Characterization of triple χ 3 sequence spaces via Orlicz functions

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1 athematica oravica Vol 20: 206), 05 4 Characterization of triple χ sequence spaces via Orlicz functions N Subramanian and A Esi Abstract In this paper we study of the characterization and general properties of triple gai sequence via Orlicz space of χ of χ establishing some inclusion relations Introduction Throughout w, χ and Λ denote the classes of all, gai and analytic scalar valued single sequences, respectively We write w for the set of all complex sequences x mnk ), where m, n, k N, the set of positive integers Then, w is a linear space under the coordinate wise addition and scalar multiplication We can represent triple sequences by matrix In case of double sequences we write in the form of a square In the case of a triple sequence it will be in the form of a box in three dimensional case Some initial work on double series is found in Apostol [] and double sequence spaces is found in Hardy [5], Subramanian et al [0-2], and many others Later on investigated by some initial work on triple sequence spaces is found in Sahiner et al [9], Esi et al [2-4], Subramanian et al [-9] and many others Let x mnk ) be a triple sequence of real or complex numbers Then the series m,n,k= x mnk is called a triple series The triple series m,n,k= x mnk is said to be convergent if and only if the triple sequence S mnk ) is convergent, where S mnk = m,n,k i,j,q= x ijq m, n, k =, 2,, ) A sequence x = x mnk ) is said to be triple analytic if sup m,n,k x mnk m+n+k < 2000 athematics Subject Classification Primary: 40A05; Secondary: 40C05,40D05 Key words and phrases Gai sequence, analytic sequence, triple sequence,dual space, Orlicz space 05 c 206 athematica oravica

2 06 Characterization of triple χ sequence spaces via Orlicz functions The vector space of all triple analytic sequences are usually denoted by Λ A sequence x = x mnk ) is called triple entire sequence if x mnk m+n+k 0 as m, n, k The vector space of all triple entire sequences are usually denoted by Γ The space Λ and Γ 2 is a metric space with the metric } ) dx, y) = sup x mnk y mnk m+n+k : m, n, k :, 2,,, m,n,k for all x = x mnk } and y = y mnk } in Γ Let φ = finite sequences} Consider a double sequence x = x mnk ) The m, n, k) th section x [m,n,k] of the sequence is defined by x [m,n,k] = m,n,k i,j,q=0 x ijqδ ijq for all m, n, k N, δ mnk = with in the m, n, k) th position and zero other wise A sequence x = x mnk ) is said to be triple gai sequence if m + n + k)! x mnk ) m+n+k 0 as m, n, k The triple gai sequences will be denoted by χ Consider a triple sequence x = x mnk ) The m, n, k) th section x [m,n,k] of the sequence is defined by x [m,n,k] = m,n,k i,j,q=0 x ijqi ijq for all m, n, k N ; where I ijq denotes the triple sequence whose only non zero term is a i+j+k)! in the i, j, k) th place for each i, j, k N An FK-spaceor a metric space)x is said to have AK property if I mnk ) is a Schauder basis for X, or equivalently x [m,n,k] x An FDK-space is a triple sequence space endowed with a complete metrizable; locally convex topology under which the coordinate mappings are continuous δ mnk = with in the m, n, k) th position and zero other wise A sequence x = x mnk ) is said to be triple gai sequence if m + n + k)! x mnk ) will be denoted by χ m+n+k 0 as m, n, k The triple gai sequences

3 N Subramanian and A Esi 07 Consider a triple sequence x = x mnk ) The m, n, k) th section x [m,n,k] of the sequence is defined by x [m,n,k] = m,n,k i,j,q=0 x ijqi ijq for all m, n, k N ; where I ijq denotes the triple sequence whose only non zero term is a i+j+k)! in the i, j, k) th place for each i, j, k N An FK-spaceor a metric space)x is said to have AK property if I mnk ) is a Schauder basis for X, or equivalently x [m,n,k] x An FDK-space is a triple sequence space endowed with a complete metrizable; locally convex topology under which the coordinate mappings are continuous If X is a sequence space, we give the following definitions: i) X is continuous dual of X; ii) X α = a = a mnk ) : m,n,k= a mnk x mnk <, for each x X } ; iii) X β = a = a mnk ) : m,n,k= a mnk x mnk is convergent, for each x X } ; iv) X γ = a = a mn ) : sup,n,k m,n m,n,k= a mnkx mnk <, for each x X } ; v) Let X be an F K-space φ; then X f = } fi mnk ) : f X ; vi) X δ = a = a mnk ) : sup m,n,k a mnk x mnk /m+n+k <, for each x X } ; X α, X β, X γ are called α - or Köthe-Toeplitz) dual of X, β-or generalized- Köthe-Toeplitz) dual of X, γ-dual of X, δ-dual of X respectively X α is defined by Gupta and Kamptan [0] It is clear that X α X β and X α X γ, but X α X γ does not hold 2 Definitions and Preliminaries m+n+k < A sequence x = x mnk )is said to be triple analytic if sup mnk x mnk The vector space of all triple analytic sequences is usually denoted by Λ A sequence x = x mnk ) is called triple entire sequence if x mnk m+n+k 0 as m, n, k The vector space of triple entire sequences is usually denoted by Γ A sequence x = x mnk ) is called triple gai sequence if m + n + k)! x mnk ) m+n+k 0 as m, n, k The vector space of triple gai sequences is usually denoted by χ The space χ is a metric space with the metric dx, y) = sup m + n + k)! xmnk y mnk ) m+n+k, 2) m,n,k m, n, k :, 2,, } for all x = x mnk } and y = y mnk } in χ Let w denote the set of all complex double sequences x = x mnk ) m,n,k= and : [0, ) [0, ) be an Orlicz function Given a triple sequence,

4 08 Characterization of triple χ sequence spaces via Orlicz functions x w Define the sets χ = x w m + n + k)! x mnk ) m+n+k : 0, as m, n, k for some > 0 } and Λ = x w : sup m,n,k x mnk m+n+k The space Λ is a metric space with the metric xmnk y mnk d x, y) = inf > 0 : sup m,n,k The space χ is a metric space with the metric d x, y) = inf > 0 : sup m,n,k m+n+k)! xmnk y mnk < for some > 0 } m+n+k } m+n+k This paper is a study of the characterization and general properties of gai sequences via triple Orlicz space of χ of χ establishing some inclusion relations ain Results Proposition If is a Orlicz function, then χ is a linear set over the set of complex numbers C Proof It is trivial Therefore, the proof is omitted Proposition 2 χ ) δ Λ xmnk y mnk Proof Let y δ dual of χ Then m+n+k for some constant > 0 and for each x χ Therefore, ymnk m+n+k for each m, n, k by taking x = I mnk ) This implies that y Λ Thus, ) ) χ δ Λ We now choose = id and define the triple sequences y mnk ) and x mnk ) by y mnk ) = for all m, n and k, and by m+2)!x m = 2 m+2)2 and m+n+k)!x mnk = 0n, k 2) for all m =, 2, Obviously, y Λ and since m + n + k)!x mnk = 0 for all m, n, k 0, m + n + k)!x mnk ) converges to zero Hence, x χ But m + 2)! a m x m ) m+n+k = 2 m+2 as m, }

5 N Subramanian and A Esi 09 hence 4) x / χ ) δ From ) and 4), we are granted χ ) δ Λ This completes the proof Proposition The dual space of χ is Λ In other words χ ) = Λ Proof We recall that I mnk = 0 0 m+n+k)! 0 with m+n+k)! in the m, n, k)th position and zero s else where, with x = I mnk, ) m + n + k)! x mnk ) m+n+k } = 0 / /) 0 /+n+k /) 0 /m+4 /) m+n+k)! ) m+n+k /) 0 /m+n++k+2 /) = m,n) th 0 /m+4 /) 0 /m+n+k+4 /) 0 0 ) m+n+k 0 m+n+k)! /), m,n,k) th which is a triple gai sequence Hence, I mnk χ, fx) = m,n,k= x mnky mnk with x χ and f χ ), where χ ) is the dual space of χ Take x = x mnk ) = I mnk χ Then, 5) y mnk f di mnk, 0) < m, n, k Thus, y mnk ) is a bounded sequence and hence an triple analytic sequence In other words, y Λ Therefore χ ) = Λ This completes the proof

6 0 Characterization of triple χ sequence spaces via Orlicz functions Proposition 4 Λ ) β χ Proof Step : Let x mnk ) Λ ) β, 6) m,n,k= x mnk y mnk < y mnk ) Λ Let us assume that x mnk ) / χ Then, there exists a strictly increasing sequence of positive integers m p + n p + k p ) such that 7) mp + n p + k p )! xmp+n p+k p) >, p =, 2,, ) 2 mp+np+kp) Let m p + n p + k p )!y mp+n p+k p) = 2 mp+np+kp) for p =, 2,, ), Then y mnk ) Λ However, = m,n,k= y mnk = 0 otherwise xmnk y mnk = mp + n p + k p )! x mpn pk p)y mpn pk p) > p= > + We know that the infinite series diverges Now we choose =id, where id is the identity and hence m,n,k= x mnky mnk / diverges This contradicts 6) Hence x mnk ) χ Therefore, 8) Λ ) β χ If we now choose = id, where id is the identity and y nk = x nk = and y mnk = x mnk = 0 m > ) for all n, k, then obviously x χ and y Λ, but m,n,k= x mnky mnk = Hence, 9) y / Λ ) β From 8) and 9), we are granted Λ ) β χ This completes the proof

7 N Subramanian and A Esi Definition 5 Let p = p mnk ) be a triple sequence of positive real numbers Then pmnk χ m+n+k)! x p) = x = x mnk ) : mnk ) m+n+k 0, 0) } m, n, k ) for some > 0 Suppose that p mnk is a constant for all m, n, k then χ p) = χ } Proposition 6 Let 0 p mnk q mnk for all m, n, k N and let qmnk p mnk be bounded Then χ q) χ p) Proof Let ) x χ q), then 2) m + n + k)! x mnk ) m+n+k qmnk 0, as m, n, k Let t mnk = m + n + k)! x mnk ) /m + n + k/ q mnk, and let γ mnk = p mnk /q mnk Since p mnk q mnk, we have 0 γ mnk Let 0 < γ < γ mnk, then t mnk, if t mnk ) u mnk = 0, if t mnk < ) ) 0, if t mnk ) v mnk = t mnk, if t mnk < ) Now, it follows that t mnk = u mnk + v mnk, t γ mnk mnk = uγ mnk mnk + vγ mnk mnk 4) u γ mnk mnk u mnk t mnk, v γ mnk mnk uγ mnk Since t γ mnk mnk = uγ mnk mnk + vγ mnk mnk, we have tγ mnk mnk t mnk + v γ mnk ) Thus, m + n + k)! x mnk ) /m+n+k qmnk ) γmnk / m + n + k)! x mnk ) /m+n+k qmnk /, 5) ) qmnk ) pmnk /q mnk which yields m + n + k)! x mnk ) /m+n+k / m + n + k)! x mnk ) / /m+n+k qmnk, m + n + k)! x mnk ) /m+n+k pmnk /

8 2 Characterization of triple χ sequence spaces via Orlicz functions m + n + k)! x mnk ) / /m+n+k qmnk However, m + n + k)! x mnk ) / /m+n+k qmnk 0 by2 Thus, m + n + k)! x mnk ) / /m+n+k pmnk 0 as m, n, k Hence, 6) x χ p) Hence ) and 6), we are granted 7) χ q) χ p) This completes the proof Proposition 7 a) Let 0 < inf p mnk p mnk, then χ p) χ b) If p mnk sup mnk <, then χ χ p) Proof The above statements are special cases of Proposition 6 Therefore, it can be proved by similar arguments Proposition 8 If 0 < p mnk q mnk < for each m, n, k then χ p) χ q) Proof Let x χ p), then 8) m + n + k)! x mnk ) /m+n+k p mnk 0, as m, n, k This implies that m + n + k)! x mnk ) /m+n+k / for sufficiently large m, n, k Since is non-decreasing, we get m + n + k)! x mnk ) /m+n+k qmnk / 9) m + n + k)! x mnk ) / /m+n+k pmnk, then m + n + k)! x mnk ) / /m+n+k qmnk 0 as m, n, k by using 8 Let x χ q) Hence, χ p) χ q) This completes the proof Competing Interests: The authors declare that there is no conflict of interests regarding the publication of this research paper Acknowledgement: The authors thank the referee s for his careful reading of the manuscript and comments that improved the presentation of the paper

9 N Subramanian and A Esi References [] T Apostol, athematical Analysis, Addison-Wesley, London, 978 [2] A Esi, On some triple almost lacunary sequence spaces defined by Orlicz functions, Research and Reviews:Discrete athematical Structures, 2), 204), 6-25 [] A Esi and Necdet Catalbas,Almost convergence of triple sequences, Global Journal of athematical Analysis, 2), 204), 6-0 [4] A Esi and E Savażs, On lacunary statistically convergent triple sequences in probabilistic normed space,applathand InfSci, 9 5), 205), [5] GH Hardy, On the convergence of certain multiple series, Proc Camb Phil Soc, 9 97), [6] PK Kamthan and Gupta, Sequence spaces and series, Lecture notes, Pure and Applied athematics, 65 arcel Dekker, In c, New York, 98 [7] IJ addox, Sequence spaces defined by a modulus, ath Proc Camb Philos Soc, 00986), 6-66 [8] WH Ruckle, FK spaces in which the sequence of coordinate vectors in bounded, Canad J ath, 2597), [9] A Sahiner, Gurdal and FK Duden, Triple sequences and their statistical convergence, Selcuk J Appl ath, Vol 8, No 22007), [0] N Subramanian and UK isra, Characterization of gai sequences via double Orlicz space, Southeast Asian Bulletin of athematics, revised) [] N Subramanian, BC Tripathy and C urugesan, The double sequence space of Γ 2, Fasciculi ath, ), 9-0 [2] N Subramanian, BC Tripathy and C urugesan, The Cesáro of double entire sequences, International athematical Forum, Vol 4, No 22009), [] N Subramanian and A Esi, The generalized triple difference of χ sequence spaces, Global Journal of athematical Analysis, 2) 205), [4] N Subramanian and A Esi, The study on χ sequence spaces, Songklanakarin Journal of Science and Technology, under review [5] N Subramanian and A Esi, Some New Semi-Normed Triple Sequence Spaces Defined by a Sequence Of oduli, Journal of Analysis & Number Theory, 2) 205), [6] TVG Shri Prakash, Chandramouleeswaran and N Subramanian, The Triple Almost Lacunary Γ sequence spaces defined by a modulus function, International Journal of Applied Engineering Research, Vol 0, No ), [7] TVG Shri Prakash, Chandramouleeswaran and N Subramanian, The triple entire sequence defined by usielak Orlicz functions over p metric space, Asian Journal of athematics and Computer Research,International Press, 5 4) 205), 96-20

10 4 Characterization of triple χ sequence spaces via Orlicz functions [8] TVG Shri Prakash, Chandramouleeswaran and N Subramanian, The Random of Lacunary statistical on Γ over metric spaces defined by usielak Orlicz functions, odern Applied Science, in press [9] TVG Shri Prakash, Chandramouleeswaran and N Subramanian, The Triple Γ of tensor products in Orlicz sequence spaces, athematical Sciences International Research Journal, Vol 4, No 2 205), N Subramanian Department of athematics SASTRA University Thanjavur-6 40 India address: nsmaths@yahoocom A Esi Adiyaman University Adiyaman Turkey address: aesi2@hotmailcom