On Double Statistical P -Convergence of Fuzzy Numbers

Abstract

Savas and Mursaleen defined the notions of statistically convergent and statistically Cauchy for double sequences of fuzzy numbers. In this paper, we continue the study of statistical convergence by proving some theorems.

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Ekrem Savas 1 and Richard F. Patterson 2

Recommended by Andrei Volodin

1, Department of Mathematics, Istanbul Commerce University, 34672 Uskudar, Istanbul, Turkey
2, Department of Mathematics and Statistics, University of North Florida, Building 11, Jacksonville, FL 32224, USA

Received 1 September 2009; Accepted 2 October 2009

1. Introduction

For sequences of fuzzy numbers, Nanda [1] studied the sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers forms a complete metric space. Kwon [2] introduced the definition of strongly p -Cesàro summability of sequences of fuzzy numbers. Savas [3] introduced and discussed double convergent sequences of fuzzy numbers and showed that the set of all double convergent sequences of fuzzy numbers is complete. Savas [4] studied some equivalent alternative conditions for a sequence of fuzzy numbers to be statistically Cauchy and he continue to study in [5, 6]. Recently Mursaleen and Basarir [7] introduced and studied some new sequence spaces of fuzzy numbers generated by nonnegative regular matrix. Quite recently, Savas and Mursaleen [8] defined statistically convergent and statistically Cauchy for double sequences of fuzzy numbers. In this paper, we continue the study of double statistical convergence and introduce the definition of double strongly p -Cesàro summabilty of sequences of fuzzy numbers.

2. Definitions and Preliminary Results

Since the theory of fuzzy numbers has been widely studied, it is impossible to find either a commonly accepted definition or a fixed notation. We therefore being by introducing some notations and definitions which will be used throughout.

Let C(^Rn )={A⊂^Rn :A compact and convex} . The spaces C(^Rn ) has a linear structure induced by the operations [figure omitted; refer to PDF] for A,B∈C(^Rn ) and λ∈R . The Hausdorff distance between A and B of C(^Rn ) is defined as [figure omitted; refer to PDF]

It is well known that (C(^Rn ),_δ∞ ) is a complete (not separable) metric space.

A fuzzy number is a function X from ^Rn to [0,1] satisfying

(1) X is normal, that is, there exists an _x0 ∈^Rn such that X(_x0 )=1 ;

(2) X is fuzzy convex, that is, for any x,y∈^Rn and 0≤λ≤1 , [figure omitted; refer to PDF]

(3) X is upper semicontinuous;

(4) the closure of {x∈^Rn :X(x)>0} ; denoted by ^X0 , is compact.

These properties imply that for each 0<α≤1 , the α -level set [figure omitted; refer to PDF] is a nonempty compact convex, subset of ^Rn , as is the support ^X0 . Let L(^Rn ) denote the set of all fuzzy numbers. The linear structure of L(^Rn ) induces the addition X+Y and scalar multiplication λX , λ∈R , in terms of α -level sets, by [figure omitted; refer to PDF] for each 0≤α≤1 .

Define for each 1≤q<∞ , [figure omitted; refer to PDF] and _d∞ =su_{p0≤α≤1δ∞} (^Xα ,^Yα ) clearly _d∞ (X,Y)=li_{mq[arrow right]∞dq} (X,Y) with _dq ≤_dr if q≤r . Moreover _dq is a complete, separable, and locally compact metric space [9].

Throughout this paper, d will denote ^dq with 1≤q≤∞ . We will need the following definitions (see [8]).

Definition 2.1.

A double sequence X=(_Xkl ) of fuzzy numbers is said to be convergent in Pringsheim's sense or P -convergent to a fuzzy number _X0 , if for every [straight epsilon]>0 there exists N∈...A9; such that [figure omitted; refer to PDF] and we denote by P-lim X=_X0 . The number _X0 is called the Pringsheim limit of _Xkl .

More exactly we say that a double sequence (_Xkl ) converges to a finite number _X0 if _Xkl tend to _X0 as both k and l tend to ∞ independently of one another.

Let ^c2 (F) denote the set of all double convergent sequences of fuzzy numbers.

Definition 2.2.

A double sequence X=(_Xkl ) of fuzzy numbers is said to be P -Cauchy sequence if for every [straight epsilon]>0, there exists N∈...A9; such that [figure omitted; refer to PDF]

Let ^C2 (F) denote the set of all double Cauchy sequences of fuzzy numbers.

Definition 2.3.

A double sequence X=(_Xkl ) of fuzzy numbers is bounded if there exists a positive number M such that d(_Xkl ,_X0 )<M for all k and l , [figure omitted; refer to PDF] We will denote the set of all bounded double sequences by ^l∞2 (F) .

Let K⊆...A9;×...A9; be a two-dimensional set of positive integers and let _Km,n be the numbers of (i,j) in K such that i≤n and j≤m . Then the lower asymptotic density of K is defined as [figure omitted; refer to PDF] In the case when the sequence (_Km,n /mn^{)m,n=1,1∞,∞} has a limit, then we say that K has a natural density and is defined as [figure omitted; refer to PDF] For example, let K={(ⁱ² ,^j2 ):i,j∈...A9;} , where ...A9; is the set of natural numbers. Then [figure omitted; refer to PDF] (i.e., the set K has double natural density zero).

Definition 2.4.

A double sequence X=(_Xkl ) of fuzzy numbers is said to be statistically convergent to _X0 provided that for each ...>0 , [figure omitted; refer to PDF]

In this case we write s_t2 -_{lim k,lXk,l} =_X0 and we denote the set of all double statistically convergent sequences of fuzzy numbers by s^t2 (F) .

Definition 2.5.

A double sequence X=(_Xkl ) of fuzzy numbers is said to be statistically P -Cauchy if for every [straight epsilon]>0, there exist N=N([straight epsilon]) and M=M([straight epsilon]) such that [figure omitted; refer to PDF] That is, d(_Xkl ,_XNM )<[straight epsilon] , a.a.(k,l ).

Let ^C2 (F) denote the set of all double Cauchy sequences of fuzzy numbers.

Definition 2.6.

A double sequence X=(_Xkl ) of fuzzy and let p be a positive real numbers. The sequence X is said to be strongly double p -Cesaro summable if there is a fuzzy number _X0 such that [figure omitted; refer to PDF] In which case we say that X is strongly p -Cesaro summable to _X0 .

It is quite clear that if a sequence X=(_Xkl ) is statistically P -convergent, then it is a statistically P -Cauchy sequence [8]. It is also easy to see that if there is a convergent sequence Y=(_Ykl ) such that _Xkl =_Ykl a.a.(k,l ), then X=(_Xkl ) is statistically convergent.

3. Main Result

Theorem 3.1.

A double sequence X=(_Xkl ) of fuzzy numbers is statistically P -Cauchy then there is a P -convergent double sequence Y=(_Ykl ) such that _Xkl =_Ykl a.a.(k,l ).

Proof.

Let us begin with the assumption that X=(_Xkl ) is statistically P -Cauchy this grant us a closed ball B=B...(_{XM1 ,N1} ,1) that contains _Xkl a.a.(k,l ) for some positive numbers _M1 and _N1 . Clearly we can choose M and N such that ^{B[variant prime]} =B...(_XM,N ,1/(2·2)) contains _XK,L a.a.(k,l ). It is also clear that _Xk,l ∈_B1,1 =B∩B... a.a.(k,l ); for [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] Thus _B1,1 is a closed ball of diameter less than or equal to 1 that contains _Xk,l a.a.(k,l ). Now we let us consider the second stage to this end we choose _M2 and _N2 such that _xk,l ∈^{B[variant prime][variant prime]} =B...(_{XM2 ,N2} ,1/(²² ·²² )). In a manner similar to the first stage we have _Xk,l ∈_B2,2 =_B1 ∩^{B[variant prime][variant prime]} , a.a.(k,l ). Note the diameter of _B2,2 is less than or equal ^21-2 ·^21-2 . If we now consider the (m,n )th general stage we obtain the following. First a sequence ^{{_Bm,n }m,n=1,1∞,∞} of closed balls such that for each (m,n ), _Bm,n ⊃ _Bm+1,n+1 , the diameter of _Bm,n is not greater than ^21-m ·^21-m with _Xk,l ∈ _Bm,n , a.a.(k,l ). By the nested closed set theorem of a complete metric space we have ^{...m,n=1,1∞,∞} _Bmn ≠ ∅ . So there exists a fuzzy number A∈^{...m,n=1,1∞,∞}_Bm,n . Using the fact that _Xk,l ∉ _Bm,n , a.a.(k,l ), we can choose an increasing sequence _Tm,n of positive integers such that [figure omitted; refer to PDF] Now define a double subsequence _Zk,l of _Xk,l consisting of all terms _Xk,l such that k,l>_T1,1 and if [figure omitted; refer to PDF] Next we define the sequence (_Yk,l ) by [figure omitted; refer to PDF] Then P-_{lim k,lYk,l} =A indeed if ...>1/m,n>0, and k,l>_Tm,n , then either _Xk,l is a term of Z . Which means _Yk,l =A or _Yk,l =_Xk,l ∈_Bm,n and d(_Yk,l -A)≤|_Bm,n |≤ diameter of ^{_Bm,n ≤21-m} ·^21-n . We will now show that _Xk,l =_Yk,l a.a.(k,l ). Note that if _Tm,n <m,n<_Tm+1,n+1 , then [figure omitted; refer to PDF] and by (3.3) [figure omitted; refer to PDF] Hence the limit as (m,n) is 0 and _Xk,l =_Yk,l a.a.(k,l ). This completes the proof.

Theorem 3.2.

If X=(_Xk,l ) is strongly p -Cesaro summable or statistically P -convergent to _X0 , then there is a P -convergent double sequences Y and a statistically P -null sequence Z such that P-_{lim k,lYk,l} =_X0 and _{st2lim k,lZk,l} =0 .

Proof.

Note that if X=[_Xk,l ] is strongly p -Cesaro summable to _X0 , then X is statistically P -convergent to _X0 . Let _N0 =0 and _M0 =0 and select two increasing index sequences of positive integers _N1 <_N2 <... and _M1 <_M2 <... such that m>_Mi and n>_Nj , we have [figure omitted; refer to PDF] Define Y and Z as follows: if _N0 <k<_N1 and _M0 <l<_M1 , set _Zk,l =0 and _Yk,l =_Xk,l . Suppose that i,j>1 and _Ni <k<_Ni+1 , _Mj <l<_Mj+1 , then [figure omitted; refer to PDF] We now show that P-_{lim k,lYk,l} =_x0 . Let ...>0 be given, pick (i,j) be given, and pick i and j such that ...>1/ij , thus for k,l>_Mi ,_Nj , since d(_Yk,l ,_X0 )<d(_Xk,l ,_X0 )<... if d(_Xk,l ,_X0 )<1/ij and d(_Yk,l ,_X0 )=0 if d(_Xk,l ,_X0 )>1/ij, we have d(_Yk,l ,_X0 )<... .

Next we show that Z is a statistically P -null double sequence, that is, we need to show that P-_{lim m,n} (1/mn)|{k≤ml≤n:_Zk,l ≠0}|=0 . Let δ>0 if (i,j)∈NxN such that 1/ij<δ, then |{k≤ml≤n:_Zk,l ≠0}|<δ for all m,n>_Mi ,_Nj . From the construction of (_Mi ,_Nj ) , if _Mi <k≤_Mi+1 and _Nj <l≤_Nj+1 , then _Zk,l ≠0 only if d(_Xk,l ,_X0 )>1/ij . It follows that if _Mi <k≤_Mi+1 and _Nj <l≤_Nj+1 , then [figure omitted; refer to PDF] Thus for _Mi <m≤_Mi+1 and _Nj <n≤_Nj+1 and p,q>i,j, then [figure omitted; refer to PDF] this completes the proof.

Corollary 3.3.

If X=(_Xk,l ) is a strongly p -Cesaro summable to _X0 or statistically P -convergent to _X0 , then X has a double subsequence which is P -converges to _X0 .

4. Conclusion

In recent years the statistical convergence has been adapted to the sequences of fuzzy numbers. Double statistical convergence of sequences of fuzzy numbers was first deduced in similar form by Savas and Mursaleen as we explain now: a double sequences X={_Xk,l } is said to be P -statistically convergent to _X0 provided that for each ...>0 , [figure omitted; refer to PDF] Since the set of real numbers can be embedded in the set of fuzzy numbers, statistical convergence in reals can be considered as a special case of those fuzzy numbers. However, since the set of fuzzy numbers is partially ordered and does not carry a group structure, most of the results known for the sequences of real numbers may not be valid in fuzzy setting. Therefore, this theory should not be considered as a trivial extension of what has been known in real case. In this paper, we continue the study of double statistical convergence and also some important theorems are proved.

References

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