The Modular of Γ 2 Defined by Asymptotically μ Invariant Modulus of Fuzzy Numbers

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1 Applie Mathematical Sciences, Vol. 7, 203, no. 27, HIKARI Lt, The Moular o Γ 2 Deine by Asymptotically μ Invariant Moulus o Fuzzy Numbers N. Subramanian Department o Mathematics SASTRA University Thanjavur-63 40, Inia nsmaths@yahoo.com P. Anbalagan Department o Mathematics Government Arts College Trichy , Inia anbalaganmaths@gmail.com P. Thirunavukarasu P.G. an Research Department o Mathematics Periyar E.V.R. College Autonomous) Tiruchirappalli , Inia ptavinash967@gmail.com Copyright c 203 N. Subramanian et al. This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. Abstract. In this paper, we introuce the moular sequence space o Γ 2 o uzzy numbers using moulus unction an examine some topological properties o these space also establish some uals results among them. Linenstrauss an Tzariri [7] use the iea o Orlicz unction to eine the sequence space l M which is calle an Orlicz sequence space. Another generalization o Orlicz an Γ2λ g, sequence spaces is ue to Woo [9]. We eine the sequence spaces Γ 2 λ where = mn ) an g =g mn ) are sequences o moulus unctions such that mn an g mn be mutually complementary or each m, n. This paper eals with the moular o Γ 2 eine by asymptotically λ an σ invariant moulus o uzzy numbers. I mn an g mn are mutually complementary unction then the

2 320 N. Subramanian, P. Anbalagan an P. Thirunavukarasu two sequences X an Y o uzzy numbers are sai to be asymptotically λ invariant statistical equivalent o multiple 0 provie that or every ɛ>0, [ ) X,σ,μ,λ SΓ2F Y = Xσ mn /m+n μ rs m, n I rs : η) Y σ mn η) ɛ} 0asr, s, uniormly in η X SΓ2Fλ g,σ,μ Y = μ rs m, n I rs :, uniormly in η. [ ) Xσ mn /m+n g η) Y σ mn η) ɛ} 0asr, s Mathematics Subject Classiication: 40A05, 46S40, 40D05 Keywors: analytic sequence, moulus unction, ouble sequences, entire sequence, moular, Fuzzy numbers, λ statistical, σ convergence. Introuction The concept o uzzy sets an uzzy set operations were irst introuce by Zaeh [30], uzzy logic has become an important area o research in various branches o Mathematics such as metric an topological spaces, theory o unctions, approximation theory etc. Subsequently several authors have iscusse various aspects o the theory an applications o uzzy sets. The concept o uzzyiness has been applie in various iels such as Statistics, Cybernetics, Artiicial intelligence, Operation research, Decision making, Agriculture, Weather orecasting, Quantum physics. Similarity relations o uzzy orerings, uzzy measures o uzzy events, uzzy mathematical programming etc. In 993, Marou presente einitions or asymptotically equivalent sequences o real numbers an asymptotic regular matrices. In 2003, Patterson extene these concepts by presenting an asymptotically statistical equivalent analog o these einitions an natural regularity conitions or nonnegative summability matrices. In 2006, Savas an Basarir introuce an stuie the concept o σ, λ) asymptotically statistical equivalent sequences. In 2008, Esi introuce an stuie the concept o asymptotically equivalent ierence sequences o uzzy numbers. In 2009, sequences o uzzy numbers, Savas introuce an stuie the concepts o strongly λ summable λ statistical convergence an asymptotically λ statistical equivalent sequences respectively. Recently, Braha eine asymptotically generalize ierence lacunary sequences. This paper is exten to the moular o Γ 2 eine by asymptotically λ invariant moulus o uzzy numbers. Throughout w, χ an Λ enote the classes o all, gai an analytic scalar value single sequences, respectively. We write w 2 or the set o all complex sequences x mn ), where m, n N, the set o positive integers. Then, w 2 is a linear space uner the coorinate wise aition an scalar multiplication.

3 Moular o Γ 2 eine by asymptotically μ invariant moulus 32 Some initial works on ouble sequence spaces is oun in Bromwich [4]. Later on, they were investigate by Hary [5], Moricz [9], Moricz an Rhoaes [0], Basarir an Solankan [2], Tripathy [7], Turkmenoglu [9], an many others. Let us eine the ollowing sets o ouble sequences: M u t) := x mn ) w 2 : sup m,n N x mn tmn < }, C p t) := x mn ) w 2 : p lim m,n x mn l tmn =or somel C }, C 0p t) := x mn ) w 2 : p lim m,n x mn tmn = }, L u t) := x mn ) w 2 : m= n= x mn tmn < }, C bp t) :=C p t) M u t) an C 0bp t) =C 0p t) M u t); where t =t mn ) is the sequence o strictly positive reals t mn or all m, n N an p lim m,n enotes the limit in the Pringsheim s sense. In the case t mn = or all m, n N; M u t), C p t), C 0p t), L u t), C bp t) an C 0bp t) reuce to the sets M u, C p, C 0p, L u, C bp an C 0bp, respectively. Now, we may summarize the knowlege given in some ocument relate to the ouble sequence spaces. Gökhan an Colak [2,22] have prove that M u t) an C p t), C bp t) are complete paranorme spaces o ouble sequences an gave the α,β,γ uals o the spaces M u t) an C bp t). Quite recently, in her PhD thesis, Zelter [23] has essentially stuie both the theory o topological ouble sequence spaces an the theory o summability o ouble sequences. Mursaleen an Eely [24] have recently introuce the statistical convergence an Cauchy or ouble sequences an given the relation between statistical convergent an strongly Cesàro summable ouble sequences. Nextly, Mursaleen [25] an Mursaleen an Eely [26] have eine the almost strong regularity o matrices or ouble sequences an applie these matrices to establish a core theorem an introuce the M core or ouble sequences an etermine those our imensional matrices transorming every boune ouble sequences x =x jk ) into one whose core is a subset o the M core o x. More recently, Altay an Basar [27] have eine the spaces BS, BS t), CS p, CS bp, CS r an BV o ouble sequences consisting o all ouble series whose sequence o partial sums are in the spaces M u, M u t), C p, C bp, C r an L u, respectively, an also examine some properties o those sequence spaces an etermine the α uals o the spaces BS, BV, CS bp an the β ϑ) uals o the spaces CS bp an CS r o ouble series. Quite recently Basar an Sever [28] have introuce the Banach space L q o ouble sequences corresponing to the well-known space l q o single sequences an examine some properties o the space L q. Quite recently Subramanian an Misra [29] have stuie the space χ 2 M p, q, u) o ouble sequences an gave some inclusion relations. Spaces are strongly summable sequences were iscusse by Kuttner [3], Maox [32], an others. The class o sequences which are strongly Cesàro summable with respect to a moulus was introuce by Maox [8] as an extension o the einition o strongly Cesàro summable sequences. Connor

4 322 N. Subramanian, P. Anbalagan an P. Thirunavukarasu [33] urther extene this einition to a einition o strong A summability with respect to a moulus where A =a n,k ) is a nonnegative regular matrix an establishe some connections between strong A summability, strong A summability with respect to a moulus, an A statistical convergence. In [34] the notion o convergence o ouble sequences was presente by A. Pringsheim. Also, in [35]-[38], an [39] the our imensional matrix transormation Ax) k,l = m= n= amn kl x mn was stuie extensively by Robison an Hamilton. We nee the ollowing inequality in the sequel o the paper. For a, b, 0 an 0 <p<, we have.) a + b) p a p + b p The ouble series m,n= x mn is calle convergent i an only i the ouble sequence s mn ) is convergent, where s mn = m,n i,j= x ijm, n N) see[]). A sequence x =x mn )is sai to be Pringsheim s ouble analytic i sup mn x mn /m+n <. The vector space o all Pringsheim s ouble analytic sequences will be enote by Λ 2. A sequence x = x mn ) is calle Pringsheim s ouble entire sequence i x mn /m+n 0asm, n. The Pringsheim s ouble entire sequences will be enote by Γ 2. Let φ = allinitesequences}. Consier a ouble sequence x =x ij ). The m, n) th section x [m,n] o the sequence is eine by x [m,n] = m,n i,j=0 x iji ij or all m, n N; where I ij enotes the ouble sequence whose only non zero term is a in the i, j) th place i+j)! or each i, j N. An FK-spaceor a metric space)x is sai to have AK property i I mn )isa Schauer basis or X. Or equivalently x [m,n] x. An FDK-space is a ouble sequence space enowe with a complete metrizable; locally convex topology uner which the coorinate mappings x =x k ) x mn )m, n N) are also continuous. Orlicz[3] use the iea o Orlicz unction to construct the space L M). Linenstrauss an Tzariri [7] investigate Orlicz sequence spaces in more etail, an they prove that every Orlicz sequence space l M contains a subspace isomorphic to l p p< ). subsequently, ierent classes o sequence spaces were eine by Parashar an Chouhary [4], Mursaleen et al. [], Bektas an Altin [3], Tripathy et al. [8], Rao an Subramanian [5], an many others. The Orlicz sequence spaces are the special cases o Orlicz spaces stuie in [6].

5 Moular o Γ 2 eine by asymptotically μ invariant moulus 323 Recalling [3] an [6], an Orlicz unction is a unction M :[0, ) [0, ) which is continuous, non-ecreasing, an convex with M 0) = 0, Mx) > 0, or x>0 an M x) as x. I convexity o Orlicz unction M is replace by subaitivity o M, then this unction is calle moulus unction, eine by Nakano [2] an urther iscusse by Ruckle [6] an Maox [8], an many others. An moulus unction M is sai to satisy the Δ 2 conition or small u or at 0 i or each k N, there exist R k > 0 an u k > 0 such that M ku) R k M u) or all u 0,u k ]. Moreover, an moulus unction M is sai to satisy the Δ 2 conition i an only i lim u 0+ sup M2u) < Mu) Two Moulus unctions M an M 2 are sai to be equivalent i there are positive constants α, β an b such that M αu) M 2 u) M βu) or all u [0,b]. An moulus unction M can always be represente in the ollowing integral orm M u) = u η t) t, 0 where η, the kernel o M, is right ierentiable or t 0,η0) = 0,ηt) > 0 or t>0,η is non-ecreasing an η t) as t whenever Mu) as u u. Consier the kernel η associate with the moulus unctionm an let μ s) =sup t : η t) s}. Then μ possesses the same properties as the unction η. Suppose now Φ= x μ s) s. 0 Then, Φ is an moulus unction. The unctions M an Φ are calle mutually complementary Orlicz unctions. Now, we give the ollowing well-known results. Let M an Φ are mutually complementary moulus unctions. Then, we have: i) For all u, y 0,.2) uy M u)+φy), Y oung sinequality) ii) For all u 0,.3) uη u) =M u)+φη u)). iii) For all u 0, an 0 <λ<,.4) M λu) λm u)

6 324 N. Subramanian, P. Anbalagan an P. Thirunavukarasu Linenstrauss an Tzariri [7] use the iea o Orlicz unction to construct Orlicz sequence space l M = x w : ) } k= M xk <, or someρ > 0, ρ The space l M with the norm x = in ρ>0: ) } k= M xk, ρ becomes a Banach space which is calle an Orlicz sequence space. For M t) = t p p< ), the spaces l M coincie with the classical sequence space l p. Any Orlicz unction M mn always has the integral representation M k x) = x p 0 mn t) t, where p mn, known as the kernel o M mn is non-ecreasing, is right continuous ort >0,p mn 0) = 0,p mn t) > 0 or t>0 an p mn t), as t. Given an Orlicz unction M mn with kernel p mn t), eine q mn s) =sup t : p mn t) s, s 0} Then q mn s) possesses the same properties as p mn t) an the unction N mn eine as N mn x) = x q 0 mn s) s is an Orlicz unction. The unctions M mn an N mn are calle mutually complementary Orlicz unctions. For a sequence M =M mn ) o Orlicz unctions, the moular sequence class l M is eine by l M = x =x mn ): m= n= M mn x mn ) < }. Using the sequence N =N mn ) o Orlicz unctions, similarly we eine l N. The class l M is eine by }.5) l M = x =x mn ): x mn y mn convergesor all y l N. m= n= For a sequence M =M mn ) o Orlicz unctions, the moular sequence class l M is also eine as l M = x =x mn ): m= n= M mn x mn ) < }. The space l M is a Banach space with respect to the norm x M eine as x M = in m= n= M mn x mn ) }. The single sequence spaces were introuce by Woo [49] aroun the year 973, an generalize the Orlicz sequence l M an the moulare sequence space consiere earlier by Nakano in [2]. An important subspace o l M, which is an AK-space is the space h M eine as h M = x l M : m= n= M mn x mn < )}

7 Moular o Γ 2 eine by asymptotically μ invariant moulus 325 A sequence M mn ) o Orlicz unctions is sai to satisy uniorm Δ 2 conition at 0 i there exist p> an k 0 N such that x 0, ) an k>k 0, we have xm mnx) M mnx) p, or equivalently, there exists a constant K> an k 0 N such that Mmn2x) K or all k>k M mnx) 0 an x 0, 2]. I the sequence Mmn ) satisies uniorm Δ 2 conition, then h M = l M an vice versa [49]. The notion o ierence sequence spaces or single sequences) was introuce by Kizmaz [30] as ollows Z Δ) = x =x k ) w :Δx k ) Z} or Z = c, c 0 an l, where Δx k = x k x k+ or all k N. Here c, c 0 an l enote the classes o convergent,null an boune sclar value single sequences respectively. The ierence space bv p o the classical space l p is introuce an stuie in the case p by BaŞar an Altay in [42] an in the case 0 <p<by Altay an BaŞar in [43]. The spaces c Δ),c 0 Δ),l Δ) an bv p are Banach spaces norme by x = x + sup k Δx k an x bvp = k= x k p ) /p, p< ). Later on the notion was urther investigate by many others. We now introuce the ollowing ierence ouble sequence spaces eine by Z Δ) = x =x mn ) w 2 :Δx mn ) Z} where Z =Λ 2,χ 2 an Δx mn =x mn x mn+ ) x m+n x m+n+ )=x mn x mn+ x m+n + x m+n+ or all m, n N. 2. Deinition an Preliminaries Throughout a ouble sequence is enote by X mn, a ouble ininite array o uzzy real numbers. Let D enote the set o all close an boune intervals X =[a,a 2 ] on the real line R. For X =[a,a 2 ] D an Y =[b,b 2 ] D, eine X, Y )=max a b, a 2 b 2 ) It is known that D, ) is a complete metric space. A uzzy real number X is a uzzy set on R, that is, a mapping X : R I = [0, ]) associating each real number t with its grae o membership X t). The α level set [X] α, o the uzzy real number X, or 0 <α ; is eine by [X] α = t R : X t) α}. The 0 level set is th closure o the strong 0 cut that is, cl t R : X t) > 0}. A uzzy real number X is calle convex i X t) X s) X r) =min X s),xr)}, where s<t<r.i there exists t 0 R such that X t 0 ) = then, the uzzy real number X is calle normal. A uzzy real number X is sai to be upper-semi continuous i, or each ɛ>0,x [0,a+ ɛ)) is open in the usual topology o R or all a I. The set o all upper-semi continuous, normal, convex uzzy real numbers is

8 326 N. Subramanian, P. Anbalagan an P. Thirunavukarasu enote by L R). The absolute value, X o X L R) is eine by max X t),x t)}, i t 0; X t) = 0, i t<0 Let : L R) L R) R be eine by X, Y )=sup 0 α [X] α, [Y ] α ). Then, eines a metric on L R) an it is well-known that L R), ) is a complete metric space. A sequence X mn LR) is sai to be null i X mn, 0) = 0. A ouble sequence X mn o uzzy real numbers is sai to be entire in Pringsheim s sense to a uzzy number 0 i lim m,n X mn ) /m+n =0. A ouble sequence X mn is sai to regularly i it converges in the Prinsheim s sense an the ollowing limts zero: lim m X mn ) /m+n = 0 or each n N, an lim n X mn ) /m+n = 0 or each m N. By the null o ouble we mean the null on the Pringsheim sense that is, a ouble sequence x =x mn ) has Pringsheim limit 0enote by P limx = 0). We shall write more briely as P null. We enote w 2 F ) the set o all sequences X =X mn ) o uzzy numbers. A sequence X =X mn ) o uzzy numbers is sai to be Pringsheim s ouble analytic i the set X mn :m, n) N} o uzzy numbers is Pringsheim s ouble analytic. A K space o sequences or which the coorinate linear unctionals are continuous. Let σ be a one-to-one mapping o the set o positive integers into itsel such that σ mn η) =σ σ m n η)),m,n=, 2, 3, 2.. Deinition. Let mn an g mn are two complementary unctions an X = X mn ),Y = Y mn ) be two sequences o uzzy numbers are sai to be σ asymptotically equivalent i X,σ,μ,λ [ Γ2F Y = in η. Xσ mn η) Y σ mn η) ) /m+n 0asm, n, uniormly 2.2. Deinition. Let mn an g mn are two complementary unctions an X = X mn ),Y =Y mn ) be two sequences o uzzy numbers are sai to be S,σ,μ,λ Γ2F i or every ɛ>0 [ S,σ,μ,λ Γ2F = /m+n μ rs m, n I rs : Xσ mn η)) ɛ} 0asr, s ), uniormly in η. In this case, we write X mn 0 S,σ,μ,λ Γ2F.

9 Moular o Γ 2 eine by asymptotically μ invariant moulus Deinition. Let mn an g mn are two complementary unctions an X = X mn ),Y =Y mn ) be two sequences o uzzy numbers are sai to be asymptotically μ invariant statistical equivalent o multiple 0 provie that or every ɛ>0, X SΓ2F,σ,μ,λ Y = μ rs m, n I rs : 0asr, s, uniormly in η. [ Xσ mn η) Y σ mn η) ) /m+n 2.4. Example. Let μ rs = rs an σ mn η) =η + or all η, r, s N. Consier the sequences o uzzy numbers X =X mn ) an Y =Y mn ) eine by X rs = rs) 2 an Y rs = rs) or [ all r, s N. Then Xσ mn lim rs μ rs m, n I rs : η) [ ) /m+n Y σ mn η) ɛ} = ɛ} =0. rs) lim rs μ rs m, n [, rs : I we take μ rs = rs or all r, s N, the above einition reuces to ollowing einition: 2.5. Deinition. Let mn an g mn are two complementary unctions an X = X mn ),Y =Y mn ) be two sequences o uzzy numbers are sai to be asymptotically invariant statistical equivalent o multiple 0 provie that or every ɛ>0, X,σ,λ SΓ2F Y = rs, uniormly in η. m, n rs) : [ ) Xσ mn /m+n η) Y σ mn η) ɛ} 0asr, s 2.6. Deinition. Let mn an g mn are two complementary unctions an X = X mn ),Y =Y mn ) be two sequences o uzzy numbers are sai to be strong V,σ,μ,λ Γ2F asymptotically equivalent o multiple 0 provie that [ X V Γ2F,σ,λ Y = ) Xσ mn /m+n λ rs m I rs n I rs η) Y σ mn η) 0asr, s, uniormly in η Example. Let μ rs = rs an σ mn η) =η + or all η, r, s N. Consier the sequences o uzzy numbers X =X mn ) an Y =Y mn ) eine by X rs = rs) 2 an Y rs = rs) [ or all r, s N. Then ) Xσ mn /m+n lim rs μ rs m I rs n I rs η) Y σ mn η) = [ lim r s rs rs m= n= rs) 2m+n) rs m+n)) = lim rs rs r m= s n= <. rs) m+n I we take λ rs = rs or all r, s N, the above einition reuces to the ollowing einition:

10 328 N. Subramanian, P. Anbalagan an P. Thirunavukarasu 2.8. Deinition. Let mn an g mn are two complementary unctions an X = X mn ),Y =Y mn ) be two sequences o uzzy numbers are sai to be strong C,σ,μ,λ Γ2F asymptotically equivalent o multiple 0 provie that [ X,σ,λ CΓ2F Y = r ) s λ rs m n= Xσ mn /m+n η) Y σ mn η) 0asr, s, uniormly in η. I we take σ mn η) =η+, the above einitions reuce the ollowing einitions: 2.9. Deinition. Let mn an g mn are two complementary unctions an X = X mn ),Y =Y mn ) be two sequences o uzzy numbers are sai to be asymptotically almost equivalent i X,σ,λ [ Y = Xm+η,n+η ) ˆF /m+n CΓ2 Y m+η,n+η 0asm, n, uniormly in η Deinition. Let mn an g mn are two complementary unctions an X =X mn ) be a sequences o uzzy numbers are sai to be statistically almost convergent 0 [ λ rs mn) I rs : X m+η,n+η ) /m+n } ] ɛ 0asr, s, uni- ormly in η. In this we write X mn 0 S Γ2 ˆF,λ ). ) 2.. Deinition. Let mn an g mn are two complementary unctions an X =X mn ) be a sequences o uzzy numbers are sai to be asymptotically almost μ statistical equivalent o multiple 0 provie that or every ɛ>0 ˆF SΓ2 X,μ,λ Y = μ rs m, n I rs : 0asr, s, uniormly in η. [ Xm+η,n+η Y m+η,n+η ) /m+n I we take μ rs = rs or all r, s N, the above einition reuces to the ollowing einition: 2.2. Deinition. Let mn an g mn are two complementary unctions an X =X mn ) be a sequences o uzzy numbers are sai to be asymptotically almost statistical equivalent o multiple 0 provie that or every ɛ>0 ˆF SΓ2 X,μ,λ Y = μ rs m, n I rs : 0asr, s, uniormly in η. [ Xm+η,n+η Y m+η,n+η ) /m+n 2.3. Deinition. Let mn an g mn are two complementary unctions an X =X mn ) be a sequences o uzzy numbers are sai to be strong asymptotically almost μ statistical equivalent o multiple 0 provie that

11 Moular o Γ 2 eine by asymptotically μ invariant moulus 329 [ V Γ2 ˆF X,μ,λ Y = ) /m+n Xm+η,n+η μ rs m I rs n I rs Y m+η,n+η 0asr, s, uniormly in η. I we take μ rs = rs or all r, s N, the above einition reuces to ollowing einition Deinition. Let mn an g mn are two complementary unctions an X = X mn ) be a sequences o uzzy numbers are sai to be strong asymptotically almost equivalent o multiple 0 provie that [ ˆF CΓ2 X,μ,λ Y = r s rs m= n=, uniormly in η. Xm+η,n+η Y m+η,n+η ) /m+n 0asr, s 3. Main Results 3.. Theorem. Let mn an g mn are two complementary unctions an X = X mn ),Y =Y mn ) be two real value sequences o uzzy numbers. Then, the ollowing conitions are satisie: i) I X V Γ2F,σ,λ Y, then XSΓ 2F,σ,λ Y ii) I X Λ 2F an X SΓ2F,σ,λ Y, then XV Γ 2F,σ,λ Y ; hence, XCΓ,σ,λ Y. 2F iii) X,σ,λ SΓ2F Y Λ 2F = X V Γ 2F,σ,λ Y Λ 2F. Proo: I ɛ>0 an X V Γ2F,σ,λ Y, then [ ) Xσ mn /m+n m I rs n I rs η) Y σ mn η) m I rs n I rs, ɛ m, n rs) : ψ λmn [ Xσ /m+n mn η),0! ɛ Y σ mn η) [ Xσ mn η) Y σ mn η) ) /m+n. Thereore, X,σ,λ SΓ2F Y. ii) Suppose that X =X mn ) an Y =Y mn ) are in Λ 2F Then, we can assume that [ Given ɛ>0, μ rs m I rs n I rs Xσ mn η) Y σ mn η) ) /m+n Xσ mn η) Y σ mn η) ) /m+n M, or all m, n an η [ Xσ mn η) Y σ mn η) ) /m+n an X SΓ2F,σ,λ Y.

12 330 N. Subramanian, P. Anbalagan an P. Thirunavukarasu [ ψ μ rs m I /m+n rs,0! μ rs m I rs n I rs, n I rs, ψ λmn M μ rs m, n rs) : Thereore, X V Γ2F,σ,λ[ s n= rs r m= r μrs s μrs rs m= n= rs m I rs n Irs r μrs s μrs μ rs m= n= μ rs m I rs n Irs 2 μ rs m I rs n Irs λmn Xσ mn η) Y σ mn η) Xσ /m+n mn η),0! <ɛ Y σ mn η) [ ɛ [ Xσ mn η) Y σ mn η) ) /m+n + Xσ mn η) Y σ mn η) ) /m+n Xσ mn η) Y σ mn η) ) /m+n + ɛ. Y. Further, we have ) Xσ mn /m+n η) Y σ mn η) = [ ) Xσ mn /m+n η) Y σ mn η) + [ ) Xσ mn /m+n η) Y σ mn η) [ ) Xσ mn /m+n η) Y σ mn η) + [ ) Xσ mn /m+n η) Y σ mn η) [ ) Xσ mn /m+n η) Y σ mn η). Hence, X,σ,λ CΓ2F Γ Y since XV,σ,λ Y. iii) Follows rom i) an ii). 2F 3.2. Theorem. Let mn an g mn are two complementary unctions an X = X mn ),Y =Y mn ) be two real value sequences o uzzy numbers, X,σ,λ SΓ2F Y X SΓ2F,σ,μ,λ Y i 3.) limin Proo: For a given [ ɛ>0, we have m, n rs) : [ m, n I rs : Thereore, [ rs m, n rs) : [ rs m, n I rs : μrs ) rs Xσ mn η) Y σ mn η) ) /m+n Xσ mn η) Y σ mn η) ) /m+n. Xσ mn η) Y σ mn η) ) /m+n Xσ mn η) Y σ mn η) ) /m+n

13 Moular o Γ 2 eine by asymptotically μ invariant moulus 33 [ μ rs ) Xσ mn /m+n rs μ rs m, n I rs : η) Y σ mn η) ɛ}. Taking the limit as r, s an using equation 3.), we get the result Conclusions. The concept o asymptotic equivalence was irst suggeste by Marou in 993. Ater that, several authors introuce an stuie some asymptotically equivalent sequences. The results are obtaine moular space o ouble entire sequences with respect o uzzy numbers. Reerences [] T.Apostol, Mathematical Analysis, Aison-wesley, Lonon, 978. [2] M.Basarir an O.Solancan, On some ouble sequence spaces, J. Inian Aca. Math., 22) 999), [3] C.Bektas an Y.Altin, The sequence space l M p, q, s) on seminorme spaces, Inian J. Pure Appl. Math., 344) 2003), [4] T.J.I A.Bromwich, An introuction to the theory o ininite series Macmillan an Co.Lt.,New York, 965). [5] G.H.Hary, On the convergence o certain multiple series, Proc. Camb. Phil. Soc., 9 97), [6] M.A.Krasnoselskii an Y.B.Rutickii, Convex unctions an Orlicz spaces, Gorningen, Netherlans, 96. [7] J.Linenstrauss an L.Tzariri, On Orlicz sequence spaces, Israel J. Math. 97), [8] I.J.Maox, Sequence spaces eine by a moulus, Math. Proc. Cambrige Philos. Soc, 00) 986), [9] F.Moricz, Extentions o the spaces c an c 0 rom single to ouble sequences, Acta. Math. Hung., 57-2), 99), [0] F.Moricz an B.E.Rhoaes, Almost convergence o ouble sequences an strong regularity o summability matrices, Math. Proc. Camb. Phil. Soc.4, 988), [] M.Mursaleen,M.A.Khan an Qamaruin, Dierence sequence spaces eine by Orlicz unctions, Demonstratio Math., Vol. XXXII 999), [2] H.Nakano, Concave moulars, J. Math. Soc. Japan, 5953), [3] W.Orlicz, Über Raume L M) Bull. Int. Aca. Polon. Sci. A, 936), [4] S.D.Parashar an B.Chouhary, Sequence spaces eine by Orlicz unctions, Inian J. Pure Appl. Math., 254)994), [5] K.Chanrasekhara Rao an N.Subramanian, The Orlicz space o entire sequences, Int. J. Math. Math. Sci., ), [6] W.H.Ruckle, FK spaces in which the sequence o coorinate vectors is boune, Cana. J. Math., 25973), [7] B.C.Tripathy, On statistically convergent ouble sequences, Tamkang J. Math., 343), 2003), [8] B.C.Tripathy,M.Et an Y.Altin, Generalize ierence sequence spaces eine by Orlicz unction in a locally convex space, J. Anal. Appl., 3)2003), [9] A.Turkmenoglu, Matrix transormation between some classes o ouble sequences, J. Inst. Math. Comp. Sci. Math. Ser., 2), 999), [20] P.K.Kamthan an M.Gupta, Sequence spaces an series, Lecture notes, Pure an Applie Mathematics, 65 Marcel Dekker, In c., New York, 98. [2] A.Gökhan an R.Çolak, The ouble sequence spaces c P 2 p) an cpb 2 p), Appl. Math. Comput., 572), 2004), [22] A.Gökhan an R.Çolak, Double sequence spaces l 2, ibi., 60), 2005),

14 332 N. Subramanian, P. Anbalagan an P. Thirunavukarasu [23] M.Zeltser, Investigation o Double Sequence Spaces by Sot an Har Analitical Methos, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ. o Tartu, Faculty o Mathematics an Computer Science, Tartu, 200. [24] M.Mursaleen an O.H.H. Eely, Statistical convergence o ouble sequences, J. Math. Anal. Appl., 288), 2003), [25] M.Mursaleen, Almost strongly regular matrices an a core theorem or ouble sequences, J. Math. Anal. Appl., 2932), 2004), [26] M.Mursaleen an O.H.H. Eely,Almost convergence an a core theorem or ouble sequences, J. Math. Anal. Appl., 2932), 2004), [27] B.Altay an F.BaŞar, Some new spaces o ouble sequences, J. Math. Anal. Appl., 309), 2005), [28] F.BaŞar an Y.Sever, The space L p o ouble sequences, Math. J. Okayama Univ, 5, 2009), [29] N.Subramanian an U.K.Misra, The semi norme space eine by a ouble gai sequence o moulus unction, Fasciculi Math., 46, 200). [30] H.Kizmaz, On certain sequence spaces, Can. Math. Bull., 242), 98), [3] B.Kuttner, Note on strong summability, J. Lonon Math. Soc., 2946), [32] I.J.Maox, On strong almost convergence, Math. Proc. Cambrige Philos. Soc., 852), 979), [33] J.Cannor, On strong matrix summability with respect to a moulus an statistical convergence, Cana. Math. Bull., 322), 989), [34] A.Pringsheim, Zurtheorie erzweiach unenlichen zahlenolgen, Math. Ann., 53, 900), [35] H.J.Hamilton, Transormations o multiple sequences, Duke Math. J., 2, 936), [36] -, A Generalization o multiple sequences transormation, Duke Math. J., 4, 938), [37] -, Change o Dimension in sequence transormation, Duke Math. J., 4, 938), [38] -, Preservation o partial Limits in Multiple sequence transormations, Duke Math. J., 4, 939), [39], G.M.Robison, Divergent ouble sequences an series, Amer. Math. Soc. Trans., 28, 926), [40] L.L.Silverman, On the einition o the sum o a ivergent series, un publishe thesis, University o Missouri stuies, Mathematics series. [4] O.Toeplitz, Über allgenmeine linear mittel briungen, Prace Matemalyczno Fizyczne warsaw), 22, 9). [42] F.BaŞar an B.Atlay, On the space o sequences o p boune variation an relate matrix mappings, Ukrainian Math. J., 55), 2003), [43] B.Altay an F.BaŞar, The ine spectrum an the matrix omain o the ierence operator Δ on the sequence space l p <p<), Commun. Math. Anal., 22), 2007), -. [44] R.Çolak,M.Et an E.Malkowsky, Some Topics o Sequence Spaces, Lecture Notes in Mathematics, Firat Univ. Elazig, Turkey, 2004, pp. -63, Firat Univ. Press, 2004), ISBN: [45] H.Dutta, F.BaŞar, A Generalization o Orlicz sequence spaces by cesàro mean o orer one, Acta Math. Univ. Commenianae, Vol. LXXX 2), 20), [46] Ayhan Esi, On some new ierence ouble sequence spaces via Orlicz unction, Journal o Avance Stuies in Topology, Vol. 2 No. 2), 20), [47] G.Goes an S.Goes. Sequences o boune variation an sequences o Fourier coeicients, Math. Z., 8, 970), [48] M.Gupta an S.Prahan, On Certain Type o Moular Sequence space, Turk J. Math., 32, 2008),

15 Moular o Γ 2 eine by asymptotically μ invariant moulus 333 [49] J.Y.T. Woo, On Moular Sequence spaces, Stuia Math., 48, 973), [50] H.I.Brown, The summability iel o a perect l l metho o summation, Journal D Analyse Mathematique, 20, 967), Receive: December, 202