STATISTICALLY CONVERGENT TRIPLE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTION

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1 Journal o athematical Analysis ISSN: , URL: Volume 4 Issue 2203), Pages STATISTICALLY CONVERGENT TRIPLE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTION AAR JYOTI DUTTA, AYHAN ESI, BINOD CHANDRA TRIPATHY Abstract. In this article we introduced the notion o statistically convergent triple sequences deined by Orlicz unction. We have established some algebraic topological properties o those sequence spaces.. Introduction A triple sequence real or complex) can be deined as a unction x : N N N RC) where N, R C denote the sets o natural numbers, real numbers complex numbers respectively. The dierent types o notions o triple sequences was introduced investigated at the initial stage by Sahiner, Gürdal Düden [9] Sahiner Tripathy [0] others. A triple sequence x nkl ) is said to be bounded i there exists K > 0 such that x nkl < K or all n, k, l. We denote the set o all bounded triple sequence is by 3 l. It is easy to prove that 3 l is a normed space, normed by x = sup x nkl. A triple sequence x nkl ) is said to be a Cauchy sequence i or every ε > 0 there exists n 0 ε) N such that x x nkl < ε or p n n 0 ; q k n 0 ; r l n 0. A triple sequence x nkl ) is said to be convergent to L in Pringsheim s sense i or every ε > 0 there exists n 0 ε) N such that x nkl L < ε, or n 0. A subset E o N N N is said to have density ρe) i the it given by ρe) = χ E n, k, l) exist. n p k q l r Thus a triple sequence x nkl ) is said to be statistically convergent to L in Pringsheim s sense i or every ε > 0, ρ n, k, l) N N N : x nkl L ε) We write stat x nkl = L. A triple sequence x = x nkl ) is said to statistically bounded i there exist H > 0 such that ρ n, k, l) N N N : x nkl > H) A triple sequence x=x nkl ) is said to statistically Cauchy i or every ε>0 there exist p=pε), q =qε) r =rε) such that ρn, k, l) N N N: x nkl x ε) 2000 athematics Subject Classiication. 40A05, 40A35, 40B05. Key words phrases. Triple sequence; Orlicz unction; statistical convergence; solid; symmetric. c 203 Ilirias Publications, Prishtinë, Kosovë. Submitted ay 2, 203. Published Jul 2,

2 STATISTICALLY CONVERGENT TRIPLE SEQUENCE SPACES... 7 Dierent classes o statistically convergent sequences were introduced investigated by Tripathy [, 2]), Tripathy Baruah [4], Tripathy Sarma [9], Tripathy Sen [23] others. 2. Deinition Results Let X, ) be a semi-normed space θ e be the zero unit elements o X, ) respectively. We introduce the ollowing deinitions or triple sequences. An X-valued triple sequence x nkl ) is said to converge regularly i it statistically convergent in Pringsheim s sense in addition the ollowing its holds: x nkl = L kl k, l N); n x nkl = L nl n, l N) k x nkl = L nk n, k N). l An X valued triple sequence x = x nkl ) is said to statistically regularly convergent i it convergent in Pringsheim s sense in addition the ollowing statistical its holds: stat n x nkl = L kl k, l N), stat k x nkl = L nl n, l N) stat l x nkl = L nk n, k N). A triple sequence space E is said to be solid or normal) i α k x nkl ) E, whenever x nkl ) E or all sequences α k ) o scalars with α k, or all k N. A triple sequence space E is said to be monotone i it contains the canonical pre-images o all its step spaces. Remark. A sequence space is solid implies that it is monotone. A triple sequence space E is said to be symmetric i x πnkl) ) E whenever x nkl ) E, where π is a permutation o N N N. A triple sequence space E is said to be convergence ree i y nkl ) E whenever x nkl ) E x nkl = θ implies y nkl = θ. An Orlicz unction is a unction : [0, ) [0, ) which is continuous, non-decreasing convex with 0) = 0, x) > 0 or x > 0 x) as x. Lindenstrauss Tzariri [4] applied the notion o Orlicz unction introduced the sequence space l, called as an Orlicz sequence space is deined by xk l = x = x k ) : <, or some ρ > 0. ρ k=

3 8 AAR JYOTI DUTTA, AYHAN ESI, BINOD CHANDRA TRIPATHY It is well known that the sequence space l becomes a Banach space, with the norm xk x = in ρ > 0 :. ρ k= The space l is closely related to the spacel p, which is an Orlicz sequence space with x) = x p, or p <. Then the Orlicz unction was applied or studying dierent classes o sequences by realized by Altin, Et Tripathy [], Esi [2], Et et.al [3], Nung Yee [7], Parashar Choudhary [8], Tripathy et.al [3], Tripathy Borgogain [5], Tripathy Dutta [7], Tripathy Hazarika [8], Tripathy Sarma [20, 2, 22]) many others in the recent years. I the convexity o is replaced by sub-additive property, x + y) x) + y) then it is called a modulus unction. The notion o modulus unction was introduced by Nakano [6]. Later on the notion was urther investigated by addox [5], Tripathy Chra [6] others. It is well known that i is an Orlicz unction, then 0) = 0 then λx) λx) or all λ with 0 < λ <. Let be an Orlicz unction X, ) denote a semi-normed space, seminormed by. We now introduce the ollowing triple sequence spaces: 3l, ) = x = x nkl ) : sup 3 c, )= x = x nkl ):stat 3 c 0, ) = ) L x = x nkl ) : stat ) <, or some > 0 ; =0, or some > 0 L = 0, or some > 0 A sequence x = x nkl ) 3 c) R, ) i x 3 c) R, ) the ollowing statistical its hold: There exist, 2, 3 > 0, such that stat n stat k stat l ) L kl = 0, or k, l N; ) L nl = 0, or n, l N 2 ) L nk = 0, or n, k N. 3 A sequence x = x nkl ) 3 c 0 ) R, ) i x 3 c 0 ) R, ) the above three conditions holds with L kl = L nl = L nk = θ, the zero element o X, or all n, k, l N. Throughout the article 3 w), 3 l ), 3 l ), 3 c), 3 c 0 ), 3 c) R ) denote the spaces o all, bounded, statistically bounded, statistically convergent, statistically null, statistically regularly convergent X-valued triple sequences. The purpose o this paper is to study statistical convergence o triple sequences deined by Orlicz unctions give some important theorems.

4 STATISTICALLY CONVERGENT TRIPLE SEQUENCE SPACES ain Results Theorem 3.. The classes o sequence 3 l, ), 3 c, ), 3 c 0, ) 3 c) R, ) are linear spaces. Proo. We proved or the sequence space 3 c 0, ). Let x, y 3 c 0, ) α, β C. Then there exists, 2 such that stat ) = 0 stat ) ynkl 2 Deine 3 = max2 α, 2 β 2 ) since is non-decreasing convex, we get ) ) α + βy nkl ) α βynkl stat stat [ ) stat 2 )] ynkl + 2 ) <stat ) ynkl + stat 2 Thus αx + βy 3 c 0, ). This completes the proo. Theorem 3.2. Let x = x nkl ) be a triple sequence let be a bounded Orlicz unction. Then x is statistically convergent to L i only i n= k= l= L) Proo. Let x = x nkl ) be a triple sequence. We suppose that n= k= l= L) I be a bounded Orlicz unction then it is easy to prove that x is statically convergent to L. Conversely suppose that x is statically convergent to L is a bounded Orlicz unction. Then ρ n, k, l) N N N : x nkl L) ε) = 0 x) < K 2

5 20 AAR JYOTI DUTTA, AYHAN ESI, BINOD CHANDRA TRIPATHY or a positive integer K. Choose δ > 0, such that δ) < ε 2. = n= k= l= + n= k= l= δ) + L) L) x nkl L <δ L) n= k= l= x nkl L δ L) n= k= l= x nkl L δ ε 2 + K 2 nkl n, k, l) N N N : x nkl L) δ <ε, since x is statistically convergent to L. Thus we have n= k= l= L Theorem 3.3. The classes o sequence 3 l, ), 3 c, ), 3 c 0, ) 3 c) R, ) are semi normed spaces, semi-normed by gx nkl ) = in > 0 : sup. Proo. Since is a semi-norm, thereore it is suicient to prove that gx nkl + y nkl ) gx nkl ) + gy nkl ). Let x nkl ), y nkl ) 3 l, ). There exist, 2 > 0, such that sup Let = + 2, we get ) + y nkl sup sup +. ynkl 2 ) sup sup + 2. ynkl 2

6 STATISTICALLY CONVERGENT TRIPLE SEQUENCE SPACES... 2 Now we have ) + y nkl gx nkl + y nkl ) = in + 2 ) = > 0 : sup ) in > 0 : sup ) ynkl + 2 > 0 : sup =gx nkl ) + gy nkl ). Thus the proo complete. 2 Theorem 3.4. The classes o sequence 3 l, ), 3 c 0, ) 3 l, ) are solid hence monotone. Proo. Consider the sequence o scalar α nkl ) with α nkl. Then we have, ) ) αnkl a nkl ankl, or all n, k, l N. The above inequality holds or all the classes o sequence hence completes the proo. Theorem 3.5. The classes o sequence 3 c 0, ), 3 c, ), 3 c) R, ) 3 l, ) are not symmetric. Proo. The proo ollows rom the ollowing example. Example 3.6. Consider the Orlicz unction x) = x p 0 < p ). Let x) = sup i x i take X to be the class o bounded convergent sequences. Deine x nkl ) by e, or n, k, l = i x nkl = 2, i N; θ, otherwise. Now consider the rearrangement o x nkl ) deined by e, or n, k even; y nkl = θ, otherwise. Then we observe that x nkl ) 3 c 0, ), 3 c, ), 3 c) R, ) 3 l, ) whereas y nkl ) / 3 c 0, ), 3 c, ), 3 c) R, ) 3 l, ). This completes the proo. Reerences [] Y. Altin;. Et B. C. Tripathy: The sequence space N p, r, q, s) on seminormed spaces; Applied athematics Computation; ), [2] A. Esi: On some sequence spaces deined by Orlicz unctions; Bull. Inst. ath. Acad. Sin; 27999), [3]. Et; Y. Altin; B. Choudhary B. C. Tripathy: On some classes o sequences deined by sequences o Orlicz unctions; athematical Inequalities Applications, 92)2006), [4] J. Lindenstrauss L. Tzariri: On Orlicz sequence spaces, Israel J. ath.;0 97), [5] I. J. addox: Sequence spaces deined by a modulus; ath. Proc. Camb. Phil. Soc.; 00)986), [6] H. Nakano: Concave modulars; J. ath. Soc. Japan; 5953),

7 22 AAR JYOTI DUTTA, AYHAN ESI, BINOD CHANDRA TRIPATHY [7] N. P. Nung L. P. Yee: Orlicz sequence spaces o a nonabsolute type; Comment. ath. Univ. St. Pauli; 262)977), [8] S. D. Parashar B. Choudhary: Sequence spaces deined by Orlicz unction, Indian J. pure Appl. ath. ; 25994), [9] A. Sahiner,. Gürdal K. Düden: Triple sequences their statistical convergence, Selcuk J. Appl. ath., 82) 2007), [0] A. Sahiner, B. C. Tripathy: Some I-related Properties o Triple Sequences, Selcuk J. Appl. ath., 92) 2008), 9-8. [] B. C. Tripathy: atrix transormations between some classes o sequences; Jour. ath. Anal. Appl., ); [2] B. C. Tripathy: On generalized dierence paranormed statistically convergent sequences; Indian Jour. Pure Appl. ath. ;355)2004), [3] B. C. Tripathy, Y. Altin. Et: Generalized dierence sequences spaces on seminormed spaces deined by Orlicz unctions, ath. Slovaca, 583)2008), [4] B. C. Tripathy A. Baruah: Lacunary statistically convergent lacunary strongly convergent generalized dierence sequences o uzzy real numbers, Kyungpook ath. Jour., 50200), [5] B. C. Tripathy S. Borgogain: The sequence space m, ϕ, n m, p)f ; ath. odel. Anal., 34) 2008), [6] B. C. Tripathy P. Chra: On some generalized dierence paranormed sequence spaces associated with multiplier sequences deined by modulus unction; Anal. Theory Appl.; 27)20), [7] B. C. Tripathy H. Dutta: On some new paranormed dierence sequence spaces deined by Orlicz unctions; Kyungpook ath. Jour; 50200), [8] B. C. Tripathy B. Hazarika: I-convergent sequences spaces deined by Orlicz unction; Acta ath. Appl. Sinica, 27)20) [9] B. C. Tripathy B. Sarma: Statistically convergent dierence double sequence spaces; Acta ath. Sinica; 245)2008), [20] B. C. Tripathy B. Sarma: Sequence spaces o uzzy real numbers deined by Orlicz unctions; ath. Slovaca, 585) 2008), [2] B. C. Tripathy B. Sarma: Vector valued double sequence spaces deined by Orlicz unction; ath. Slovaca; 596)2009), [22] B. C. Tripathy B. Sarma: Double sequence spaces o uzzy numbers deined by Orlicz unction; Acta ath. Scientia, 3B) 20), [23] B. C. Tripathy. Sen: On generalized statistically convergent sequences, Indian Jour. Pure Appl. ath.; 32) 200); Amar Jyoti Dutta athematical Sciences Division; Institute o Advanced Study in Science Technology; Paschim Boragaon; GARCHUK; GUWAHATI-78035, ASSA, INDIA. address: amar iasst@yahoo.co.in Ayhan Esi Department o athematics; Faculty o Science Art; Adiyaman University; 02040, ADIYAAN, TURKEY. address: aesi23@hotmail.com Binod Chra Tripathy athematical Sciences Division; Institute o Advanced Study in Science Technology; Paschim Boragaon; GARCHUK; GUWAHATI-78035, ASSA, INDIA. address: tripathybc@redimail.com, tripathybc@yahoo.com