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On some A I -convergent difference sequence spaces of fuzzy numbers defined by the sequence of Orlicz functions

Abstract

In this paper, using the difference operator of order m, the sequences of Orlicz functions, and an infinite matrix, we introduce and examine some classes of sequences of fuzzy numbers defined by I-convergence. We study some basic topological and algebraic properties of these spaces. In addition, we shall establish inclusion theorems between these sequence spaces.

MSC:40A05, 40G15, 46A45.

1 Introduction

The notion of ideal convergence was introduced first by Kostyrko et al. [1] as a generalization of statistical convergence [2, 3], which was further studied in topological spaces [4]. More applications of ideals can be seen in [59].

The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh [10]. Subsequently, several authors have discussed various aspects of the theory and applications of fuzzy sets such as fuzzy topological spaces, similarity relations and fuzzy ordering, fuzzy measures of fuzzy events, and fuzzy mathematical programming. In particular, the concept of fuzzy topology has very important applications in quantum particle physics, especially in connection with both string and ε theory, which were given and studied by El Naschie [11]. The theory of sequences of fuzzy numbers was first introduced by Matloka [12]. Matloka 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 [13], Nanda studied sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers forms a complete metric space. Different classes of sequences of fuzzy real numbers have been discussed by Nuray and Savas [14], Altinok, Colak, and Et [15], Savas [1620], Savas and Mursaleen [21], and many others.

The study of Orlicz sequence spaces was initiated with a certain specific purpose in Banach space theory. Lindenstrauss and Tzafriri [22] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space l M contains a subspace isomorphic to l p (1p<). The Orlicz sequence spaces are the special cases of Orlicz spaces studied in [23]. Orlicz spaces find a number of useful applications in the theory of nonlinear integral equations. Although the Orlicz sequence spaces are the generalization of l p spaces, the l p -spaces find themselves enveloped in Orlicz spaces [24]. Recently, Savas [19] generalized c(Δ) and l (Δ) for a single sequence of fuzzy numbers by using the Orlicz function and also established some inclusion theorems.

In the later stage, different classes of Orlicz sequence spaces were introduced and studied by Parashar and Choudhary [25], Savas [2629], and many others.

Throughout the article, w F denotes the class of all fuzzy real-valued sequence spaces. Also, N and R denote the set of positive integers and the set of real numbers, respectively.

The operator Δ n : w F w F is defined by ( Δ 0 X ) k = X k ; ( Δ 1 X ) k =Δ X k = X k X k + 1 ; ( Δ n X ) k = Δ n X k = Δ n 1 X k Δ n 1 X k + 1 (n2) for all nN. The generalized difference has the following binomial expression for n0:

Δ n x k = ν = 0 n ( n ν ) ( 1 ) ν x k + ν .
(1)

In this paper, we study some new sequence spaces of fuzzy numbers using I-convergence, the sequence of Orlicz functions, an infinite matrix, and the difference operator. We establish the inclusion relation between the sequence spaces w I ( F ) [A,M, Δ m ,p], w I ( F ) [ A , M , Δ m , p ] 0 , w F [ A , M , Δ m , p ] , and w I ( F ) [ A , M , Δ m , p ] , where p=( p k ) denotes the sequence of positive real numbers for all kN and M=( M k ) is a sequence of Orlicz functions. In addition, we study some algebraic and topological properties of these new spaces.

2 Definitions and notations

Before continuing with this paper, we present some definitions and preliminaries which we shall use throughout this paper.

Let X and Y be two nonempty subsets of the space w of complex sequences. Let A=( a n k ) (n,k=1,2,) be an infinite matrix of complex numbers. We write Ax=( A n (x)) if A n (x)= k a n k x k converges for each n. (Throughout, k denotes summation over k from k=1 to k=). If x=( x k )XAx=( A n (x))Y, we say that A defines a (matrix) transformation from X to Y and we denote it by A:XY.

Let X be a nonempty set, then a family of sets I 2 X (the class of all subsets of X) is called an ideal if and only if for each A,BI, we have ABI, and for each AI and each BA, we have BI. A nonempty family of sets F 2 X is a filter on X if and only if ΦF, for each A,BF, we have ABF, and for each AF and each AB, we have BF. An ideal I is called non-trivial ideal if IΦ and XI. Clearly, I 2 X is a non-trivial ideal if and only if F=F(I)={XA:AI} is a filter on X. A non-trivial ideal I 2 X is called admissible if and only if {{x}:xX}I. A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal JI containing I as a subset. Further details on ideals of 2 X can be found in Kostyrko et al. [1].

Let D denote the set of all closed and bounded intervals X=[ x 1 , x 2 ] on the real line R. For X,YD, we define XY if and only if x 1 y 1 and x 2 y 2 ,

d(X,Y)=max { | x 1 y 1 | , | x 2 y 2 | } ,where X=[ x 1 , x 2 ] and Y=[ y 1 , y 2 ].

Then it can be easily seen that d defines a metric on D and (D,d) is a complete metric space (see [30]). Also, the relation ‘≤’ is a partial order on D. A fuzzy number X is a fuzzy subset of the real line R, i.e., a mapping X:RJ (=[0,1]) associating each real number t with its grade of membership X(t). A fuzzy number X is convex if X(t)X(s)X(r)=min{X(s),X(r)}, where s<t<r. If there exists t 0 R such that X( t 0 )=1, then the fuzzy number X is called normal. A fuzzy number X is said to be upper semicontinuous if for each ϵ>0, X 1 ([0,a+ϵ)) for all a[0,1] is open in the usual topology in R. Let R(J) denote the set of all fuzzy numbers which are upper semicontinuous and have compact support, i.e., if XR(J), then for any α[0,1], [ X ] α is compact, where

The set R of real numbers can be embedded in R(J) if we define r ¯ R(J) by

r ¯ (t)={ 1 , if  t = r ; 0 , if  t r .

The additive identity and multiplicative identity of R(J) are defined by 0 ¯ and 1 ¯ respectively.

The arithmetic operations on R(J) are defined as follows:

Let X,YR(J) and the α-level sets be [ X ] α =[ x 1 α , x 2 α ], [ Y ] α =[ y 1 α , y 2 α ], α[0,1]. Then the above operations can be defined in terms of α-level sets as follows:

For rR and XR(J), the product rX is defined as follows:

rX(t)={ X ( r 1 t ) , if  r 0 ; 0 , if  r = 0 .

The absolute value |X| of XR(J) is defined by

|X|(t)={ max { X ( t ) , X ( t ) } , if  t 0 ; 0 , if  t < 0 .

Define a mapping d ¯ :R(J)×R(J) R + {0} by

d ¯ (X,Y)= sup 0 α 1 d ( [ X ] α , [ Y ] α ) .

A metric d ¯ on R(J) is said to be a translation invariant if d ¯ (X+Z,Y+Z)= d ¯ (X,Y) for X,Y,ZR(J).

Proposition 2.1 If d ¯ is a translation invariant metric on R(J), then

  1. (i)

    d ¯ (X+Z,0) d ¯ (X,0)+ d ¯ (Y,0),

  2. (ii)

    d ¯ (λX,0)|λ| d ¯ (X,0), |λ|>1.

The proof is easy and so it is omitted.

A sequence X=( X k ) of fuzzy numbers is said to converge to a fuzzy number X 0 if for every ϵ>0, there exists a positive integer n 0 such that d ¯ ( X k , X 0 )<ϵ for all n n 0 .

A sequence X=( X k ) of fuzzy numbers is said to be bounded if the set { X k :kN} of fuzzy numbers is bounded. A sequence X=( X k ) of fuzzy numbers is said to be I-convergent to a fuzzy number X 0 if for each ϵ>0 such that

A= { k N : d ¯ ( X k , X 0 ) ϵ } I.

The fuzzy number X 0 is called I-limit of the sequence ( X k ) of fuzzy numbers, and we write I-lim X k = X 0 .

A sequence X=( X k ) of fuzzy numbers is said to be I-bounded if there exists M>0 such that

{ k N : d ¯ ( X k , 0 ¯ ) > M } I.

Example 2.1 If we take I= I f ={AN:A is a finite subset}, then I f is a non-trivial admissible ideal of N, and the corresponding convergence coincides with the usual convergence.

Example 2.2 If we take I= I δ ={AN:δ(A)=0}, where δ(A) denotes the asymptotic density of the set A, then I δ is a non-trivial admissible ideal of N, and the corresponding convergence coincides with the statistical convergence.

Lemma 2.1 (Kostyrko, Salat, and Wilczynski [1], Lemma 5.1)

If I 2 N is a maximal ideal, then for each AN, we have either AI or NAI.

Recall in [23] that the Orlicz function M:[0,)[0,) is a continuous, convex, non-decreasing function such that M(0)=0 and M(x)>0 for x>0, and M(x) as x. If convexity of the Orlicz function is replaced by M(x+y)M(x)+M(y), then this function is called the modulus function and characterized by Ruckle [31]. An Orlicz function M is said to satisfy the Δ 2 -condition for all values of u, if there exists K>0 such that M(2u)KM(u), u0.

The following well-known inequality will be used throughout the article. Let p=( p k ) be any sequence of positive real numbers with 0 p k sup k p k =G, H=max{1, 2 G 1 }, then

| a k + b k | p k H ( | a k | p k + | b k | p k )

for all kN and a k , b k C. Also, | a k | p k max{1,|a | G } for all aC.

3 Some new sequence spaces of fuzzy numbers

In this section, using the sequence of Orlicz functions, an infinite matrix, the difference operator Δ m , and I-convergence, we introduce the following new sequence spaces and examine some properties of the resulting sequence spaces. Let I be an admissible ideal of N, and let p=( p k ) be a sequence of positive real numbers for all kN and A=( a n k ) be an infinite matrix. Let M=( M k ) be a sequence of Orlicz functions and X=( X k ) be a sequence of fuzzy numbers. We define the following new sequence spaces:

and

Let us consider a few special cases of the above sets.

  1. (i)

    If m=0, then the above classes of sequences are denoted by w I ( F ) [A,M,p], w I ( F ) [ A , M , p ] 0 , w F [ A , M , p ] , and w I ( F ) [ A , M , p ] , respectively.

  2. (ii)

    If M k (x)=x for all kN, then the above classes of sequences are denoted by w I ( F ) [A, Δ m ,p], w I ( F ) [ A , Δ m , p ] 0 , w F [ A , Δ m , p ] , and w I ( F ) [ A , Δ m , p ] , respectively.

  3. (iii)

    If p=( p k )=(1,1,1,), then we denote the above spaces by w I ( F ) [A,M, Δ m ], w I ( F ) [ A , M , Δ m ] 0 , w F [ A , M , Δ m ] , and w I ( F ) [ A , M , Δ m ] .

  4. (iv)

    If we take A=(C,1), i.e., the Cesàro matrix, then the above classes of sequences are denoted by w I ( F ) [M, Δ m ,p], w I ( F ) [ M , Δ m , p ] 0 , w F [ M , Δ m , p ] , and w I ( F ) [ M , Δ m , p ] , respectively.

  5. (v)

    If we take A=( a n k ) is a de la Valée Poussin mean, i.e.,

    a n k ={ 1 λ n , if  k I n = [ n λ n + 1 , n ] , 0 , otherwise ,

where ( λ n ) is a non-decreasing sequence of positive numbers tending to ∞ and λ n + 1 λ n +1, λ 1 =1, then the above classes of sequences are denoted by w λ I ( F ) [A,M, Δ m ,p], w λ I ( F ) [ M , Δ m , p ] 0 , w λ F [ M , Δ m , p ] , and w λ I ( F ) [ M , Δ m , p ] , respectively (see [32]).

  1. (vi)

    By a lacunary θ=( k r ), r=0,1,2, where k 0 =0, we shall mean an increasing sequence of non-negative integers with k r k r 1 as r. The intervals determined by θ will be denoted by I r =( k r 1 , k r ] and h r = k r k r 1 . As a final illustration, let

    a n k ={ 1 h r , if  k r 1 < k k r , 0 , otherwise .

Then we denote the above classes of sequences by w θ I ( F ) [M, Δ m ,p], w θ I ( F ) [ M , Δ m , p ] 0 , w θ F [ M , Δ m , p ] , and w θ I ( F ) [ M , Δ m , p ] , respectively.

  1. (vii)

    If I= I f , then we obtain

If X=( X k ) w F [A,M, Δ m ,p], then we say that X=( X k ) is strongly A-convergent with respect to the sequence of Orlicz functions M.

  1. (viii)

    If I= I δ is an admissible ideal of N, then we obtain

and

4 Main results

In this section, we examine the basic topological and algebraic properties of the new sequence spaces and obtain the inclusion relation related to these spaces.

Theorem 4.1 Let ( p k ) be a bounded sequence. Then the sequence spaces w I ( F ) [A,M, Δ m ,p], w I ( F ) [ A , M , Δ m , p ] 0 , and w I ( F ) [ A , M , Δ m , p ] are linear spaces.

Proof We will prove the result for the space w θ I ( F ) [ M , Δ m , p ] 0 only and others can be proved in a similar way.

Let X=( X k ) and Y=( Y k ) be two elements in w θ I ( F ) [ M , Δ m , p ] 0 . Then there exist ρ 1 >0 and ρ 2 >0 such that

A ε 2 = { r N : k = 1 a n k [ M k ( d ¯ ( Δ m X k , 0 ¯ ) ρ 1 ) ] p k ε 2 } I

and

B ε 2 = { r N : k = 1 a n k [ M k ( d ¯ ( Δ m Y k , 0 ¯ ) ρ 2 ) ] p k ε 2 } I.

Let α, β be two scalars. By the continuity of the function M=( M k ), the following inequality holds:

where K=max{1, ( | α | | α | ρ 1 + | β | ρ 2 ) G , ( | α | | β | ρ 1 + | β | ρ 2 ) G }.

From the above relation, we obtain the following:

This completes the proof. □

Theorem 4.2 w I ( F ) [A,M, Δ m ,p], w I ( F ) [ A , M , Δ m , p ] 0 , and w I ( F ) [ A , M , Δ m , p ] are linear topological spaces with the paranorm g Δ defined by

g Δ ( X ) = inf { ρ p n H : ( k = 1 a n k [ M k ( d ¯ ( Δ m X k , 0 ¯ ) ρ ) ] p k ) 1 / H 1 , for some ρ > 0 , n = 1 , 2 , 3 , } ,

where H=max{1, sup k p k }.

Proof Clearly, g Δ (X)= g Δ (X) and g Δ (θ)=0. Let X=( X k ) and Y=( Y k ) be two elements in w I ( F ) [ A , M , Δ m , p ] . Then for every ρ>0, we write

A 1 = { ρ > 0 : ( k = 1 a n k [ M k ( d ¯ ( Δ m X k , 0 ¯ ) ρ ) ] p k ) 1 H 1 }

and

A 2 = { ρ > 0 : ( k = 1 a n k [ M k ( d ¯ ( Δ m X k , 0 ¯ ) ρ ) ] p k ) 1 H 1 } .

Let ρ 1 A 1 and ρ 2 A 2 . If ρ= ρ 1 + ρ 2 , then we get the following:

( k = 1 a n k [ M k ( d ¯ ( Δ m X k , 0 ¯ ) ρ ) ] ) ρ 2 ρ 1 + ρ 2 ( k = 1 a n k [ M k ( d ¯ ( Δ m X k , 0 ¯ ) ρ ) ] ) + ρ 2 ρ 1 + ρ 2 ( k = 1 a n k [ M k ( d ¯ ( Δ m X k , 0 ¯ ) ρ ) ] ) .

Hence, we obtain

k = 1 a n k [ M k ( d ¯ ( Δ m X k , 0 ¯ ) ρ ) ] p k 1

and

g ( x + y ) = inf { ( ρ 1 + ρ 2 ) p n H : ρ 1 A 1 , ρ 2 A 2 } inf { ( ρ 1 ) p n H : ρ 1 A 1 } + inf { ( ρ 2 ) p n H : ρ 2 A 2 } = g ( x ) + g ( y ) .

 □

Let u k m t, where u k m , uC, and let g( X k m X k )0 as m. To prove that g( u k m X k m u X k )0 as m, let u k u, where u k ,uC and g( X k m X k )0 as m.

We have

A 3 = { ρ k > 0 : k = 1 a n k [ M k ( d ¯ ( Δ m X k , 0 ¯ ) ρ k ) ] p k 1 }

and

A 4 = { ρ k > 0 : k = 1 a n k [ M k ( d ¯ ( Δ m X k , 0 ¯ ) ρ k ) ] p k 1 } .

If ρ k A 3 and ρ k A 4 and by continuity of the function M= M k , we have that

From the above inequality, it follows that

k = 1 a n k [ M k ( d ¯ ( Δ m ( u m X k m u X ) , 0 ¯ ) | u m u | ρ k + | u | ρ k ) ] p k 1,

and consequently

g ( u k m x k u x k ) = inf { ( | u k m u | ρ k + | u | ρ k ) p n G : ρ k A 3 , ρ k A 4 } | u k m u | p n G inf { ( ρ k ) p n G : ρ k A 3 } + | u | p n G inf { ( ρ k ) p n G : ρ k A 4 } max { | u | , | u | p n G } g ( x k m x k ) .

Note that g( x m )g(x)+g( x m x) for all mN.

Hence, by our assumption, the right-hand side tends to 0 as m. This completes the proof of the theorem.

Theorem 4.3 Let I be an admissible ideal and M=( M k ) be a sequence of Orlicz functions. Then the following hold:

w I ( F ) [ A , M , Δ m 1 , p ] 0 w I ( F ) [ A , M , Δ m , p ] 0 ; w I ( F ) [A,M, Δ m 1 ,p] w I ( F ) [A,M, Δ m ,p]; w I ( F ) [ A , M , Δ m 1 , p ] w I ( F ) [ A , M , Δ m , p ] for m1 and the inclusions are strict.

In general, for all i=1,2,3,,m1, the following hold:

w I ( F ) [ A , M , Δ i , p ] 0 w I ( F ) [ A , M , Δ m , p ] 0 ; w I ( F ) [A,M, Δ i ,p] w I ( F ) [A,M, Δ m ,p]; w I ( F ) [ A , M , Δ i , p ] w I ( F ) [ A , M , Δ m , p ] and the inclusions are strict.

Proof Let X=( X k ) be an element in w I ( F ) [ A , M , Δ m 1 , p ] . Then there exists K>0, and for given ε>0, ρ>0, we have

{ n N : k = 1 a n k [ M k ( d ¯ ( Δ m 1 X k , 0 ¯ ) ρ ) ] p k K } I.

Since M=( M k ) is non-decreasing and convex, it follows that

Hence, we have

 □

The inclusion is strict, it follows from the following example.

Example 4.1 Let M k (x)=x, p k =1 for all kN and A=(C,1), i.e., the Cesàro matrix, m=3. Consider the sequence X=( X k ) of fuzzy numbers as follows:

X k (t)={ t k 3 1 1 , for  k 3 1 t 0 ; t k 3 + 1 1 , for  0 < t k 3 + 1 ; 0 , otherwise .

For α(0,1], α-level sets of X k , Δ X k , Δ 2 X k , and Δ 3 X k are

respectively. It is easy to see that the sequence [ Δ 2 X k ] α is not I-bounded although [ Δ 3 X k ] α is I-bounded.

Theorem 4.4 (a) Let 0<inf p k p k 1. Then

w I ( F ) [A,M, Δ m ,p] w I ( F ) [A,M, Δ m ]; w I ( F ) [ A , M , Δ m , p ] 0 w I ( F ) [ A , M , Δ m ] 0 .

(b) Let 1 p k sup p k <. Then

w I ( F ) [A,M, Δ m ] w I ( F ) [A,M, Δ m ,p]; w I ( F ) [ A , M , Δ m ] 0 w I ( F ) [ A , M , Δ m , p ] 0 .

Proof (a) Let X=( X k ) be an element in w I ( F ) [A,M, Δ m ,p]. Since 0<inf p k p k 1, we have

k = 1 a n k [ M k ( d ¯ ( Δ m X k , X 0 ) ρ ) ] k = 1 a n k [ M k ( d ¯ ( Δ m X k , X 0 ) ρ ) ] p k .

Therefore,

The other part can be proved in a similar way.

  1. (b)

    Let X=( X k ) be an element in w I ( F ) [A,M, Δ m ,p]. Since 1 p k sup p k <, then for each 0<ε<1, there exists a positive integer n 0 such that

    k = 1 a n k [ M k ( d ¯ ( Δ m X k , X 0 ) ρ ) ] ε<1for all n n 0 .

This implies that

k = 1 a n k [ M k ( d ¯ ( Δ m X k , X 0 ) ρ ) ] p k k = 1 a n k [ M k ( d ¯ ( Δ m X k , X 0 ) ρ ) ] .

Therefore, we have

 □

The other part can be proved in a similar way.

The following corollary follows immediately from the above theorem.

Corollary 4.5 Let A=(C,1), i.e., the Cesàro matrix, and M=( M k ) be a sequence of Orlicz functions.

(a) Let 0<inf p k p k 1. Then

w I ( F ) [M, Δ m ,p] w I ( F ) [M, Δ m ]; w I ( F ) [ M , Δ m , p ] 0 w I ( F ) [ M , Δ m ] 0 .

(b) Let 1 p k sup p k <. Then

w I ( F ) [M, Δ m ] w I ( F ) [M, Δ m ,p]; w I ( F ) [ M , Δ m ] 0 w I ( F ) [ M , Δ m , p ] 0 .

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Savas, E. On some A I -convergent difference sequence spaces of fuzzy numbers defined by the sequence of Orlicz functions. J Inequal Appl 2012, 261 (2012). https://doi.org/10.1186/1029-242X-2012-261

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  • DOI: https://doi.org/10.1186/1029-242X-2012-261

Keywords

  • ideal
  • I-convergent
  • infinite matrix
  • Orlicz function
  • fuzzy number
  • difference space