SUMMABLE SEQUENCES OF STRONGLY FUZZY NUMBERS

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1 TWMS J. App. Eng. Math. V., N., 20, pp THE χ 2F SUMMABLE SEQUENCES OF STRONGLY FUZZY NUMBERS N.SUBRAMANIAN, U.K.MISRA 2 Abstract. We introduce the classes of χ 2F A, p) summable sequences of strongly fuzzy numbers and give some relations between these classes. We also give a natural relationship between χ 2F A) statistical convergence of sequences of fuzzy numbers. χ 2F summable sequences of strongly fuzzy numbers and strongly Keywords: Fuzzy numbers, de la Vallee-Poussin mean, statistical convergence,analytic sequence, gai sequence. AMS Subject Classification: 40A05,40C05,40D05. Introduction Throughout the paper, a double sequence is denoted by X = X mn ) of fuzzy numbers and denote w 2 F ) all double sequences of fuzzy numbers. Nanda [6] studied single sequence of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers form a complete metric space. In [2], Savas studied the concept double convergent sequences of fuzzy numbers. Savas [] studied the classes of difference sequences of fuzzy [ ] numbers c, F ) and l, F ). In [3] Savas studied the concepts of strongly double V, summable and double S convergent sequences for double sequences of fuzzy numbers. Recently, Esi [5] studied the concepts of strongly double, F ) summable and S 2 ) convergence for double sequences of fuzzy numbers. Right through this paper w, χ and Λ denote the classes of all, gai and analytic scalar valued single sequences, respectively. We write w 2 for the set of all complex sequences x mn ), where m, n N, the set of positive integers. Then, w 2 is a linear space under the coordinate-wise addition and scalar multiplication. Some initial work on double sequence spaces is found in Bromwich [8]. Later on, they were investigated by Hardy [], Moricz [5], Moricz and Rhoades [6], Basarir and Department of Mathematics, SASTRA University, Thanjavur-63 40, India, nsmaths@yahoo.com 2 Department of Mathematics, Berhampur University, Berhampur ,Odissa, India, umakanta misra@yahoo.com Manuscript received 20 May 20. TWMS Journal of Applied and Engineering Mathematics Vol. No. c Işık University, Department of Mathematics 20; all rights reserved. 98

2 N.SUBRAMANIAN, U.K.MISRA: THE χ 2F SUMMABLE SEQUENCES OF STRONGLY FUZZY Solankan [9], Tripathy [23], Turkmenoglu [25], and many others. Let us define the following sets of double sequences: M u t) := x mn ) w 2 : sup m,n N x mn t mn < }, C p t) := x mn ) w 2 : p lim m,n x mn l t mn = for some l C }, C 0p t) := x mn ) w 2 : p lim m,n x mn t mn = }, } L u t) := x mn ) w 2 : x mn t mn <, m= n= C bp t) := C p t) M u t) and C 0bp t) = C 0p t) M u t) ; where t = t mn ) is the sequence of strictly positive reals t mn for all m, n N and p lim m,n denote the limit in the Pringsheim s sense. In that case, t mn = for all m, n N; M u t), C p t), C 0p t), L u t), C bp t) and C 0bp t) reduce to the sets M u, C p, C 0p, L u, C bp and C 0bp, respectively. Now, we may summarize the knowledge given in some document related to the double sequence spaces. Gökhan and Colak [27,28] have proved that M u t) and C p t), C bp t) are complete paranormed spaces of double sequences and gave the α, β, γ duals of the spaces M u t) and C bp t). Quite recently, in her Ph.D thesis, Zelter [29] has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely [30] have recently introduced the statistical convergence and Cauchy the double sequences and given the relation between statistical convergent and strongly Cesàro summable double sequences. Next, Mursaleen [3] and Mursaleen and Edely [32] have defined the almost strong regularity of matrices for double sequences and applied these matrices to establish a core theorem and introduced the M core for double sequences and determined those four dimensional matrices transforming every bounded double sequence x = x jk ) into one whose core is a subset of the M core of x. More recently, Altay and Basar [33] have defined the spaces BS, BS t), CS p, CS bp, CS r and BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces M u, M u t), C p, C bp, C r and L u, respectively, and also have examined some properties of those sequence spaces and determined the α duals of the spaces BS, BV, CS bp and the β ϑ) duals of the spaces CS bp and CS r of double series. Quite recently Basar and Sever [34] have introduced the Banach space L q of double sequences corresponding to the well-known space l q of single sequences and have examined some properties of the space L q. Quite recently Subramanian and Misra [35] have studied the space χ 2 M p, q, u) of double sequences and have given some inclusion relations, of late. Spaces are strongly summable sequences and is discussed by Kuttner [37], Maddox [38], and others. The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox [4] as an extension of the definition of strongly Cesàro summable sequences. Connor [39] further extended this definition to a definition of strong A summability with respect to a modulus where A = a n,k ) is a nonnegative regular matrix and established some connections between strong A summability, strong A summability with respect to a modulus, and A statistical convergence. In [40] the

3 00 TWMS J. APP. ENG. MATH. V., N., 20 notion of convergence of double sequences is presented by a Pringsheim. Also, in [4]- [44], and [45] the four dimensional matrix transformation Ax) k,l = m= n= amn kl x mn was studied extensively by Robison and Hamilton. In their work and throughout this paper, the four dimensional matrices and double sequences have real-valued entries unless specified otherwise. In this paper we extend a few results known in the literature for ordinarysingle) sequence spaces to multiply sequence spaces. The double series m,n= x mn is called convergent if and only if the double sequence s mn ) is convergent, where s mn = m,n i,j= x ijm, n N) see[7]). A sequence x = x mn )is said to be double analytic if sup mn x mn /m+n <. The vector space of all double analytic sequences will be denoted by Λ 2. A sequence x = x mn ) is called double gai sequence if m + n)! x mn ) /m+n 0 as m, n. Double gai sequences are denoted by χ 2. Let φ = all finite sequences}. Consider a double sequence x = x ij ). The m, n) th section x [m,n] of the sequence is defined by x [m,n] = m,n i,j=0 x iji ij for all m, n N ; where I ij denotes the double sequence whose only non zero term is a i+j)! in the i, j) th place for each i, j N. An FK-spaceor a metric space)x is said to have AK property if I mn ) is a Schauder basis for X. Or equivalently x [m,n] x. An FDK-space is a double sequence space endowed with a complete metrizable; locally convex topology under which the coordinate mappings x = x k ) x mn )m, n N) are also continuous. Orlicz[9] uses the idea of Orlicz function to construct the space L M). Lindenstrauss and Tzafriri [3] investigates Orlicz sequence spaces in more detail, and they prove that every Orlicz sequence space l M contains a subspace isomorphic to l p p < ). Subsequently, different classes of sequence spaces are defined by Parashar and Choudhary [20], Mursaleen et al. [7], Bektas and Altin [0], Tripathy et al. [24], Rao and Subramanian [2], and many others. The Orlicz sequence spaces are the special cases of Orlicz spaces studied in [2]. Recalling [9] and [2], an Orlicz function is a function M : [0, ) [0, ) which is continuous, non-decreasing, and convex with M 0) = 0, M x) > 0, for x > 0 and M x) as x. If convexity of Orlicz function M is replaced by subadditivity of M, then this function is called modulus function, defined by Nakano [8] and further discussed by Ruckle [22] and Maddox [4], and many others. An Orlicz function M is said to satisfy the 2 condition for all values of u if there exists a constant K > 0 such that M 2u) KM u) u 0). The 2 condition is equivalent to M lu) KlM u), for all values of u and for l >.

4 N.SUBRAMANIAN, U.K.MISRA: THE χ 2F SUMMABLE SEQUENCES OF STRONGLY FUZZY... 0 Lindenstrauss and Tzafriri [3] use the idea of Orlicz function to construct Orlicz sequence space l M = x w : ) } k= M xk ρ <, for some ρ > 0, The space l M with the norm x = inf ρ > 0 : ) } k= M xk ρ, becomes a Banach space which is called an Orlicz sequence space. For M t) = t p p < ), the spaces l M coincide with the classical sequence space l p. If X is a sequence space, we give the following definitions: i)x = the continuous dual of X; ii)x α = a = a mn ) : m,n= a mn x mn <, for each x X } ; iii)x β = a = a mn ) : m,n= a mn x mn is convegent, for each x X } ; iv)x γ = a = a mn ) : sup mn } M,N m,n= a mnx mn <, for each x X ; v)let X be anf K space φ; then X f = fi mn ) : f X } ; vi)x δ = } a = a mn ) : sup mn a mn x mn /m+n <, for each x X ; X α.x β, X γ are called α orköthe T oeplitz)dual of X, β or generalized Köthe T oeplitz) dual ofx, γ dual of X, δ dual ofx respectively.x α is defined by Gupta and Kamptan [26]. It is clear that X α X β and X α X γ, but X β X γ does not hold, since the sequence of partial sums of a double convergent series need not to be bounded. The notion of difference sequence spaces for single sequences) is introduced by Kizmaz [36] as follows Z ) = x = x k ) w : x k ) Z} for Z = c, c 0 and l, where x k = x k x k+ for all k N. Here c, c 0 and l denote the classes of convergent,null and bounded sclar valued single sequences respectively. The difference space bv p of the classical space l p is introduced and studied in the case p by Başar and Altay in [48] and in the case 0 < p < by Altay and Başar in [49]. The spaces c ), c 0 ), l ) and bv p are Banach spaces normed by x = x + sup k x k and x bvp = k= x k p ) /p, p < ). Later on the notion has been further investigated by many others. We now introduce the following difference double sequence spaces defined by Z ) = x = x mn ) w 2 : x mn ) Z } where Z = Λ 2, χ 2 and x mn = x mn x mn+ ) x m+n x m+n+ ) = x mn x mn+ x m+n + x m+n+ for all m, n N

5 02 TWMS J. APP. ENG. MATH. V., N., Definition and Preliminaries Let D be the set of all bounded intervals A = [ A, A ] on the real line R. For A, B D, define A B if and only if A B and A B, da, B) = max A B, A B }. Then it can be easily see that d defines a metric on D and D, d) is complete metric space. A fuzzy number is fuzzy subset of the real line R which is bounded, convex and normal. Let LR) denote the set of all fuzzy numbers which are upper semi continuous and have compact support, i.e if X LR) then for any α [0, ], X α is compact where t : Xt) α if 0 < α, X α) = t : Xt) > 0, ifα = 0 For each 0 < α, the α level set X α is a nonempty compact subset of R. The linear structure of LR) includes addition X + Y and scalar multiplication X, a scalar)in terms of α level sets, by [X + Y ] α = [X] α + [Y ] α and [X] α = [X] α. for each 0 α. The additive identity andmultiplicative identity of LR) are denoted by 0 and respectively. The zero sequence of fuzzy numbers is denoted by θ. Define a map d : LR) LR) R by d X, Y ) = sup 0 α d X α, Y α ). For X, Y LR) define X Y if and only if X α Y α for any α [0, ]. It is known that LR), d ) is a complete metric space. A sequence X = X mn ) of fuzzy numbers is a function X from the set N of natural numbers into LR). The fuzzy number X mn denotes the value of the function at m, n N and is called the m, n) th term of the sequence. A metric on L R n ) is said to be translation invariant if d X + Z, Y + Z) = d X, Y ) for all X, Y, Z L R n ) A real sequence X = X mn ) is said to be statistically convergent to 0 if, lim m, n p, q : m + n)! X mn ) /m+n, 0} = 0, pq pq where the vertical bars denote the cardinality of the set which they enclose, in which case we write S limx = 0. Let = pq ) be a nondecreasing sequence of positive real numbers tending to infinity with =, p+,q+ pq +. The generalized de la Vallee-Poussin mean is defined by t rs X) = rs X mn n I rs where I rs = [r, s rs +, r, s]. A real sequence X = X mn ) is said to be χ 2 ) summable to a number 0 if t rs X) 0 as r, s. 3. The summable sequences of strongly: Let A = a mn kl ) be an infinite four dimensional matrix of fuzzy numbers and p = p mn) be a double analytic sequence of positive real numbers, i.e., 0 < h < inf mn p mn p mn sup mn p mn = H <, and let X = X mn be a sequence of fuzzy numbers. Then we write

6 N.SUBRAMANIAN, U.K.MISRA: THE χ 2F SUMMABLE SEQUENCES OF STRONGLY FUZZY A mn X) = m=0 n=0 amn kl m + n)! X mn ) /m+n if the series converges for each m, n N. we now define χ X 2F A, p) = = X mn ) w 2F : lim rs rs Λ 2F A, p) = X = X mn ) w 2F : sup rs lim rs rs n I rs d A mn X), 0) pmn = 0 A sequence X = X mn ) of fuzzy numbers is said to be strongly χ 2F } n I rs d A mn X), 0) pmn < to a fuzzy number 0 if there is a fuzzy number such that X = X mn ) χ 2F case we write X mn 0 χ 2F A, p)). If p mn = for m, n N, then we write the classes χ 2F A, p) and Λ2F A, p), respectively. classes χ 2F, } A, p) convergent A, p). In this A) and Λ2F A) in place of the In this section we examine some topological properties of these classes of sequence of fuzzy numbers and investigate some inclusion relations between them. 3.. Theorem. i) χ 2F A, p) Λ2F A, p) ii) χ2f A, p) and Λ2F A, p) are closed under the operations of addition and scalar multiplication if d is a translation invariant four dimensional metric. Proof: i) For χ 2F A, p) χ2f A, p), we use the triangle inequality d A mn X), 0) pmn [d A mn X), 0) + d 0, 0)] pmn K d A mn X), 0) pmn + K max, 0 ), where K = max, sup mn p mn < ). So, X = X mn ) Λ 2F A, p). Proof: ii) We consider only χ 2F A, p). The others can be treated similarly. Suppose that X = X mn ), Y = Y mn ) χ 2F A, p). By combining Minknowski s inequality with property and of the metric d we derive that Therefore, d X + Y, Z + W ) d X, Z) + d Y, W ) ) d cx, cy ) = c d X, Y ) 2) d A mn X) + A mn Y ), 0 + 0) d A mn X), 0) + d A mn Y ), 0). d A mn X) + A mn Y ), 0 + 0) p mn K d A mn X), 0) p mn + K d A mn Y ), 0) p mn where K = max, sup mn p mn < ). This implies that. Let α = α mn ) χ 2F X + Y = X mn ) + Y mn ) χ 2F A, p). A, p) and α R. Then by taking into account properties ) and 2) of the metric d, d A mn α X), α 0) pmn α p mn d A mn α), 0) pmn max, α ) d A mn X), 0) pmn

7 04 TWMS J. APP. ENG. MATH. V., N., 20 since α p mn max, α ). Hence α X χ 2F 3.2. Theorem. The space χ 2F X uv = X uv mn) mn = rs A, p). This completes the proof. A, p) is a complete metric space with the metric ρ X, Y ) = sup r,s,m,n n I rs d A mn X) A mn Y )) mn) p /pmn, pmn <. Proof: Let X uv ) be a Cauchy sequence in χ 2F A, p), where x uv x uv 2 x uv 3 xuv n 0 ρ X uv, X st) = sup r,s,m,n as u, v, s, t. Hence x uv 2 x uv 22 x uv 23 xuv 2n 0. x uv m x uv m2 x uv m3 xuv mn rs d A mn X uv ) A mn X st )) = d m=0 n=0 amn kl as u, v, s, t for each m, n N. m=0 n=0 amn kl χ 2F A, p) for each u, v N. Then n I rs d A mn X uv ) A mn X st )) p mn ) /pmn 0 lim uvst d m + n)! X uv mn X st mn m + n)! X uv mn Xmn st )) ) /m+n 0 )) ) /m+n = 0, for each m, n N. Hence lim uvst d Xmn uv Xmn) st = 0, for each m, n N. Therefore X uv ) uv is a Cauchy sequence in L R). Since L R) is complete, it is convergent, lim uv Xmn uv = X mn say), for each m, n N. Since X uv ) uv is a Cauchy sequence, for each ɛ > 0, there exists p 0 q 0 = p 0 q 0 ɛ) such that ρ X uv, X st) < ɛ for all u, v, s, t p 0 q 0. So, we have lim s,t d [ A mn X uv ) A mn X st )) p mn ] [ = d Amn X uv ) A mn X st ))] p mn < ɛ p mn, for allu, v p 0 q 0. This implies that ρ X uv, X st) < ɛ, for all u, v p 0 q 0, that is X uv X as u, v, where X = X mn ). Since mn n I rs d [A mn X), 0] pmn 2 p mn mn n I rs d [Amn X p 0q 0 ), 0] p mn + d [A mn X p 0q 0 ) A mn X)] pmn} 0 as m, n. So, we obtain X = X mn ) χ 2F A, p). Therefore χ2f A, p) is a complete metric space. It can be also shown that χ 2F A, p) is a complete metric space with the metric ρ X, Y ) = sup ) r,s,m,n rs n I rs d A mn X) A mn Y )) p mn, 0 < p mn <. This completes the proof. ) 3.3. Theorem. Let 0 < p mn q mn and pmn q mn be double analytic. Then χ 2F A, q) χ 2F A, p). ) pmn q mn Proof: Let X = X mn ) χ 2F A, q) and w mn = d [A mn X), 0] q mn and γ mn = for all m, n N. Then 0 < γ mn for all m, n N. Let 0 < γ γ mn for all m, n N. We define the sequences u mn ) and v mn ) as follows: For w mn, let u mn = w mn and v mn = 0 and for w mn <, let u mn = 0 and v mn = w mn. Then it is clear that for all

8 N.SUBRAMANIAN, U.K.MISRA: THE χ 2F SUMMABLE SEQUENCES OF STRONGLY FUZZY m, n N, we have w mn = u mn + v mn and w γ mn mn u γ mn mn u mn w mn and v γ mn mn vmn. γ Therefore rs n I rs wmn γmn = rs rs n I rs w mn + rs n I rs vmn. γ Now for each rs, rs and so n I rs v γ mn = = n I rs [ n I rs ) γ I rs v mn rs n I rs wmn γmn rs Hence X = X mn ) χ 2F A, p), i.e., χ2f = u γ mn mn n I rs u mn + v mn ) γmn ) γ ) γ vmn γ rs rs ) γ ) ] /γ vmn γ rs n I rs w mn + A, q) χ2f rs + v γ mn mn. Now it follows that n I rs 4. χ 2 Statistical convergence: rs ) γ ) / γ ) γ n I rs v mn. A, p). This completes the proof. A sequence X = X mn ) is said to be statistically convergent to 0 if, mn) pq) : d m + n)! X mn ) /m+n, 0)} = 0, lim pq pq The set of all statistically convergence sequences of fuzzy numbers is denoted by χ 2F S. 4.. Definition. A sequence X = X mn ) of fuzzy numbers is said to be χ 2F S A) statistically convergent to a fuzzy number 0, lim rs rs mn) I rs : d A mn X), 0)} = 0 where A mn X) = m + n)! X mn ) /m+n. The set of all χ 2F s A) statistically χ2 convergent sequences of fuzzy numbers is de- s A)) A) statistical convergence and strongly χ2f noted by χ 2F s A) In this case we write X mn 0 χ 2F Now we give the relations between χ 2F s convergence Theorem. The following statement are valid: i) χ 2F X mn ) Λ 2F χ 2F s A), then X = X mn) χ 2F where / γ A, p) A, p) χ2f s A), ii) If X = A, p), iii) Λ2F A) χ 2F s A) = χ2f A) χ 2F Λ 2F A) = X = X mn ) w 2F : sup mn d [A mn X), 0] < } Proof: i) Let X = X mn ) χ 2F rs A, p). Then we have n I rs d [A mn X), 0] p mn rs mn) I rs : d A mn X), 0) ɛ} min ɛ h, ɛ H). Hence x = X mn ) χ 2F A). ii) Suppose that X = X mn ) Λ 2F χ 2F A). Since X = X mn) Λ 2F, we can write d A mn X), 0) T for all m, n N. Given ɛ > 0, let A, p),

9 06 TWMS J. APP. ENG. MATH. V., N., 20 G rs = mn) I rs : d A mn X), 0) ɛ} and H rs = mn) I rs : d A mn X), 0) < ɛ}. Then we have d [A mn X), 0] pmn = d [A mn X), 0] pmn rs rs n I rs m G rs n G rs + d [A mn X), 0] pmn rs m H rs n H rs max T h, T H) G rs + max ɛ h.ɛ H) rs Taking the limit as ɛ 0 and r, s, it follows that X = X mn ) χ 2f iii) Follows from i) and ii). This completes the proof. References A, p). [] Savas, E., 2000),A note on sequences of fuzzy numbers, Information Sciences, 24, [2] Savas, E., 996), A note on double sequences of fuzzy numbers, Turkish Journal of Mathematics, 202), [3] Savas, E., 2008), On statistically convergent double sequences of fuzzy numbers, Journal of Inequalities and Applications, Article ID 47827, 6 pages, doi:0.55/2008/ [4] Savas, E., 2000), On strongly summable sequences of fuzzy numbers, Information Science, 25, [5] Esi, A., 20), On Some Double, F ) Statistical Convergence of Fuzzy numbers, Acta Universittis Apulensis, 25, [6] Nanda, S., 989), On sequences of fuzzy numbers, Fuzzy Sets System, 33, [7] Apostol, T., 978), Mathematical Analysis, Addison-wesley, London. [8] Bromwich, T. J. I A., 965), An introduction to the theory of infinite series, Macmillan and Co.Ltd., New York. [9] Basarir, M. and Solancan, O., 999), On some double sequence spaces, J. Indian Acad. Math., 22), [0] Bektas, C. and Altin, Y., 2003), The sequence space l M p, q, s) on seminormed spaces, Indian J. Pure Appl. Math., 344), [] Hardy, G. H., 97), On the convergence of certain multiple series, Proc. Camb. Phil. Soc., 9, [2] Krasnoselskii, M. A. and Rutickii, Y. B., 96), Convex functions and Orlicz spaces, Gorningen, Netherlands. [3] Lindenstrauss, J. and Tzafriri, L., 97), On Orlicz sequence spaces, Israel J. Math., 0, [4] Maddox, I. J., 986), Sequence spaces defined by a modulus, Math. Proc. Cambridge Philos. Soc, 00), [5] Moricz, F., 99), Extentions of the spaces c and c 0 from single to double sequences, Acta. Math. Hung., 57-2), [6] Moricz, F. and Rhoades, B.E., 988), Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Camb. Phil. Soc., 04, [7] Mursaleen, M., 999),Khan, M. A. and Qamaruddin, Difference sequence spaces defined by Orlicz functions, Demonstratio Math., Vol. XXXII, [8] Nakano, H., 953), Concave modulars, J. Math. Soc. Japan, 5, [9] Orlicz, W., 936), Über Raume `L M, Bull. Int. Acad. Polon. Sci. A, [20] Parashar, S. D. and Choudhary, B., 994), Sequence spaces defined by Orlicz functions, Indian J. Pure Appl. Math., 254), [2] Chandrasekhara Rao, K. and Subramanian, N., 2004), The Orlicz space of entire sequences, Int. J. Math. Math. Sci., 68, [22] Ruckle, W. H., 973), FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math., 25,

10 N.SUBRAMANIAN, U.K.MISRA: THE χ 2F SUMMABLE SEQUENCES OF STRONGLY FUZZY [23] Tripathy, B. C., 2003), On statistically convergent double sequences, Tamkang J. Math., 343), [24] Tripathy, B. C., Et, M. and Altin, Y., 2003), Generalized difference sequence spaces defined by Orlicz function in a locally convex space, J. Anal. Appl., 3), [25] Turkmenoglu, A., 999), Matrix transformation between some classes of double sequences, J. Inst. Math. Comp. Sci. Math. Ser., 2), [26] Kamthan, P. K. and Gupta, M., 98), Sequence spaces and series, Lecture notes, Pure and Applied Mathematics, 65 Marcel Dekker, In c., New York. [27] Gökhan, A. and Çolak, R., 2004), The double sequence spaces c P 2 p) and c P 2 B p), Appl. Math. Comput., 572), [28] Gökhan, A. and Çolak, R., 2005), Double sequence spaces l 2, ibid., 60), [29] Zeltser, M., 200), Investigation of Double Sequence Spaces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu. [30] Mursaleen, M. and Edely, O. H. H., 2003), Statistical convergence of double sequences, J. Math. Anal. Appl., 288), [3] Mursaleen, M., 2004), Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl., 2932), [32] Mursaleen, M. and Edely, O. H. H., 2004), Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl., 2932), [33] Altay, B. and Basar, F., 2005), Some new spaces of double sequences, J. Math. Anal. Appl., 309), [34] Basar, F. and Sever, Y., 2009), The space L p of double sequences, Math. J. Okayama Univ, 5, [35] Subramanian, N. and Misra, U. K., 200), The semi normed space defined by a double gai sequence of modulus function, Fasciculi Math., 46. [36] Kizmaz, H., 98), On certain sequence spaces, Cand. Math. Bull., 242), [37] Kuttner, B., 946), Note on strong summability, J. London Math. Soc., 2, [38] Maddox, I. J., 979), On strong almost convergence, Math. Proc. Cambridge Philos. Soc., 852), [39] Cannor, J., 989), On strong matrix summability with respect to a modulus and statistical convergence, Canad. Math. Bull., 322), [40] Pringsheim, A., 900), Zurtheorie derzweifach unendlichen zahlenfolgen, Math. Ann., 53, [4] Hamilton, H. J., 936), Transformations of multiple sequences, Duke Math. J., 2, [42] -, 938), A Generalization of multiple sequences transformation, Duke Math. J., 4, [43] -, 938), Change of Dimension in sequence transformation, Duke Math. J., 4, [44] -, 939), Preservation of partial Limits in Multiple sequence transformations, Duke Math. J., 4, [45], Robison, G. M., 926), Divergent double sequences and series, Amer. Math. Soc. Trans., 28, [46] Silverman, L. L., On the definition of the sum of a divergent series, unpublished thesis, University of Missouri studies, Mathematics series. [47] Toeplitz, O., 9), Über allgenmeine linear mittel bridungen, Prace Matemalyczno Fizyczne warsaw), 22. [48] Basar, F. and Altay, B., 2003), On the space of sequences of p bounded variation and related matrix mappings, Ukrainian Math. J., 55), [49] Altay, B. and Basar, F., 2007), The fine spectrum and the matrix domain of the difference operator on the sequence space l p, 0 < p < ), Commun. Math. Anal., 22), -. [50] Çolak, R., Et, M. and Malkowsky, E., 2004), Some Topics of Sequence Spaces, Lecture Notes in Mathematics, Firat Univ. Elazig, Turkey, 2004, pp. -63, Firat Univ. Press, ISBN:

11 08 TWMS J. APP. ENG. MATH. V., N., 20 N. Subramanian, Senior Assistant Professor of Mathematics has been working SASTRA University, India, since His area of research is summability through functional analysis. He has been actively involved in helping research scholar in that area. He has published more than seventy six research papers in refereed journals. Details of some of the published articles. ) The sequence spaces defined by a modulus, Kragujevac Journal of Mathematics. 2) The Double χ- sequences, Selcuk J. Appl. Math. 3) The generalized semi normed difference of double gai sequence spaces defined by a modulus function, Stud. Univ. Babes, Bolyai Math. 4) Characterization of gai sequences via double Orlicz space, Southeast Asian Bulletin of Mathematics. The present article deals with double χ sequences of strongly fuzzy numbers.