Some I-convergent Sequence Spaces of Fuzzy Numbers Defined by Infinite Matrix

In this paper we introduce and study some new sequence spaces of fuzzy numbers defined by I-convergence using the sequences of Orlicz functions, infinite matrix. We study some basic topological and algebraic properties of these spaces. Also we investigate the relations related to these spaces.


INTRODUCTION
The theory of sequence of fuzzy numbers was first introduced by Matloka [10].Matloka introduced bounded and convergent sequence of fuzzy numbers and studied some of their properties and showed that every convergent sequence of fuzzy numbers is bounded.Nanda [12] studied the sequence of fuzzy numbers and showed that the set of all convergent sequence of fuzzy numbers forms a complete metric space.J. S. Kwon [8] introduced the definition of strongly p-Cesaro summability of sequence of fuzzy numbers.Savas [17] introduced and discussed double convergent sequence of fuzzy numbers and showed that the set of all double convergent sequence of fuzzy numbers is complete.Savas [24] studied some equivalent alternative conditions for a sequence of fuzzy numbers to be statistically Cauchy and he continued to study the statistical convergence in [22,25].Recently, Mursaleen and Basarir [11] introduced and studied some new sequence space of fuzzy numbers generated by non-negative regular matrix.Also Savas and Mursaleen [21] defined statistically convergent and statistically Cauchy for double sequence of fuzzy numbers.
Let X be a non-empty set, then a family of sets . A non-trivial ideal I is maximal if there cannot exists any non-trivial ideal I J  containing I as a subset.Further details on ideals of X 2 can be found in Kostyrko et al. [22].Recall in [5] that an Orlicz function M is continuous, convex, nondecreasing function such that M(0)=0 and M(x)>0 for x>0 and M(x)   as x   .If convexity of Orlicz function is replaced by M(x+y)=M(x)+M(y) then this function is called the modulus function and characterized by Ruckle [16].An Orlicz function M is said to satisfy 2  -condition for all values of u, if there exists K>0 such that M(2u)≤KM(u), u≥0.
Lindenstrauss and Tzafriri [9] studied some Orlicz type sequence spaces defined as follows: The following well-known inequality will be used throughout the article.Let p=(p k )be any sequence of positive real numbers with for all aC.
In the later stage different classes of Orlicz sequence spaces were introduced and studied by Parashar and Choudhary [14], Savas ([26]- [29]) and many others.
Throughout the article w F denote the class of all fuzzy real-valued sequence space.Also N and R denote the set of positive integers and set of real numbers respectively.
In this paper, we study some new sequence spaces of fuzzy numbers defined by using I-convergence, the sequence of Orlicz functions and an infinite matrix.We establish inclusion relations between the sequence spaces and where p=(p k ) denote the sequence of positive real numbers for all nN and M=(M k ) be a sequence of Orlicz functions.

DEFINITIONS AND NOTATIONS
Before continuing with this paper we present a few definitions and preliminaries.
Given any interval A, we shall denote its end points by A A, and by D set of all closed bounded intervals on real line R i.e.,

   
It is easy to see that d defines a Hausdorff metric on D and (D,d) is a complete metric space.Also ≤ partial order on D.
A fuzzy number is a function X from R to [0, 1] which satisfying the following conditions (i) X is normal, i.e. there exists an x 0 R such that X(x 0 )=1; (ii) X is fuzzy convex, i.e. for any x,yR and [0, 1] , X(x+(1-)y)min{X(x), Y(y)}; (iii) X is upper semi continuous; (iv) The closure of the set {xR : X(x)>0}, denoted by X 0 is compact.
The properties (i) to (iv) imply that for each [0,1] , the -level set X  ={xR: and Ralescu [15] 1], we say that X<Y if X≤Y and there exist .The fuzzy number X and Y are said to be incomparable if neither X≤Y nor Y≤X.
For any X,Y,ZL(R), the linear structure of L(R) induced addition X+Y and scalar multiplication X,R, in terms of -level sets, by fuzzy numbers is said to converge to a fuzzy number X 0 if for every >0, there exists a positive integer n 0 such that d (X k ,X 0 ) < for all nn 0 .A sequence X=(X k ) of fuzzy numbers is said to be bounded if the set {X k : kN} of fuzzy numbers is bounded.
A sequence X=(X k ) of fuzzy numbers is said to be I-convergent to a fuzzy numberX 0 if for each > 0 such that A={kN : d (X k ,X 0 ) }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> 0such that {kN : d (X k ,0) >M}I.
Let E F be denote the sequence space of fuzzy numbers.
A sequence space E F is said to be solid A sequence space E F is said to be monotone if E F contains the canonical preimages of all its step spaces.
Example 2.1.If we take I=I f ={AN : A is a finite subset}.Then I f is a nontrivial admissible ideal of N and the corresponding convergence coincide with the usual convergence.
Example 2.2.If we take I=I  ={AN :(A)=0} where (A) denote the asymptotic density of the set A. Then I  is a non-trivial admissible ideal of N and the corresponding convergence coincide with the statistical convergence.Lemma 2.1.A sequence space E F is normal implies E F is monotone.(For the crisp set case, one may refer to Kamthan and Gupta (see, [4]).

SOME NEW SEQUENCE SPACES OF FUZZY NUMBERS
In this section, using the sequence of Orlicz functions, an infinite matrix and Iconvergence; we introduced 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 ) bea sequence of positive real numbers for all kN, and A=(a nk ) an infinite matrix.Let M=(M k ) be a sequence Orlicz functions and X=(X k ) be a sequence of fuzzy numbers, we define the following new sequence spaces: (iv) If we take A=(C,1),i.e., the Cesaro matrix, then the above classes of sequences are denoted by w I(F) [w,M,p], w I(F) [w,M,p] 0 , w F [w,M,p] ∞ , and w I(F) [w,M,p] ∞ , respectively.
v) If we take A=(a nk ) is a de la Valeepoussin mean, i.e., where ( n ) is a non-decreasing sequence of positive numbers tending to ∞ and  n+1 ≤ n +1,  1 = 1, then the above classes of sequences are denoted by w  (vi) By a lacunary=(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 will be denoted by I r =(k r-1 ,k r ] and h r =k r -k r-1 .As a final illustration let  X  w [ A,M , , p ]  X  w : , n N : a M I , for some 0 and X L( R ) , ,p] then we say that X=(X k ) is strongly (p)-Cesaro convergent with respect to the sequence of Orlicz functions M.

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.Proof.We shall prove the result for the space w 

I(F)
[A,M,p] 0 only and the others can be proved in similar way.
Let X=(X k ) and Y=(Y k ) be two elements in w 

I(F)
[A,M,p] 0 .Then there exist  1 >0 and Let , ß be two scalars.By the continuity of the function M=(M k ) the following inequality holds: From the above relation we obtain the following: This completes the proof.Theorem 4.2.w I(F) [A,M,p], w I(F) [A,M,p] 0 , and w I(F) [A,M,p] ∞ are linear are linear topological space spaces with the paranorm g defined by   Proof.This can be easily verified by using standard techniques and so is omitted.
The other part can be proved in similar way.
(b) Let X=(X k ) be an element in w I(F) [A,M,p].Since 1≤p k ≤sup p k <∞.Then for each 0<<1 there exists a positive integer n 0 such that The other part can be proved in similar way.
Hence B 1 I and so Y=(Y k )w I(F) [A,M,p] 0 .Thus the space w I(F) [A,M,p] 0 is normal.Also from the Lemma 2.1, it follows that w I(F) [A,M,p] 0 is monotone.

X I 2  2 F 2 
(the class of all subsets of X) is called an ideal if and only if for each we have I B  .A non-empty family of sets X F is a filter on X if and only if F  , for each A and each B A  , we have F B  .An ideal I is called non-trivial ideal if I I and I X  .Clearly X I is a non-trivial ideal if and only if


becomes a Banach space which is called an Orlicz sequence space.The space M  is closely related to the space p  which is an Orlicz sequence space with M(t)= |t| p , for 1≤p<∞.
Let us consider a few special cases of the above sets.(i)If M k (x)=x for all kN, then the above classes of sequences are denoted byw I(F) [A, p], w I(F) [A, p] 0 , w F [A, p] ∞ ,and w I(F) [A, p] ∞ , respectively.(iii) If p=(p k )=(1,1,1,...), then we denote the above spaces by w I(F) [A, M], w I(F) [A, M] 0 , E. Savaş w F [A, M] ∞ , and w I(F) [A, M] ∞.
the above classes of sequences by w  I(F) [A,M,p], w  I(F) [M,p] 0 , w  F [M,p] ∞ , and w  I(F) [M,p] ∞ , respectively.(vii) If I=I f , then we obtain

Proposition 4 . 4 .
The sequence spaces wI(F) [A,M,p] 0 .andwI(F) [A,M,p] ∞ are normal as well as monotone.Proof.We give the proof of the theorem for wI(F) [A,M,p] 0 only.Let X = (X k )w I(F) [A,M,p] 0 and Y = (Y k ) be such that  