), F ∞ ( , M, p), S ( , M, p) are defined as
d(m X k , X 0 ) pk cF (m , M, p) = X = (X k ) ∈ w F : lim M = 0 for some > 0 , k→∞ pk
m ¯ d( X , 0) k = 0 for some > 0 , c0F (m , M, p) = X = (X k ) ∈ w F : lim M k→∞ pk
m ¯ d( X , 0) k F ∞ (m , M, p) = X = (X k ) ∈ w F : sup M < ∞ for some > 0 , k ⱖ0
d(m X k , X 0 ) pk F m F = 0 (statistically) for some > 0 . S ( , M, p) = X = (X k ) ∈ w : lim M k→∞ If we take M(x) = x and pk = 1(for all k ∈ N) in the spaces Z F (m , M, p), then we get the sequence spaces Z = c, c0 , ∞ , S. In the case pk = 1 for all k ∈ N we shall write Z F (m , M) instead of Z F (m , M, p).
Z F (m ), for
Definition 3.2. Let M be an Orlicz function, p = ( pk ) be any sequence of strictly positive real numbers and = (n ) be a nondecreasing sequence of positive real numbers such that n+1 ⱕ n + 1, 1 = 1, n → ∞ (n → ∞). We define
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the following sequence spaces: 1 d(m X k ,X 0 ) pk F lim = 0 for some > 0 M , w F (m , M, p, ) = X = (X k ) ∈ w : n→∞ n k∈In pk 1 d(m X k ,0) ¯ F M lim = 0 for some > 0 w0F (m , M, p, ) = X = (X k ) ∈ w : n→∞ , n k∈In pk 1 d(m X k ,0) ¯ F F m M < ∞ for some > 0 w∞ ( , M, p, ) = X = (X k ) ∈ w : sup , n n k∈In where In = [n − n + 1, n]. If X ∈ w F (m , M, p, ), then we say that the sequence X = (X k ) is strongly m ( p) -Cesàro convergent with respect
to the Orlicz function M. In this case, we write X k → X 0 (w F (m , M, p, )).
Definition 3.3. Let M be an Orlicz function. A sequence X = (X k ) of fuzzy numbers is said to be m ( p) -statistically convergent to the fuzzy number X 0 with respect to the Orlicz function M if there is a fuzzy number X 0 such that
1 d(m X k , X 0 ) pk ⱖ = 0 lim k ∈ In : M n→∞ n for all > 0 and some > 0. In this case, we write SF( p) − lim m X k = X 0 . F m The set of all such m ( p) -statistically convergent sequences will be denoted by S ( , M, p, ). We obtain the set
SF
of all statistically convergent sequences of fuzzy numbers in the case m = 0, M(x) = x, pk = 1 for each k ∈ N and n = n for each n ∈ N in S F (m , M, p, ). If the sequence X = (X k ) ∈ S F is statistically convergent to the fuzzy number X 0 , then we write S F − lim X k = X 0 . For the definition of statistical convergence of sequences of fuzzy numbers, see [22]. Throughout the paper m will be a natural number, p = ( pk ) will be a sequence of strictly positive real numbers, = (n ) will be a nondecreasing sequence of positive real numbers such that n+1 ⱕ n + 1, 1 = 1, n → ∞ F and S F . (n → ∞) and Y F will denote any one of w F , w0F , w∞ We get the following sequence spaces from the above sequence spaces on giving particular values to number m, sequences and p, and the function M. (1) (2) (3) (4) (5)
If n = n for all n ∈ N, we obtain Y F (m , M, p) instead of Y F (m , M, p, ). If pk = 1 for all k ∈ N, we obtain Y F (m , M, ) instead of Y F (m , M, p, ). If M(x) = x, we obtain Y F (m , , p) instead of Y F (m , M, p, ). If M(x) = x and pk = p0 for all k ∈ N, we obtain Y F (m , p0 , ) instead of Y F (m , M, p, ). If we take m = 0 and pk = 1 for all k ∈ N, we obtain Y F (M, ) instead of Y F (m , M, p, ).
Now, for example F
w ( , M, p) = m
X = (X k ) ∈
wF
m pk n 1 d( X k ,X 0 ) M : lim = 0 for some > 0 n→∞ n k=1
for Y F = w F . The sequence spaces Y F (m , M, p, ) contain some unbounded sequences of fuzzy numbers which are divergent, too. To show this let M(x) = x, pk = 1 for all k ∈ N and n = n for all n ∈ N and consider the following example. Example 1. Consider the sequence (X k ) of fuzzy numbers as follows: ⎧ if x ∈ [2k − 1, 2k], ⎪ ⎨ x − 2k + 1 X k (x) = −x + 2k + 1 if x ∈ [2k, 2k + 1], ⎪ ⎩ 0 otherwise. This sequence is neither convergent nor bounded.
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(for m=1) (for m=3)
σn
σn
1
X1 X2
Xk
σn
2k-1
-2m
-8
-4
-2 0 σn (for m=2)
2
4
8
2k+1
2k
2m
Fig. 1. A fuzzy sequence ( n ) which is both convergent and bounded.
Now we calculate -level sets of (X k ) and (m X k ) as [X k ] = [2k − 1 + , 2k + 1 − ] and
[ X k ] = m
[ − 4 + 2, −2]
if m = 1,
[ − 2m (1 − ), 2m (1 − )]
if m ⱖ 2
for ∈ (0, 1]. So, we find the -level sets of arithmetic mean sequence ( n ) as follows: n 1 m 1 [ X k ] = [(−4 + 2)n, −2n] = [−4 + 2, −2] for m = 1, [ n ] = n n k=1
[ n ] =
n 1 m 1 [ X k ] = [−2m (1 − )n, 2m (1 − )n] = [−2m (1 − ), 2m (1 − )] for m ⱖ 2. n n k=1
These sums are usual addition of fuzzy real numbers through -level sets. Thus, it follows that each sequence ( n ) of fuzzy numbers for different values of m is both convergent and bounded (see Fig. 1). It is known that for the sequence (xk ) with real or complex terms, S − lim x k = implies S − lim m xk = 0 for any natural number m. But this is not valid for the sequences of fuzzy numbers. To show this let us give the following example. Example 2. Define the sequence (X k ) as follows: ⎧ ⎫ x −k+1 if k − 1 ⱕ x ⱕ k ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ −x + k + 1 if k < x ⱕ k + 1 if k = n 2 (n = 1, 2, 3, . . .), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎨ 0 otherwise X k (x) = ⎫ ⎪ if 1 ⱕ x ⱕ 2 ⎪ ⎪ ⎪ ⎪ x −1 ⎬ ⎪ ⎪ ⎪ ⎪ −x + 3 if 2 < x ⱕ 3 otherwise. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0 otherwise Then we obtain [X k ] = and
[k − 1 + , k + 1 − ] if k = n 2 , [1 + , 3 − ]
otherwise
⎧ [k − 4 + 2, k − 2] if k = n 2 (n = 1, 2, 3, . . .), ⎪ ⎪ ⎨ [X k ] = [ − k − 1 + 2, −k + 3 − 2] if k + 1 = n 2 (n = 2, 3, . . .), ⎪ ⎪ ⎩ [ − 2 + 2, 2 − 2] otherwise.
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Fig. 2. A fuzzy sequence which is m -statistically convergent, but not m -convergent.
Thus we have S F −lim X k = 1 , where [1 ] = [1+, 3−] and S F −lim X k = 2 , where [2 ] = [−2+2, 2−2]. ¯ where [3 ] = [−2m (1−), 2m (1−)] Moreover, taking difference of order m, we have S F −lim m X k = 3 and 3 0, (see Fig. 2). It can be shown that if a sequence X = (X k ) of fuzzy numbers is m -convergent to the fuzzy number X 0 , then it is also m -statistically convergent to the fuzzy number X 0 , but the converse does not hold. For this consider the following example. Example 3. Let us consider the sequence (X k ) defined in Example 2. Then if we take the difference of order m using the -level set [X k ] , we conclude that (m X k ) is statistically convergent to , where [] = [−2m (1 − ), 2m (1 − )], but (m X k ) is not convergent. Thus, (X k ) is m -statistically convergent, but not m -convergent (see Fig. 2). F (m , M, p, ) Theorem 3.4. Let the sequence ( pk ) be bounded. Then w0F (m , M, p, ) ⊂ w F (m , M, p, ) ⊂ w∞ and the inclusions are strict.
Proof. The inclusion w0F (m , M, p, ) ⊂ w F (m , M, p, ) is obvious. So, we will only show that w F (m , M, p, ) ⊂ F (m , M, p, ). Let X = (X ) ∈ w F (m , M, p, ). Define = 2 . Since M is nondecreasing and convex, we w∞ k 1 have pk
pk
¯ ¯ d(m X k , 0) 1 d(m X k , X 0 ) pk D 1 d(X 0 , 0) M M ⱕ + M n n 2 pk 1 1 k∈In
k∈In
H
¯ d(m X k , X 0 ) pk D d(X 0 , 0) M , ⱕ + D max 1, sup M n 1 1 k∈In
F (m , M, p, ). To show that the inclusion where supk pk = H , and D = max(1, 2 H −1 ). Thus we get X = (X k ) ∈ w∞ is strict, consider the following example.
Example 4. Consider the sequence (X k ) of fuzzy numbers as follows: ⎫ ⎧ k 2 ⎪ ⎪ ⎪ ⎪ x + 1, − ⱕ x ⱕ 0 ⎪ ⎪ ⎪ ⎪ k 2 ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎨ 2 if k = 10 j ( j = 1, 2, 3, . . .), k − x + 1, 0 < x ⱕ 2 ⎪ X k (x) = ⎪ k ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ 0 otherwise ⎪ ⎪ ⎪ ⎪ ⎩¯ 0 otherwise.
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Then, for ∈ (0, 1], we have -level sets of X k and X k as follows: ⎧
⎪ ⎨ k ( − 1), k (1 − ) if k = 10 j ( j = 1, 2, 3, . . .), 2 2 [X k ] = ⎪ ⎩ [0, 0] otherwise and ⎧
k k ⎪ ⎪ ( − 1), (1 − ) if k = 10 j ( j = 1, 2, 3, . . .), ⎪ ⎪ 2 2 ⎪ ⎪ ⎨ [X k ] = (k + 1) (k + 1) ⎪ ⎪ ( − 1), (1 − ) if k + 1 = 10 j ( j = 1, 2, 3, . . .), ⎪ ⎪ 2 2 ⎪ ⎪ ⎩ [0, 0] otherwise. Now it is easy to see that −2¯ < [ n ] < 2¯ for ∈ (0, 1] and all n ∈ N, where [ n ] = (1/n) nk=1 [X k ] . Thus, the sequence ( n ) of fuzzy numbers is bounded. On the other hand, this sequence is not convergent. We give the following theorems without proof. Theorem 3.5. Let the sequence ( pk ) be bounded. Then the sequence spaces w0F (m , M, p, ), w F (m , M, p, ), F (m , M, p, ) and S F (m , p, ) are closed under the operations of addition and scalar multiplication. w∞ Theorem 3.6. Let p = ( pk ) and t = (tk ) be two sequences of positive real numbers and assume that for all k ∈ N, 0 < pk ⱕ tk and sequence (tk / pk ) is bounded. Then w F (m , M, t, ) ⊂ wF (m , M, p, ). Theorem 3.7. Let M1 and M2 be two Orlicz functions. Then we have (i) w F (m , M1 , p, ) ∩ w F (m , M2 , p, ) ⊂ w F (m , M1 + M2 , p, ), (ii) w0F (m , M1 , p, ) ∩ w0F (m , M2 , p, ) ⊂ w0F (m , M1 + M2 , p, ), F (m , M , p, ) ∩ w F (m , M , p, ) ⊂ w F (m , M + M , p, ). (iii) w∞ 1 2 1 2 ∞ ∞ Theorem 3.8. Let po be a positive real number. Then the space w F (m , po , ) is complete metric space with the metric ⎛ ⎞1/ po m 1 d(X k , Yk ) + sup ⎝ [d(m X k , m Yk )] po ⎠ (X, Y ) = n n k=1
k∈In
for 1 ⱕ po < ∞. Proof. Let (X s ) be a Cauchy sequence such that X s = (X ks )k = (X 1s , X 2s , . . .) ∈ w F (m , po , ) for each s ∈ N. Then we have ⎛ ⎞1/ po m 1 d(X ks , X kt ) + sup ⎝ [d(m X ks , m X kt )] po ⎠ →0 (X s , X t ) = n n k=1
k∈In
as s, t → ∞ and so that m k=1
⎛ d(X ks , X kt ) → 0 and sup ⎝−1 n n
k∈In
⎞1/ po [d(m X ks , m X kt )] po ⎠
→0
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as s, t → ∞. Now for k ⱕ m we get d(X ks , X kt ) → 0 as s, t → ∞, and for all k ∈ N we get d(m X ks , m X kt ) → 0 as s, t → ∞. If we consider the definition of d and since the inequality m m s t m s m t s t s t d(X k+m , X k+m ) ⱕ d( X k , X k ) + , X k+m−1 ) d(X k , X k ) + · · · + d(X k+m−1 0 m−1 holds for all k ∈ N, we get d(X ks , X kt ) → 0 as s, t → ∞ and for each k ∈ N. This implies that the sequence (X ks )k = (X k1 , X k2 , . . .) is a Cauchy in L(Rn ) for each k ∈ N. Also, it is convergent, because L(Rn ) is complete. Assume that lims X ks = X k for each k ∈ N. Since (X s ) is a Cauchy sequence, then for every > 0 there exists a number n 0 = n 0 () such that (X s , X t ) < for all s, t ⱖ n 0 . Hence for all n ∈ N and all s, t ⱖ n 0 we get m
d(X ks , X kt ) < and
k=1
1 [d(m X ks , m X kt )] po < po . n k∈In
Taking limit as t → ∞ in the last inequalities, we get lim t
m
d(X ks , X kt ) =
k=1
m
d(X ks , X k ) <
k=1
and lim t
1 1 [d(m X ks , m X kt )] po = [d(m X ks , m X k )] po < po n n k∈In
k∈In
for s ⱖ n 0 and all n ∈ N. This implies that ⎛ ⎞1/ po m d(X ks , X k ) + sup ⎝−1 [d(m X ks , m X k )] po ⎠ →0 (X s , X ) = n k=1
n
k∈In
and hence X s → X as s → ∞. Since ⎧ ⎫ ⎨1 ⎬ 1 1 [d(m X k , X 0 )] po ⱕ 2 po [d(m X kn 0 , X 0 )] po + [d(m X kn 0 , m X k )] po ⎩ n ⎭ n n k∈In
k∈In
k∈In
we get X ∈ w F (m , po , ). It can be shown that w F (m , po , ) is a complete metric space with the metric (X, Y ) =
m k=1
for 0 < po < 1.
[d(X k , Yk )] po + sup n
1 [d(m X k , m Yk )] po n k∈In
Theorem 3.9. If lim inf k pk > 0 and (X k ) is strongly m ( p) -Cesàro convergent to the fuzzy number X 0 with respect to
the Orlicz function M, then X k → X 0 (w F (m , M, p, )) uniquely.
Proof. Let lim inf pk = s > 0. Suppose that X k → X 0 (w F (m , M, p, )), and X k → X 1 (w F (m , M, p, )). Then d(m X k , X 0 ) pk M = 0 for some 1 > 0 lim −1 n→∞ n 1 k∈In
H. Altınok et al. / Fuzzy Sets and Systems 160 (2009) 3128 – 3139
and lim −1 n
n→∞
M
k∈In
d(m X k , X 1 ) 2
pk
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= 0 for some 2 > 0.
Define = max(2 1 , 2 2 ). Since M is nondecreasing and convex, we have
d(X 0 , X 1 ) pk d(m X k , X 0 ) pk D 1 d(m X k , X 1 ) pk −1 n M M ⱕ + M n 2 pk 1 2 k∈In
k∈In
d(m X k , X 0 ) pk D M ⱕ n 1 k∈In
+
d(m X k , X 1 ) pk D M → 0 (n → ∞), n 2 k∈In
where supk pk = H ve D = max(1, 2 H −1 ). Hence d(X 0 , X 1 ) pk M lim −1 =0 n n k∈In
and so X 0 = X 1 . Thus the limit is unique. We give the following two theorems without proof. Theorem 3.10. Let M be an Orlicz function and p = ( pk ) be a sequence such that 0 < inf pk = h < H = sup pk < ∞. Then we have (i) If a sequence X = (X k ) of fuzzy numbers is strongly m ( p) -Cesàro convergent to the fuzzy number X 0 with respect m to the Orlicz function M, then it is -statistically convergent. (ii) If a sequence X = (X k ) of fuzzy numbers is m -bounded and m -statistically convergent, then the sequence -Cesàro convergent to the fuzzy number X 0 with respect to the Orlicz function M. X = (X k ) is strongly m ( p) Theorem 3.11. Assume that lim inf n→∞ (n /n) > 0. If a sequence X = (X k ) of fuzzy numbers is m ( p) -statistically convergent to X 0 with respect to the Orlicz function M, then it is also m -statistically convergent to X 0. ( p) F (m , M, ) = F (m , M). Theorem 3.12. Let X = (X k ) be a sequence of fuzzy numbers, then we have w∞ ∞ F (m , M, ). Then there exists a constant K > 0 such that Proof. Let X ∈ w∞ 1
¯ ¯ 1 d(m X k , 0) d(m X k , 0) ⱕ sup ⱕ K1 M M n n k∈In
F
and so we have X ∈ ∞ (m , M). m m ¯ Conversely, let X ∈ F ∞ ( , M). Then there exists a constant K 2 > 0 such that M(d( X k , 0)/ ) ⱕ K 2 for all k ∈ N, and so ¯ K2 d(m X k , 0) 1 ⱕ M 1 ⱕ K 2 for all k ∈ N. n n k∈In
F (m , M, ). Thus X ∈ w∞
k∈In
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F F Theorem 3.13. The sequence spaces c0F (M), F ∞ (M), c0 (M, ) and ∞ (M, ) are solid and hence monotone, but the m m sequence spaces Z F ( , M, p) and Y F ( , M, p, ) are neither solid nor symmetric.
¯ ¯ Proof. Let (X k ) ∈ F ∞ (M, ) and (Yk ) be sequences such that d(Yk , 0) ⱕ d(X k , 0) for each k ∈ N. Then we get ¯ d X k , 0¯ 1 d(Yk , 0) 1 ⱕ . M M n n k∈In
k∈In
F m Hence F ∞ (M, ) is solid and hence monotone. To show that the space w∞ ( , M, p, ) is not solid, let pk = 1 for ¯ 2, ¯ 3, ¯ . . .) ∈ all k ∈ N, n = n for each n ∈ N and m = 1. Let us consider the sequences X = (X n ) = (n) ¯ = (1, F (m , M, p, ) and w∞ 0 if n is odd, n = 1 if n is even. F (m , M, p, ). Hence w F (m , M, p, ) is not solid. ¯ 2, ¯ 0, ¯ 4, ¯ 0, ¯ 6, ¯ . . .) ∈ / w∞ Then (n X n ) = (0, ∞ m F ¯ 4, ¯ 3, ¯ 2, ¯ 5, ¯ 6, ¯ . . .) be a rearrangement of the To show that the space w∞ ( , M, p, ) is not symmetric, let (Yk ) = (1, F (m , M, p, ) in the case M(x) = x, p = 1 for all k ∈ N. / w∞ sequence (X k ) defined above. Then (Yk ) ∈ k The proof for the other spaces is similar.
4. Conclusion The sequences of fuzzy numbers were introduced and studied by Matloka [19] and the first difference sequences of fuzzy numbers studied by Sava¸s [24] and Bilgin [4]. Now in this paper we study the mth difference sequences of fuzzy numbers for some sequence classes. The results which we obtained in this study are much more general than those obtained by others. To do this we introduce some of fairly wide classes of sequences of fuzzy numbers using the generalized difference operator m and a nondecreasing sequence = (n ) of positive real numbers such that n+1 ⱕ n + 1, 1 = 1, n → ∞ as n → ∞. Furthermore using these concepts we establish some inclusion relations between wF (m , M, p, ) and S F (m , M, ), between S F (m , M, p, ) and S F (m , M, p) and show that the sequence space wF (m , po , ) is a complete metric space with a suitable metric. In addition, we study the solidity, monotonicity and symmetricity of the spaces. Furthermore, in this study we add an example with its figure which contains a sequence ( n ) of fuzzy numbers that is both convergent and bounded. Let v = (vk ) be a sequence of nonzero scalars. Then for a space E of sequences of fuzzy numbers, the multiplier sequence space E(v), associated with the multiplier sequence v can be defined as E(v) = {X = (X k ) : (vk X k ) ∈ E}. The basic properties and v-invariance of this space can be studied. v-invariance of some complex (real) sequence spaces have been studied by various mathematicians. This is an open problem to work for researchers. Acknowledgment The authors are thankful to the referees for their carefully reading the manuscript and valuable suggestions which improved the presentation of the paper. References [1] Y. Altin, M. Et, R. Çolak, Lacunary statistical and lacunary strongly convergence of generalized difference sequences of fuzzy numbers, Comput. Math. Appl. 52 (6–7) (2006) 1011–1020. [2] F. Ba¸sar, B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Math. J. 55 (1) (2003) 136–147. [3] M. Ba¸sarır, M. Mursaleen, Some sequence spaces of fuzzy numbers generated by infinite matrices, J. Fuzzy Math. 11 (3) (2003) 757–764. [4] T. Bilgin, Lacunary strongly -convergent sequences of fuzzy numbers, Inf. Sci. 160 (1–4) (2004) 201–206. [5] M. Burgin, Theory of fuzzy limits, Fuzzy Sets and Systems 115 (2000) 433–443. [6] R. Çolak, H. Altinok, M. Et, Generalized difference sequences of fuzzy numbers, Chaos Solitons Fractals 40 (2009) 1106–1117. [7] J. Connor, A topological and functional analytic approach to statistical convergence, in: Analysis of Divergence, Orono, ME, 1997, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Boston, MA, 1999, pp. 403–413.
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J. Connor, M.A. Swardson, Measures and ideals of C ∗ (X ), Ann. N. Y. Acad. Sci. 704 (1993) 80–91. P. Diamond, P. Kloeden, Metric spaces of fuzzy sets, Fuzzy Sets and Systems 35 (2) (1990) 241–249. M. Et, R. Çolak, On some generalized difference sequence spaces, Soochow J. Math. 21 (4) (1995) 377–386. J.X. Fang, H. Huang, On the level convergence of a sequence of fuzzy numbers, Fuzzy Sets and Systems 147 (2004) 417–435. H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244. J.A. Fridy, On statistical convergence, Analysis 5 (4) (1985) 301–313. P.K. Kamthan, M. Gupta, Sequence Spaces and Series, Marcel Dekker Inc., New York, 1981. H. Kızmaz, On certain sequence spaces, Canad. Math. Bull. 24 (2) (1981) 169–176. V. Lakshmikantham, R.N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor & Francis, New York, 2003. K. Lindberg, On subspaces of Orlicz sequence spaces, Stud. Math. 45 (1973) 379–390. J. Lindenstrauss, L. Tzafriri, On Orlicz sequence spaces, Israel J. Math. 10 (1971) 379–390. M. Matloka, Sequences of fuzzy numbers, BUSEFAL 28 (1986) 28–37. S. Nanda, On sequences of fuzzy numbers, Fuzzy Sets and Systems 33 (1989) 123–126. F. Nuray, Lacunary statistical convergence of sequences of fuzzy numbers, Fuzzy Sets and Systems 99 (3) (1998) 353–356. F. Nuray, E. Sava¸s, Statistical convergence of sequences of fuzzy real numbers, Math. Slovaca 45 (3) (1995) 269–273. T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca 30 (2) (1980) 139–150. E. Savas, A note on sequence of fuzzy numbers, Inf. Sci. 124 (1–4) (2000) 297–300. I.J. Schoenberg, The integrability of certain functions and related summability methods, Am. Math. Mon. 66 (1959) 361–375. L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353.
3139
-Difference sequence spaces of fuzzy numbers Hıfsı Altınok∗ , Rifat Çolak, Mikail Et Department of Mathematics, Firat University, 23119 Elazı˘g, Turkey Received 8 December 2007; received in revised form 3 June 2009; accepted 4 June 2009 Available online 12 June 2009
Abstract In this paper, using the difference operator of order m and an Orlicz function, we introduce and examine some classes of sequences of fuzzy numbers. We give the relations between the strongly Cesàro type convergence and statistical convergence in these spaces. Furthermore, we study some of their properties like completeness, solidity, symmetricity, etc. We also give some inclusion relations related to these classes. © 2009 Elsevier B.V. All rights reserved. MSC: 40A05; 40C05; 46A45; 03E72 Keywords: Orlicz function; Fuzzy number; Statistical convergence; Difference sequence
1. Introduction The notion of statistical convergence was introduced by Fast [12] and Schoenberg [25], 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 [7], Fridy [13], Šalát [23] 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 ˇ of the Stone–Cech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability [8]. The existing literature on statistical convergence appears to have been restricted to real or complex sequences, but Nuray and Sava¸s [22], Nuray [21] and Bilgin [4] extended the idea to apply to sequences of fuzzy numbers and also Matloka [19], Nanda [20], Ba¸sarır and Mursaleen [3], Fang and Huang [11], Burgin [5] and Sava¸s [24] studied the sequences of fuzzy numbers. The study of Orlicz sequence spaces was initiated with a certain specific purpose in Banach space theory. Indeed, Lindberg [17] got interested in Orlicz spaces in connection with finding Banach spaces with symmetric Schauder bases having complementary subspaces isomorphic to c0 or p (1 ⱕ p < ∞). Subsequently Lindenstrauss and Tzafriri [18] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space M contains a subspace isomorphic to p (1 ⱕ p < ∞) [14]. ∗ Corresponding author.
E-mail addresses: [email protected] (H. Altınok), [email protected] (R. Çolak), [email protected] (M. Et). 0165-0114/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2009.06.002
H. Altınok et al. / Fuzzy Sets and Systems 160 (2009) 3128 – 3139
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In the present paper we study some generalized sequence spaces of fuzzy numbers. The main purpose of this paper is to study the difference sequences of order m of fuzzy numbers for some sequence spaces defined by an Orlicz function so as to fill up the existing gaps in the literature. In Section 2 we give a brief overview about statistical convergence, fuzzy numbers and sequences of fuzzy numbers. In Section 3, using the generalized difference operator m , an Orlicz function and a nondecreasing sequence = (n ) of positive real numbers such that n+1 ⱕ n + 1, 1 = 1, n → ∞ (n → ∞), we introduce the concepts -statistical convergence and strong -Cesàro convergence of generalized difference F (m , M, p, ) and sequences of fuzzy numbers, define the sequence spaces w0F (m , M, p, ), w F (m , M, p, ), w∞ m F S ( , M, p, ) and establish some relations about these spaces. In addition, we study the solidity, monotonicity and symmetricity of the spaces. It is particularly interesting to use an Orlicz function for introducing a sequence space of fuzzy numbers. The spaces we define by using Orlicz functions are much more general then those we already have in the literature and these spaces contain them in the special choices of Orlicz functions. 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 (E) = lim
n→∞
n 1 E (k) provided the limit exists, n k=1
where E is the characteristic function of E. It is clear that any finite subset of N has zero natural density and (E c ) = 1 − (E). A sequence (xk ) of complex numbers is said to be statistically convergent to if for every > 0, ({k ∈ N : |xk − | ⱖ }) = 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 ∈ 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 [26] a fuzzy subset of X is a nonempty subset {(x, u(x)) : x ∈ X } of X × [0, 1] for some function u : X −→ [0, 1]. The function u itself is often used for the fuzzy set. Let C(Rn ) denote the family of all nonempty, compact, convex subsets of Rn . If , ∈ R and A, B ∈ C(Rn ), then (A + B) = A + B, ()A = (A), 1A = A and if , ⱖ 0, then ( + )A = A + A. The distance between A and B is defined by the Hausdorff metric ∞ (A, B) = max sup inf a − b , sup inf a − b , a∈A b∈B
b∈B a∈A
where · denotes the usual Euclidean norm in Rn . It is well known that (C(Rn ), ∞ ) is a complete metric space. Denote L(Rn ) = {u : Rn −→ [0, 1] : u satisfies (i)–(iv) below}, where (i) (ii) (iii) (iv)
u is normal, that is, there exists an x0 ∈ Rn such that u(x0 ) = 1; u is fuzzy convex, that is, for x, y ∈ Rn and 0 ⱕ ⱕ 1, u(x + (1 − )y) ⱖ min[u(x), u(y)]; u is upper semicontinuous; the closure of {x ∈ Rn : u(x) > 0}, denoted by [u]0 , is compact.
If u ∈ L(Rn ), then u is called a fuzzy number, and L(Rn ) is said to be a fuzzy number space.
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For 0 < ⱕ 1, the -level set [u] is defined by [u] = {x ∈ Rn : u(x) ⱖ }. Then from (i) to (iv), it follows that the -level sets [u] ∈ C(Rn ). Some arithmetic operations for -level sets are defined as follows: Let u, v ∈ L(Rn ) and the -level sets be [u]a = [u 1 , u 2 ], [v]a = [v1 , v2 ], ∈ [0, 1]. Then we have [u + v]a = [u 1 + v1 , u 2 + v2 ], [u − v]a = [u 1 − v2 , u 2 − v1 ], [ku 1 , ku 2 ] if k ⱖ 0, [ku] = [ku 2 , ku 1 ] otherwise. Define, for each 1 ⱕ q < ∞, dq (u, v) = 0
1
1/q [∞ ([u] , [v] )] d
q
and d∞ (u, v) = sup0 ⱕ ⱕ 1 ∞ ([u] , [v] ), where ∞ is the Hausdorff metric. Clearly d∞ (u, v) = limq→∞ dq (u, v) with dq ⱕ ds if q ⱕ s [9,16]. For simplicity in notation, throughout the paper d will denote the notation dq with 1 ⱕ q ⱕ ∞. A sequence X = (X k ) of fuzzy numbers is a function X from the set N of all positive integers into L(Rn ). Thus, a sequence of fuzzy numbers (X k ) 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 X k rather than X (k) and to denote the sequence by (X k ) rather than X. The fuzzy number X k is called the k-th term of the sequence. Let X = (X k ) be a sequence of fuzzy numbers. The sequence X = (X k ) of fuzzy numbers is said to be bounded if the set {X k : k ∈ N} of fuzzy numbers is bounded and convergent to the fuzzy number X 0 , written as limk X k = X 0 , F if for every > 0 there exists a positive integer k0 such that d(X k , X 0 ) < for k > k0 . Let F ∞ and c denote the set of all bounded sequences and all convergent sequences of fuzzy numbers, respectively [19,20]. Recall that an Orlicz function is a function M : [0, ∞) → [0, ∞), which is continuous, nondecreasing and convex with M(0) = 0, M(x) > 0 for x > 0 and M(x) → ∞ as x → ∞. Let w be the space of all sequences (xk ) of complex numbers. Lindenstrauss and Tzafriri [18] used the idea of Orlicz function to construct the sequence space ∞ |xk | < ∞ for some > 0 . M M = x = (xk ) ∈ w : k=1
The space M is a Banach space with the norm ∞ |xk | ⱕ1 M x = inf > 0 : k=1
and this space is called an Orlicz sequence space. For M(t) = t p , 1 ⱕ p < ∞, the space M coincides with the classical sequence space p . The difference sequence spaces ∞ (), c() and c0 (), consisting of all real valued sequences x = (xk ) such that 1 x = (xk − xk+1 ) in the sequence spaces ∞ , c and c0 , were defined by Kızmaz [15]. Continuing on this way, Ba¸sar and Altay [2] have recently introduced the difference sequence space bv p of real sequences whose -transforms are in the space p , where x = (xk − xk−1 ) and 1 ⱕ p ⱕ ∞. The idea of difference sequences is generalized by Et and Çolak [10], Altin et al. [1], and Çolak et al. [6]. Let w F be the set of all sequences of fuzzy numbers. The operator m : w F → w F is defined by (0 X )k = X k , (1 X )k = 1 X k = X k − X k+1 , (m X )k = 1 (m−1 X )k (m ⱖ 2) for all k ∈ N.
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¯ ⱕ d(X k , 0) ¯ for all k ∈ N, implies (Yk ) ∈ E F , A sequence space E F ⊂ w F is said to be normal (or solid), if d(Yk , 0) F whenever (X k ) ∈ E , where 1, t = (0, 0, . . . , 0), ¯ = 0(t) 0 otherwise. 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 = {kn : kn < kn+1 , n ∈ N} ⊆ N and E F be a sequence space. A K -step space of E F is a sequence space E F = {(X ) ∈ w F : (X ) ∈ E F }.
K kn n E F is a sequence (Y ) ∈ w F defined as A canonical pre-image of a sequence (X kn ) ∈ K n X n if n ∈ K , Yn = 0¯ otherwise. F
F
E is a set of canonical pre-images of all elements in E , i.e., Y is in A canonical pre-image of a step space K K F E EF . canonical pre-image K if and only if Y is canonical pre-image of some X ∈ K
Remark. If a sequence space E F is solid, then E F is monotone [14]. A sequence space E F is said to be symmetric if (X (n) ) ∈ E F , whenever (X k ) ∈ E F , where is a permutation of N. 3. Main results In this section using an Orlicz function M, a sequence p = ( pk ) of positive real numbers, a nondecreasing sequence = (n ) of positive real numbers such that n+1 ⱕ n + 1, 1 = 1, n → ∞ (n → ∞) and the operator m , we F (m , M, p, ) and S F (m , M, p, ). Then we define the sequence spaces w0F (m , M, p, ), w F (m , M, p, ), w∞ establish some inclusion relations about these spaces and show that the space w F (m , po , ) is complete metric space. m Furthermore we give the relation between the strongly m ( p) -Cesàro convergence and -statistical convergence of a sequence X = (X k ) of fuzzy numbers with respect to the Orlicz function M. Definition 3.1 (Çolak et al. [6]). Let M be an Orlicz function, p = ( pk ) be any sequence of strictly positive real m F m numbers and m be a natural number. Then the sequence spaces cF (m , M, p), c0F (m , M, p), F ∞ ( , M, p), S ( , M, p) are defined as
d(m X k , X 0 ) pk cF (m , M, p) = X = (X k ) ∈ w F : lim M = 0 for some > 0 , k→∞ pk
m ¯ d( X , 0) k = 0 for some > 0 , c0F (m , M, p) = X = (X k ) ∈ w F : lim M k→∞ pk
m ¯ d( X , 0) k F ∞ (m , M, p) = X = (X k ) ∈ w F : sup M < ∞ for some > 0 , k ⱖ0
d(m X k , X 0 ) pk F m F = 0 (statistically) for some > 0 . S ( , M, p) = X = (X k ) ∈ w : lim M k→∞ If we take M(x) = x and pk = 1(for all k ∈ N) in the spaces Z F (m , M, p), then we get the sequence spaces Z = c, c0 , ∞ , S. In the case pk = 1 for all k ∈ N we shall write Z F (m , M) instead of Z F (m , M, p).
Z F (m ), for
Definition 3.2. Let M be an Orlicz function, p = ( pk ) be any sequence of strictly positive real numbers and = (n ) be a nondecreasing sequence of positive real numbers such that n+1 ⱕ n + 1, 1 = 1, n → ∞ (n → ∞). We define
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the following sequence spaces: 1 d(m X k ,X 0 ) pk F lim = 0 for some > 0 M , w F (m , M, p, ) = X = (X k ) ∈ w : n→∞ n k∈In pk 1 d(m X k ,0) ¯ F M lim = 0 for some > 0 w0F (m , M, p, ) = X = (X k ) ∈ w : n→∞ , n k∈In pk 1 d(m X k ,0) ¯ F F m M < ∞ for some > 0 w∞ ( , M, p, ) = X = (X k ) ∈ w : sup , n n k∈In where In = [n − n + 1, n]. If X ∈ w F (m , M, p, ), then we say that the sequence X = (X k ) is strongly m ( p) -Cesàro convergent with respect
to the Orlicz function M. In this case, we write X k → X 0 (w F (m , M, p, )).
Definition 3.3. Let M be an Orlicz function. A sequence X = (X k ) of fuzzy numbers is said to be m ( p) -statistically convergent to the fuzzy number X 0 with respect to the Orlicz function M if there is a fuzzy number X 0 such that
1 d(m X k , X 0 ) pk ⱖ = 0 lim k ∈ In : M n→∞ n for all > 0 and some > 0. In this case, we write SF( p) − lim m X k = X 0 . F m The set of all such m ( p) -statistically convergent sequences will be denoted by S ( , M, p, ). We obtain the set
SF
of all statistically convergent sequences of fuzzy numbers in the case m = 0, M(x) = x, pk = 1 for each k ∈ N and n = n for each n ∈ N in S F (m , M, p, ). If the sequence X = (X k ) ∈ S F is statistically convergent to the fuzzy number X 0 , then we write S F − lim X k = X 0 . For the definition of statistical convergence of sequences of fuzzy numbers, see [22]. Throughout the paper m will be a natural number, p = ( pk ) will be a sequence of strictly positive real numbers, = (n ) will be a nondecreasing sequence of positive real numbers such that n+1 ⱕ n + 1, 1 = 1, n → ∞ F and S F . (n → ∞) and Y F will denote any one of w F , w0F , w∞ We get the following sequence spaces from the above sequence spaces on giving particular values to number m, sequences and p, and the function M. (1) (2) (3) (4) (5)
If n = n for all n ∈ N, we obtain Y F (m , M, p) instead of Y F (m , M, p, ). If pk = 1 for all k ∈ N, we obtain Y F (m , M, ) instead of Y F (m , M, p, ). If M(x) = x, we obtain Y F (m , , p) instead of Y F (m , M, p, ). If M(x) = x and pk = p0 for all k ∈ N, we obtain Y F (m , p0 , ) instead of Y F (m , M, p, ). If we take m = 0 and pk = 1 for all k ∈ N, we obtain Y F (M, ) instead of Y F (m , M, p, ).
Now, for example F
w ( , M, p) = m
X = (X k ) ∈
wF
m pk n 1 d( X k ,X 0 ) M : lim = 0 for some > 0 n→∞ n k=1
for Y F = w F . The sequence spaces Y F (m , M, p, ) contain some unbounded sequences of fuzzy numbers which are divergent, too. To show this let M(x) = x, pk = 1 for all k ∈ N and n = n for all n ∈ N and consider the following example. Example 1. Consider the sequence (X k ) of fuzzy numbers as follows: ⎧ if x ∈ [2k − 1, 2k], ⎪ ⎨ x − 2k + 1 X k (x) = −x + 2k + 1 if x ∈ [2k, 2k + 1], ⎪ ⎩ 0 otherwise. This sequence is neither convergent nor bounded.
H. Altınok et al. / Fuzzy Sets and Systems 160 (2009) 3128 – 3139
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(for m=1) (for m=3)
σn
σn
1
X1 X2
Xk
σn
2k-1
-2m
-8
-4
-2 0 σn (for m=2)
2
4
8
2k+1
2k
2m
Fig. 1. A fuzzy sequence ( n ) which is both convergent and bounded.
Now we calculate -level sets of (X k ) and (m X k ) as [X k ] = [2k − 1 + , 2k + 1 − ] and
[ X k ] = m
[ − 4 + 2, −2]
if m = 1,
[ − 2m (1 − ), 2m (1 − )]
if m ⱖ 2
for ∈ (0, 1]. So, we find the -level sets of arithmetic mean sequence ( n ) as follows: n 1 m 1 [ X k ] = [(−4 + 2)n, −2n] = [−4 + 2, −2] for m = 1, [ n ] = n n k=1
[ n ] =
n 1 m 1 [ X k ] = [−2m (1 − )n, 2m (1 − )n] = [−2m (1 − ), 2m (1 − )] for m ⱖ 2. n n k=1
These sums are usual addition of fuzzy real numbers through -level sets. Thus, it follows that each sequence ( n ) of fuzzy numbers for different values of m is both convergent and bounded (see Fig. 1). It is known that for the sequence (xk ) with real or complex terms, S − lim x k = implies S − lim m xk = 0 for any natural number m. But this is not valid for the sequences of fuzzy numbers. To show this let us give the following example. Example 2. Define the sequence (X k ) as follows: ⎧ ⎫ x −k+1 if k − 1 ⱕ x ⱕ k ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ −x + k + 1 if k < x ⱕ k + 1 if k = n 2 (n = 1, 2, 3, . . .), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎨ 0 otherwise X k (x) = ⎫ ⎪ if 1 ⱕ x ⱕ 2 ⎪ ⎪ ⎪ ⎪ x −1 ⎬ ⎪ ⎪ ⎪ ⎪ −x + 3 if 2 < x ⱕ 3 otherwise. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0 otherwise Then we obtain [X k ] = and
[k − 1 + , k + 1 − ] if k = n 2 , [1 + , 3 − ]
otherwise
⎧ [k − 4 + 2, k − 2] if k = n 2 (n = 1, 2, 3, . . .), ⎪ ⎪ ⎨ [X k ] = [ − k − 1 + 2, −k + 3 − 2] if k + 1 = n 2 (n = 2, 3, . . .), ⎪ ⎪ ⎩ [ − 2 + 2, 2 − 2] otherwise.
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Fig. 2. A fuzzy sequence which is m -statistically convergent, but not m -convergent.
Thus we have S F −lim X k = 1 , where [1 ] = [1+, 3−] and S F −lim X k = 2 , where [2 ] = [−2+2, 2−2]. ¯ where [3 ] = [−2m (1−), 2m (1−)] Moreover, taking difference of order m, we have S F −lim m X k = 3 and 3 0, (see Fig. 2). It can be shown that if a sequence X = (X k ) of fuzzy numbers is m -convergent to the fuzzy number X 0 , then it is also m -statistically convergent to the fuzzy number X 0 , but the converse does not hold. For this consider the following example. Example 3. Let us consider the sequence (X k ) defined in Example 2. Then if we take the difference of order m using the -level set [X k ] , we conclude that (m X k ) is statistically convergent to , where [] = [−2m (1 − ), 2m (1 − )], but (m X k ) is not convergent. Thus, (X k ) is m -statistically convergent, but not m -convergent (see Fig. 2). F (m , M, p, ) Theorem 3.4. Let the sequence ( pk ) be bounded. Then w0F (m , M, p, ) ⊂ w F (m , M, p, ) ⊂ w∞ and the inclusions are strict.
Proof. The inclusion w0F (m , M, p, ) ⊂ w F (m , M, p, ) is obvious. So, we will only show that w F (m , M, p, ) ⊂ F (m , M, p, ). Let X = (X ) ∈ w F (m , M, p, ). Define = 2 . Since M is nondecreasing and convex, we w∞ k 1 have pk
pk
¯ ¯ d(m X k , 0) 1 d(m X k , X 0 ) pk D 1 d(X 0 , 0) M M ⱕ + M n n 2 pk 1 1 k∈In
k∈In
H
¯ d(m X k , X 0 ) pk D d(X 0 , 0) M , ⱕ + D max 1, sup M n 1 1 k∈In
F (m , M, p, ). To show that the inclusion where supk pk = H , and D = max(1, 2 H −1 ). Thus we get X = (X k ) ∈ w∞ is strict, consider the following example.
Example 4. Consider the sequence (X k ) of fuzzy numbers as follows: ⎫ ⎧ k 2 ⎪ ⎪ ⎪ ⎪ x + 1, − ⱕ x ⱕ 0 ⎪ ⎪ ⎪ ⎪ k 2 ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎨ 2 if k = 10 j ( j = 1, 2, 3, . . .), k − x + 1, 0 < x ⱕ 2 ⎪ X k (x) = ⎪ k ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ 0 otherwise ⎪ ⎪ ⎪ ⎪ ⎩¯ 0 otherwise.
H. Altınok et al. / Fuzzy Sets and Systems 160 (2009) 3128 – 3139
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Then, for ∈ (0, 1], we have -level sets of X k and X k as follows: ⎧
⎪ ⎨ k ( − 1), k (1 − ) if k = 10 j ( j = 1, 2, 3, . . .), 2 2 [X k ] = ⎪ ⎩ [0, 0] otherwise and ⎧
k k ⎪ ⎪ ( − 1), (1 − ) if k = 10 j ( j = 1, 2, 3, . . .), ⎪ ⎪ 2 2 ⎪ ⎪ ⎨ [X k ] = (k + 1) (k + 1) ⎪ ⎪ ( − 1), (1 − ) if k + 1 = 10 j ( j = 1, 2, 3, . . .), ⎪ ⎪ 2 2 ⎪ ⎪ ⎩ [0, 0] otherwise. Now it is easy to see that −2¯ < [ n ] < 2¯ for ∈ (0, 1] and all n ∈ N, where [ n ] = (1/n) nk=1 [X k ] . Thus, the sequence ( n ) of fuzzy numbers is bounded. On the other hand, this sequence is not convergent. We give the following theorems without proof. Theorem 3.5. Let the sequence ( pk ) be bounded. Then the sequence spaces w0F (m , M, p, ), w F (m , M, p, ), F (m , M, p, ) and S F (m , p, ) are closed under the operations of addition and scalar multiplication. w∞ Theorem 3.6. Let p = ( pk ) and t = (tk ) be two sequences of positive real numbers and assume that for all k ∈ N, 0 < pk ⱕ tk and sequence (tk / pk ) is bounded. Then w F (m , M, t, ) ⊂ wF (m , M, p, ). Theorem 3.7. Let M1 and M2 be two Orlicz functions. Then we have (i) w F (m , M1 , p, ) ∩ w F (m , M2 , p, ) ⊂ w F (m , M1 + M2 , p, ), (ii) w0F (m , M1 , p, ) ∩ w0F (m , M2 , p, ) ⊂ w0F (m , M1 + M2 , p, ), F (m , M , p, ) ∩ w F (m , M , p, ) ⊂ w F (m , M + M , p, ). (iii) w∞ 1 2 1 2 ∞ ∞ Theorem 3.8. Let po be a positive real number. Then the space w F (m , po , ) is complete metric space with the metric ⎛ ⎞1/ po m 1 d(X k , Yk ) + sup ⎝ [d(m X k , m Yk )] po ⎠ (X, Y ) = n n k=1
k∈In
for 1 ⱕ po < ∞. Proof. Let (X s ) be a Cauchy sequence such that X s = (X ks )k = (X 1s , X 2s , . . .) ∈ w F (m , po , ) for each s ∈ N. Then we have ⎛ ⎞1/ po m 1 d(X ks , X kt ) + sup ⎝ [d(m X ks , m X kt )] po ⎠ →0 (X s , X t ) = n n k=1
k∈In
as s, t → ∞ and so that m k=1
⎛ d(X ks , X kt ) → 0 and sup ⎝−1 n n
k∈In
⎞1/ po [d(m X ks , m X kt )] po ⎠
→0
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as s, t → ∞. Now for k ⱕ m we get d(X ks , X kt ) → 0 as s, t → ∞, and for all k ∈ N we get d(m X ks , m X kt ) → 0 as s, t → ∞. If we consider the definition of d and since the inequality m m s t m s m t s t s t d(X k+m , X k+m ) ⱕ d( X k , X k ) + , X k+m−1 ) d(X k , X k ) + · · · + d(X k+m−1 0 m−1 holds for all k ∈ N, we get d(X ks , X kt ) → 0 as s, t → ∞ and for each k ∈ N. This implies that the sequence (X ks )k = (X k1 , X k2 , . . .) is a Cauchy in L(Rn ) for each k ∈ N. Also, it is convergent, because L(Rn ) is complete. Assume that lims X ks = X k for each k ∈ N. Since (X s ) is a Cauchy sequence, then for every > 0 there exists a number n 0 = n 0 () such that (X s , X t ) < for all s, t ⱖ n 0 . Hence for all n ∈ N and all s, t ⱖ n 0 we get m
d(X ks , X kt ) < and
k=1
1 [d(m X ks , m X kt )] po < po . n k∈In
Taking limit as t → ∞ in the last inequalities, we get lim t
m
d(X ks , X kt ) =
k=1
m
d(X ks , X k ) <
k=1
and lim t
1 1 [d(m X ks , m X kt )] po = [d(m X ks , m X k )] po < po n n k∈In
k∈In
for s ⱖ n 0 and all n ∈ N. This implies that ⎛ ⎞1/ po m d(X ks , X k ) + sup ⎝−1 [d(m X ks , m X k )] po ⎠ →0 (X s , X ) = n k=1
n
k∈In
and hence X s → X as s → ∞. Since ⎧ ⎫ ⎨1 ⎬ 1 1 [d(m X k , X 0 )] po ⱕ 2 po [d(m X kn 0 , X 0 )] po + [d(m X kn 0 , m X k )] po ⎩ n ⎭ n n k∈In
k∈In
k∈In
we get X ∈ w F (m , po , ). It can be shown that w F (m , po , ) is a complete metric space with the metric (X, Y ) =
m k=1
for 0 < po < 1.
[d(X k , Yk )] po + sup n
1 [d(m X k , m Yk )] po n k∈In
Theorem 3.9. If lim inf k pk > 0 and (X k ) is strongly m ( p) -Cesàro convergent to the fuzzy number X 0 with respect to
the Orlicz function M, then X k → X 0 (w F (m , M, p, )) uniquely.
Proof. Let lim inf pk = s > 0. Suppose that X k → X 0 (w F (m , M, p, )), and X k → X 1 (w F (m , M, p, )). Then d(m X k , X 0 ) pk M = 0 for some 1 > 0 lim −1 n→∞ n 1 k∈In
H. Altınok et al. / Fuzzy Sets and Systems 160 (2009) 3128 – 3139
and lim −1 n
n→∞
M
k∈In
d(m X k , X 1 ) 2
pk
3137
= 0 for some 2 > 0.
Define = max(2 1 , 2 2 ). Since M is nondecreasing and convex, we have
d(X 0 , X 1 ) pk d(m X k , X 0 ) pk D 1 d(m X k , X 1 ) pk −1 n M M ⱕ + M n 2 pk 1 2 k∈In
k∈In
d(m X k , X 0 ) pk D M ⱕ n 1 k∈In
+
d(m X k , X 1 ) pk D M → 0 (n → ∞), n 2 k∈In
where supk pk = H ve D = max(1, 2 H −1 ). Hence d(X 0 , X 1 ) pk M lim −1 =0 n n k∈In
and so X 0 = X 1 . Thus the limit is unique. We give the following two theorems without proof. Theorem 3.10. Let M be an Orlicz function and p = ( pk ) be a sequence such that 0 < inf pk = h < H = sup pk < ∞. Then we have (i) If a sequence X = (X k ) of fuzzy numbers is strongly m ( p) -Cesàro convergent to the fuzzy number X 0 with respect m to the Orlicz function M, then it is -statistically convergent. (ii) If a sequence X = (X k ) of fuzzy numbers is m -bounded and m -statistically convergent, then the sequence -Cesàro convergent to the fuzzy number X 0 with respect to the Orlicz function M. X = (X k ) is strongly m ( p) Theorem 3.11. Assume that lim inf n→∞ (n /n) > 0. If a sequence X = (X k ) of fuzzy numbers is m ( p) -statistically convergent to X 0 with respect to the Orlicz function M, then it is also m -statistically convergent to X 0. ( p) F (m , M, ) = F (m , M). Theorem 3.12. Let X = (X k ) be a sequence of fuzzy numbers, then we have w∞ ∞ F (m , M, ). Then there exists a constant K > 0 such that Proof. Let X ∈ w∞ 1
¯ ¯ 1 d(m X k , 0) d(m X k , 0) ⱕ sup ⱕ K1 M M n n k∈In
F
and so we have X ∈ ∞ (m , M). m m ¯ Conversely, let X ∈ F ∞ ( , M). Then there exists a constant K 2 > 0 such that M(d( X k , 0)/ ) ⱕ K 2 for all k ∈ N, and so ¯ K2 d(m X k , 0) 1 ⱕ M 1 ⱕ K 2 for all k ∈ N. n n k∈In
F (m , M, ). Thus X ∈ w∞
k∈In
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F F Theorem 3.13. The sequence spaces c0F (M), F ∞ (M), c0 (M, ) and ∞ (M, ) are solid and hence monotone, but the m m sequence spaces Z F ( , M, p) and Y F ( , M, p, ) are neither solid nor symmetric.
¯ ¯ Proof. Let (X k ) ∈ F ∞ (M, ) and (Yk ) be sequences such that d(Yk , 0) ⱕ d(X k , 0) for each k ∈ N. Then we get ¯ d X k , 0¯ 1 d(Yk , 0) 1 ⱕ . M M n n k∈In
k∈In
F m Hence F ∞ (M, ) is solid and hence monotone. To show that the space w∞ ( , M, p, ) is not solid, let pk = 1 for ¯ 2, ¯ 3, ¯ . . .) ∈ all k ∈ N, n = n for each n ∈ N and m = 1. Let us consider the sequences X = (X n ) = (n) ¯ = (1, F (m , M, p, ) and w∞ 0 if n is odd, n = 1 if n is even. F (m , M, p, ). Hence w F (m , M, p, ) is not solid. ¯ 2, ¯ 0, ¯ 4, ¯ 0, ¯ 6, ¯ . . .) ∈ / w∞ Then (n X n ) = (0, ∞ m F ¯ 4, ¯ 3, ¯ 2, ¯ 5, ¯ 6, ¯ . . .) be a rearrangement of the To show that the space w∞ ( , M, p, ) is not symmetric, let (Yk ) = (1, F (m , M, p, ) in the case M(x) = x, p = 1 for all k ∈ N. / w∞ sequence (X k ) defined above. Then (Yk ) ∈ k The proof for the other spaces is similar.
4. Conclusion The sequences of fuzzy numbers were introduced and studied by Matloka [19] and the first difference sequences of fuzzy numbers studied by Sava¸s [24] and Bilgin [4]. Now in this paper we study the mth difference sequences of fuzzy numbers for some sequence classes. The results which we obtained in this study are much more general than those obtained by others. To do this we introduce some of fairly wide classes of sequences of fuzzy numbers using the generalized difference operator m and a nondecreasing sequence = (n ) of positive real numbers such that n+1 ⱕ n + 1, 1 = 1, n → ∞ as n → ∞. Furthermore using these concepts we establish some inclusion relations between wF (m , M, p, ) and S F (m , M, ), between S F (m , M, p, ) and S F (m , M, p) and show that the sequence space wF (m , po , ) is a complete metric space with a suitable metric. In addition, we study the solidity, monotonicity and symmetricity of the spaces. Furthermore, in this study we add an example with its figure which contains a sequence ( n ) of fuzzy numbers that is both convergent and bounded. Let v = (vk ) be a sequence of nonzero scalars. Then for a space E of sequences of fuzzy numbers, the multiplier sequence space E(v), associated with the multiplier sequence v can be defined as E(v) = {X = (X k ) : (vk X k ) ∈ E}. The basic properties and v-invariance of this space can be studied. v-invariance of some complex (real) sequence spaces have been studied by various mathematicians. This is an open problem to work for researchers. Acknowledgment The authors are thankful to the referees for their carefully reading the manuscript and valuable suggestions which improved the presentation of the paper. References [1] Y. Altin, M. Et, R. Çolak, Lacunary statistical and lacunary strongly convergence of generalized difference sequences of fuzzy numbers, Comput. Math. Appl. 52 (6–7) (2006) 1011–1020. [2] F. Ba¸sar, B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Math. J. 55 (1) (2003) 136–147. [3] M. Ba¸sarır, M. Mursaleen, Some sequence spaces of fuzzy numbers generated by infinite matrices, J. Fuzzy Math. 11 (3) (2003) 757–764. [4] T. Bilgin, Lacunary strongly -convergent sequences of fuzzy numbers, Inf. Sci. 160 (1–4) (2004) 201–206. [5] M. Burgin, Theory of fuzzy limits, Fuzzy Sets and Systems 115 (2000) 433–443. [6] R. Çolak, H. Altinok, M. Et, Generalized difference sequences of fuzzy numbers, Chaos Solitons Fractals 40 (2009) 1106–1117. [7] J. Connor, A topological and functional analytic approach to statistical convergence, in: Analysis of Divergence, Orono, ME, 1997, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Boston, MA, 1999, pp. 403–413.
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