Some Ideal Convergent Sequence Spaces Defined by a Sequence of Modulus Functions Over n-normed Spaces

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1 MATEMATIKA, 203, Volue 29, Nube 2, c Depatent of Matheatical Sciences, UTM. Soe Ideal Convegent Sequence Spaces Defined by a Sequence of Modulus Functions Ove n-noed Spaces Kuldip Raj and 2 Sunil K. Shaa,2 School of Matheatics, Shi Mata Vaishno Devi Univesity, Kata-82320, J&K, INDIA. e-ail: kuldipaj68@gail.co, 2 sunilkshaa42@yahoo.co.in Abstact In the pesent pape we study soe ideal convegent sequence spaces defined by a sequence of odulus functions in n-noed spaces and exaine soe topological popeties and inclusion elations between these spaces. Keywods Lacunay sequence, Diffeence sequence space, Modulus function, Ideal convegence, n-noed space. 200 Matheatics Subject Classification 40A05, 40C05, 46A45. Intoduction The concept of 2-noed spaces was initially developed by Gähle ] in the id of 960 s, while that of n-noed spaces one can see in Misiak 2]. Since then, any othes have studied this concept and obtained vaious esults, see Gunawan (3], 4]) and Gunawan and Mashadi 5]. Definition Let n N and X be a linea space ove the field K, whee K is the field of eal o coplex nubes of diension d, whee d n 2. A eal valued function,, on X n satisfying the following fou conditions: (i) x, x 2,, x n = 0 if and only if x, x 2,, x n ae linealy dependent in X; (ii) x, x 2,, x n is invaiant unde peutation; (iii) αx, x 2,, x n = α x, x 2,, x n fo any α K, and (iv) x + x, x 2,, x n x, x 2,, x n + x, x 2,, x n is called a n-no on X and the pai (X,,, ) is called a n-noed space ove the field K. Fo exaple, we ay take X = R n being equipped with the Euclidean n-no x, x 2,, x n E to be equal to the volue of the n-diensional paallelopiped spanned by the vectos x, x 2,, x n which ay be given explicitly by the foula x, x 2,, x n E = det(x ij ),

2 88 Kuldip Raj and Sunil K. Shaa whee x i = (x i, x i2,, x in ) R n fo each i =, 2,, n. Let (X,,, ) be an n- noed space of diension d n 2 and a, a 2,, a n } be linealy independent set in X. Then the following function,, on X n defined by x, x 2,, x n = ax x, x 2,, x n, a i : i =, 2,, n} defines an (n )-no on X with espect to a, a 2,, a n }. The standad n-no on X, a eal inne poduct space of diension d n is as follows: x, x 2,, x n S = x, x... x, x n x n, x... x n, x n whee.,. denotes the inne poduct on X. If X = R n, then this n-no is exactly the sae as the Euclidean n-no x, x 2,, x n E entioned ealie. Fo n =, this n-no is the usual no x = x, x 2. Definition 2 A sequence (x k ) in a n-noed space (X,,, ) is said to convege to soe L X if li x k L, z,, z n = 0 fo evey z,, z n X. k 2, Definition 3 A sequence (x k ) in a n-noed space (X,,, ) is said to be Cauchy if li x k x p, z,, z n = 0 fo evey z,, z n X. k p If evey Cauchy sequence in X conveges to soe L X, then X is said to be coplete with espect to the n-no. Any coplete n-noed space is said to be n-banach space. The notion of ideal convegence was intoduced fist by Kostyko 6] as a genealization of statistical convegence which was futhe studied in topological spaces by Das et al. 7]. Moe applications of ideals can be seen in Das et al. 7] and Das et al. 8]. Let (X,. ) be a noed space. Recall that a sequence (x n ) n N of eleents of} X is called statistically convegent to x X if the set A(ɛ) = n N : x n x ɛ has natual density zeo fo each ɛ > 0. Definition 4 A faily I 2 Y of subsets of a non epty set Y is said to be an ideal in Y if (i) φ I (ii) A, B I iply A B I (iii) A I, B A iply B I,

3 Soe Ideal Convegent Sequence Spaces 89 while an adissible ideal I of Y futhe satisfies x} I fo each x Y 9]. Let I 2 N be a non tivial ideal in N. A sequence (x n ) n N in X is said } to be I- convegent to x X, if fo each ɛ > 0 the set A(ɛ) = n N : x n x ɛ belongs to I 6]. A sequence of positive integes θ = (k ) is called lacunay if k 0 = 0, 0 < k < k + and h = k k as. The intevals deteined by θ will be denoted by I = (k, k ) and q = k k. The space of lacunay stongly convegent sequences N θ was defined by Feedan et al. 0] as: N θ = } x w : li x k l = 0, fo soe l. h Stongly alost convegent sequence was intoduced and studied by Maddox ] and Feedan et al. 0]. Paasha and Choudhay 2] have intoduced and exained soe popeties of fou sequence spaces defined by using an Olicz function M, which genealized the well-known Olicz sequence spaces C,, p], C,, p] 0 and C,, p]. It ay be noted hee that the space of stongly suable sequences wee discussed by Maddox 3]. The notion of diffeence sequence spaces was intoduced by Kızaz 4], who studied the diffeence sequence spaces l ( ), c( ) and c 0 ( ). The notion was futhe genealized by Et and Çolak 5] by intoducing the spaces l ( n ), c( n ) and c 0 ( n ). Let w be the space of all eal o coplex sequences x = (x k ). Let, n be non-negative integes, then fo Z = l, c and c 0, we have sequence spaces, Z( n ) = x = (x k) w : ( n x k) Z} whee n x = ( n x k ) = ( n x k n x k+ ) and 0 x k = x k fo all k N, which is equivalent to the following binoial epesentation n x k = n ( ( ) v n v v=0 ) x k+v. Taking = n =, we get the spaces l ( ), c( ) and c 0 ( ) intoduced and studied by Kızaz 4]. Definition 5 A odulus function is a function f : 0, ) 0, ) such that (i) f(x) = 0 if and only if x = 0, (ii) f(x + y) f(x) + f(y) fo all x 0, y 0, (iii) f is inceasing, (iv) f is continuous fo ight at 0. It follows fo (i) and (iv) that f ust be continuous eveywhee on 0, ). Fo a sequence of odulus function F = (f k ), we give the following conditions: (v) sup f k (x) < fo all x > 0, k

4 90 Kuldip Raj and Sunil K. Shaa (vi) li x 0 f k (x) = 0 unifoly in k. We eak that in case f = (f k ) fo all k, whee f is a odulus, the conditions (v) and (vi) ae autoatically fulfilled. The odulus function ay be bounded o unbounded. Fo exaple, if we take f(x) = x x+, then f(x) is bounded. If f(x) = xp, 0 < p <, then the odulus f(x) is unbounded. Subsequently, odulus function has been discussed 6 2]. Let I be an adissible ideal, F = (f k ) be a sequence of odulus functions, (X,,, ) be an n-noed space, p = (p k ) be a sequence of positive eal nubes and u = ( ) be a sequence of stictly positive eal nubes. By w(n X) we denote the space of all sequences defined ove n- noed space (X,,, ). In the pesent pape, we define the following classes of sequences: N θ, n, F, u, p,,,, X S ] I = ( x = (x k ) w(n X) : N : fk h n x k L, z, z 2,, z n )] p k ε, fo soe L and fo evey z, z 2,, z n X I, N θ, n, F, u, p,,,, X S ] I 0 = ( x = (x k ) w(n X) : N : fk h n x k, z, z 2,, z n )] pk ε, fo evey z, z 2,, z n X I. If F(x) = x, we get N θ, n, u, p,,,, X S ] I = ( x = (x k ) w(n X) : N : h n x k L, z, z 2,, z n ) p k ε, fo soe L and fo evey z, z 2,, z n X I, N θ, n, u, p,,,, X S ] I 0 = ( x = (x k ) w(n X) : N : h n x k, z, z 2,, z n ) p k ε, fo evey z, z 2,, z n X I. If p = (p k ) =, we get N θ, n, F, u,,,, X S ] I = ( x = (x k ) w(n X) : N : fk h n x k L, z, z 2,, z n )] ε, fo soe L and fo evey z, z 2,, z n X I,

5 Soe Ideal Convegent Sequence Spaces 9 N θ, n, F, u,,,, X S ] I 0 = ( x = (x k ) w(n X) : N : fk h n x k, z, z 2,, z n )] ε, fo evey z, z 2,, z n X I. If p = (p k ) = and u = ( ) =, we get N θ, n, F,,,, X S ] I = ( x = (x k ) w(n X) : N : fk h n x k L, z, z 2,, z n )] ε, fo soe L and fo evey z, z 2,, z n X I, N θ, n, p,,,, X S] I 0 = ( x = (x k ) w(n X) : N : fk h n x k, z, z 2,, z n )] ε, fo evey z, z 2,, z n X I. The following inequality will be used thoughout the pape. If 0 p k sup p k = H, D = ax(, 2 H ) then a k + b k pk D a k pk + b k pk } () fo all k and a k, b k C. Also a pk ax(, a H ) fo all a C. The ai of this pape is to study diffeence I-convegent sequence spaces defined by a sequence of odulus functions in n-noed spaces and exaine soe topological popeties and inclusion elations between the spaces N θ, n, F, u, p,,,, X S ] I and N θ, n, F, u, p,,,, X S ] I 0. 2 Main Results Theoe Let F = (f k ) be a sequence of odulus functions, p = (p k ) be a bounded sequence of positive eal nubes and u = ( ) be any sequence of stictly positive eal nubes, then N θ, n, F, u, p,,,, X S] I and N θ, n, F, u, p,,,, X S] I 0 ae linea spaces ove the field of coplex nube C. Poof Let x = (x k ), y = (y k ) N θ, n, F, u, p,,,, X S ] I 0 and α, β C, then thee exist positive integes M α and N β such that α M α and β N β. Since,, is a n-no and f k is a odulus function fo all k and also by using (), the following inequality holds

6 92 Kuldip Raj and Sunil K. Shaa ( fk h n (αx k + βy k ), z, z 2,, z n )] p k D(M α ) H h fk ( n x k, z, z 2,, z n )] p k + D(N β ) H ( fk h n y k, z, z 2,, z n )] pk. On the othe hand fo the above inequality, we get ( N : h fk n (αx k + βy k ), z, z 2,, z n ) p k ε N : D(M α ) H h N : D(N β ) H h fk ( n x k, z, z 2,, z n )] p k ε } fk ( n y k, z, z 2,, z n )] p k ε }. Two sets on the ight side belongs to I, so this copletes the poof. Lea Let f be a odulus function and let 0 < δ <. Then fo each x > δ, we have f(x) 2f()δ x. Fo detail see 3]. Theoe 2 Let F = (f k ) be a sequence of odulus functions and 0 < inf k p k = h p k sup k p k = H <. Then (i) N θ, n, u, p,,,, X S ] I N θ, n, F, u, p,,,, X S ] I and (ii) N θ, n, u, p,,,, X S] I 0 N θ, n, F, u, p,,,, X S] I 0. Poof If x = (x k ) N θ, n, u, p,,,, X S ] I, then fo soe L > 0 and fo evey z, z 2,, z n X. Thus ( N : h n x k L, z, z 2,, z n ) } pk ε. Now let ε > 0 be given. We can choose 0 < δ < such that fo evey t with 0 t δ we have f k (t) < ε fo all k. Now, using Lea, we get ( N : fk h n x k, z, z 2,, z n ) p k ε = ( N : h axε h, ε H } ) ε } h N : ax(2f k ()δ ) h, (2f k ()δ ) H } ( h n x k L, z, z 2,, z n ) } p k.

7 Soe Ideal Convegent Sequence Spaces 93 This copletes the poof of (i). Siilaly we can pove (ii). Theoe 3 Let F = (f k ) be a sequence of odulus functions. If li t sup f k(t) t = A > 0 fo all k, then N θ, n, F, u, p,,,, X S ] I 0 = N θ, n, u, p,,,, X S ] I 0 and N θ, n, u, p,,,, X S] I = N θ, n, F, u, p,,,, X S] I. Poof To pove N θ, n, u, p,,,, X S ] I = N θ, n, F, u, p,,,, X S ] I. It is sufficient to show that N θ, n, F, u, p,,,, X S ] I N θ, n, u, p,,,, X S ] I. Let x N θ, n, F, u, p,,,, X S ] I. Since A > 0, fo evey t > 0 we wite f k (t) At fo all k. Fo this inequality ( fk h n x k, z, z 2,, z n )] p k A H ( h n x k, z, z 2,, z n ) p k we get the esult. Siilaly we can pove the othe pat. Coollay Let F = (f k ) and F = (f k ) be sequences of odulus functions. If li t sup f k (t) f k (t) < iplies N θ, n, F, u, p,,,, X S ] I 0 N θ, n, F, u, p,,,, X S ] I 0 and N θ, n, F, u, p,,,, X S ] I N θ, n, F, u, p,,,, X S ] I. Poof It is tivial. Theoe 4 Let (X,,, XS ) and (X,,, XE ) be standad and Euclid n-noed spaces, espectively. Then N θ, n, F, u, p,,,, X S ] I N θ, n, u, p,,,, X E ] I N θ, n, F, u, p, (,, XS +,, XE )] I.

8 94 Kuldip Raj and Sunil K. Shaa Poof We have the following inclusion (( )( N : h fk,, XS +,, XE n x k L, z, z 2,, z n ))] pk N : D ( ) fk pk h n x k L, z, z 2,, z n XS ε N : D h by using inequality (). This copletes the poof. fk ( n x k L, z, z 2,, z n XE )] pk ε } Theoe 5 Let F = (f k ) and F = (f k ) be two sequences of odulus functions. Then (i) N θ, n, F, u, p,,,, X S ] I 0 N θ, n, F F, u, p,,,, X S ] I 0 and (ii) N θ, n, F, u, p,,,, X S ] I N θ, n, F F, u, p,,,, X S ] I. N θ, n, F, u, p,,,, X S ] I 0 N θ, n, F, u, p,,,, X S ] I 0 } ε and N θ, n, F + F, u, p,,,, X S ] I 0 N θ, n, F, u, p,,,, X S ] I 0 N θ, n, F, u, p,,,, X S ] I 0 N θ, n, F + F, u, p,,,, X S ] I 0. Poof (i) Let x = (x k ) N θ, n, F, u, p,,,, X S ] I. Let 0 < ε < and δ with 0 < δ < such that f k (t) < ε fo 0 < t < δ. Let y k = f k ( n x k L, z, z 2,, z n ). Let f h k (y k ) ] p k = f h k (y k ) ] p k + f h k (y k ) ] p k, 2 whee the fist suation is ove y k δ and the second suation is ove y k > δ. Then f k (y k ) ] p k ε H and fo y k > δ, we use the fact that h y k < y k δ < + y k δ ], whee z ] denotes the intege pat of z. So that fo the popeties of odulus function, we have fo y k > δ ( f k (y k) < + y ]) k δ f k () 2f k ()y k δ fo all k. Hence f h k (y k ) ] p k 2 f k () ] H y k ] pk δ h 2 2

9 Soe Ideal Convegent Sequence Spaces 95 which togethe with h f k (y k ) ] p k ε H yields f h k (y k ) ] pk ε H + ax(, (2f k()δ ) H ) f k (y k ) ] p k and this copletes the poof of (i). (ii) Let x = (x k ) N θ, n, F, u, p,,,, X S ] I 0 N θ, n, F, u, p,,,, X S ] I 0. The fact that ] (f k + f k )( pk n x k L, z, z 2,, z n ) h gives the esult. Conclusion h + h ] f k ( n x pk k L, z, z 2,, z n ) ] f k ( pk n x k L, z, z 2,, z n ) In this pape we have constucted I-convegent sequence spaces defined by a sequence of odulus functions ove n- noed spaces. We have studied soe topological popeties and inteesting inclusion elation between these sequence spaces. The solutions obtained ae potentially significant and ipotant fo the explanation of soe pactical physical pobles. The ethod ay also be applied to othe sequence spaces. Acknowledgents We thank the anonyous efeee(s) fo the caeful eading, valuable suggestions which ipoved the pesentation of the pape. Refeences ] Gähle, S. Linea 2-noiete Rue. Math. Nach : ] Misiak, A. n-inne poduct spaces. Math. Nach : ] Gunawan, H. On n-inne poduct, n-nos, and the Cauchy-Schwatz inequality. Sci. Math. Jpn : ] Gunawan, H. The space of p-suable sequence and its natual n-no. Bull. Aust. Math. Soc : ] Gunawan, H. and Mashadi, M. On n-noed spaces. Int. J. Math. Math. Sci :

10 96 Kuldip Raj and Sunil K. Shaa 6] Kostyko, P., Salat, T. and Wilczynski, W. I-Convegence. Real Anal. Exchange (2): ] Das, P., Kostyko, P., Wilczynski, W. and Malik, P. I and I* convegence of double sequences. Math. Slovaca : ] Das, A. and Malik, P., On the statistical and I- vaiation of double sequences. Real Anal. Exchange (2): ] Gudal, M. and Pehlivan, S. Statistical convegence in 2-noed spaces. Southeast Asian Bull. Math (2): ] Feedan, A. R., Sebe, J. J. and Raphael, M. Soe Cesao-type suability spaces. Poc. London Math. Soc : ] Maddox, I. J. On stong alost convegence. Math. Poc. Cab. Phil. Soc : ] Paasha, S. D. and B. Choudhay. Sequence spaces defined by Olicz functions. Indian J Pue Appl. Math : ] Maddox, I. J. Spaces of stongly suable sequences. Quat. J. Math : ] Kızaz, H. On cetain sequence spaces. Canad. Math. Bull : ] Et, M. and Çolak, R. On soe genealized diffeence sequence spaces. Soochow J. Math : ] Bilgen, T. On statistical convegence. An. Univ. Tiisoaa Se. Math. Info : ] Malkowsky, E. and Savaş, E. Soe λ-sequence spaces defined by a odulus. Ach. Math. (Bno), : ] Nuten, F., Mehet, A. and Esi, A. I-Lacunay genealized diffeence convegent sequences in n-noed spaces. J. Math. Analysis, 20. 2: ] Raj, K., Shaa, S. K. and Shaa, A. K. Soe new sequence spaces defined by a sequence of odulus functions in n-noed spaces. Int. J Math. Sci. Engg. Appl : ] Raj, K. and Shaa, S. K. Diffeence sequence spaces defined by sequence of odulus functions. Poyecciones : ] Savaş, E. On soe genealized sequence spaces defined by a odulus. Indian J. Pue Appl. Math :