The strongly generalized triple difference Γ3 sequence spaces defined

Mathematica Moravica Vol. 20-1 (2016), 115–123

The strongly generalized triple difference Γ3 sequence spaces defined by a modulus T.V.G. Shri Prakash, M. Chandramouleeswaran, and N. Subramanian Abstract. In this paper we introduce the strongly generalized difference sequence spaces using non-negative four dimensional matrix of complex numbers. We also give natural relationship between strongly generalized difference V 3 Γ3 λ3 [A, ∆m , p, f ] − summable sequences with respect to f . We examine some topological properties of V 3 Γ3 λ3 [A, ∆m , p, f ] − spaces and investigate some inclusion relations between these spaces.

1. Introduction Throughout w, Γ and Λ denote the classes of all, entire and analytic scalar valued single sequences, respectively. We write w3 for the set of all complex triple sequences (xmnk ), where m, n, k ∈ N, the set of positive integers.Then, w3 is a linear space under the coordinate wise addition and scalar multiplication. Let P∞Then the seP(xmnk ) be a triple sequence of real or complex numbers. x is called a triple series. The triple series ries ∞ mnk m,n,k=1 xmnk is m,n,k=1 said to be convergent if and only if the triple sequence (Smnk )is convergent, where m,n,k X Smnk = xijq , (m, n, k = 1, 2, 3, . . . ). i,j,q=1

A sequence x = (xmnk ) is said to be triple analytic if 1

sup |xmnk | m+n+k < ∞. m,n,k

The vector space of all triple analytic sequences are usually denoted by Λ3 . A sequence x = (xmnk ) is called triple entire sequence if 1

|xmnk | m+n+k → 0 as m, n, k → ∞. 2000 Mathematics Subject Classification. Primary: 40A05; Secondary: 40C05, 40D05. Key words and phrases. entire sequence, analytic sequence, triple sequence,difference sequence. 115

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The strongly generalized triple difference Γ3 sequence spaces. . .

The vector space of all triple entire sequences are usually denoted by Γ3 . The space Λ3 and Γ3 is a metric space with the metric o n 1 (1) d(x, y) = sup |xmnk − ymnk | m+n+k : m, n, k : 1, 2, 3, . . . , m,n,k

forall x = {xmnk } and y = {ymnk } in Γ3 . Let φ = {finite sequences}. 2. Definitions and Preliminaries Consider a triple sequence x = (xmnk ). The (m, n, k)th section x[m,n,k] P m,n,k of the sequence is defined by x[m,n,k] = i,j,q=0 xijq =ijq for all m, n, k ∈ N; where =ijq denotes the triple sequence whose only non zero term is a 1 in the (i, j, k)th place for each i, j, q ∈ N. If X is a sequence space, we give the following definitions: (i) X 0 is continuous dual of X; n o P α (ii) X = a = (amnk ) : ∞ |a x | < ∞, for each x ∈ X ; mnk mnk m,n,k=1 n P ∞ (iii) X β = a = (amnk ) : m,n,k=1 amnk xmnk is convergent, for each o x∈X ; P n (iv) X γ = a = (amn ) : supm,n≥1 M,N,K a x m,n,k=1 mnk mnk < ∞, for each o x∈X ; o n 0 (v) Let X be an FK-space ⊃ φ; then X f = f (=mnk ) : f ∈ X ; n (vi) X δ = a = (amnk ) : supm,n,k |amnk xmnk |1/m+n+k < ∞, for each o x∈X ; X α , X β , X γ are called α- (or Köthe-Toeplitz)dual of X, β-(or generalizedKöthe-Toeplitz)dual of X, γ-dual of X, δ-dual of X respectively. X α is defined by Gupta and Kamptan [10]. It is clear that X α ⊂ X β and X α ⊂ X γ , but X α ⊂ X γ does not hold. A sequence x = (xmnk ) is said to be triple analytic if Consider a triple sequence x = (xmnk ). The (m, n, k)th section x[m,n,k] P m,n,k of the sequence is defined by x[m,n,k] = i,j,q=0 xijq =ijq for all m, n, k ∈ N; where =ijq denotes the triple sequence whose only non zero term is a 1 in the (i, j, k)th place for each i, j, q ∈ N. If X is a sequence space, we give the following definitions: (i) X 0 is continuous dual of X; n o P α (ii) X = a = (amnk ) : ∞ |a x | < ∞, for each x ∈ X ; m,n,k=1 mnk mnk n P ∞ (iii) X β = a = (amnk ) : m,n,k=1 amnk xmnk is convergent, for each o x∈X ;

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P n a = (amn ) : supm,n≥1 M,N,K a x < ∞, for each mnk mnk m,n,k=1 o x∈X ; n o (v) Let X be an FK-space ⊃ φ; then X f = f (=mnk ) : f ∈ X 0 ; n (vi) X δ = a = (amnk ) : supm,n,k |amnk xmnk |1/m+n+k < ∞, for each o x∈X ;

(iv) X γ =

X α , X β , X γ are called α-(or Köthe-Toeplitz)dual of X, β-(or generalizedKöthe-Toeplitz)dual of X, γ-dual of X, δ-dual of X respectively. X α is defined by Gupta and Kamptan [6]. It is clear that X α ⊂ X β and X α ⊂ X γ , but X α ⊂ X γ does not hold. The vector space of all triple analytic sequences is usually denoted by 1 3 Λ and is defined by supmnk |xmnk | m+n+k < ∞. A sequence x = (xmnk ) is 1 called triple entire sequence if |xmnk | m+n+k → 0 as m, n, k → ∞. The vector space of triple entire sequences is usually denoted by Γ3 . Throughout the article w3 , Γ3 (∆), Λ3 (∆) denote the spaces of all, triple entire difference sequence spaces and triple analytic difference sequence spaces respectively. For a triple sequence x ∈ w3 , we define the sets n o Γ3 (∆) = x ∈ w3 : |∆xmnk |1/m+n+k → 0 as m, n, k → ∞ , ( ) Λ3 (∆) =

x ∈ w3 : sup |∆xmnk |1/m+n+k < ∞ . m,n,k

The space Λ3 (∆) is a metric space with the metric n o d (x, y) = sup |∆xmnk − ∆ymnk |1/m+n : m, n, k = 1, 2, . . . m,n,k

for all x = (xmnk ) and y = (ymnk ) in Λ3 (∆). The triple sequence λ3 = {(βr , µs , ηt )} is called triple λ3 sequence if there exist three non-decreasing sequences of positive numbers tending to infinity such that βr+1 ≤ βr + 1, β1 = 1, µs+1 ≤ µs + 1, µ1 = 1 and βr+1 ≤ ηt + 1, η1 = 1. The generalized double de Vallee-Poussin mean is defined by X 1 trst = trst (xmnk ) = xmnk , λrst (m,n,k)∈Irst

where λrst = βr · µs · ηt and Irst = {(mnk) : r − βr + 1 ≤ m ≤ r, s − µs + 1 ≤ n ≤ s, t − ηt + 1 ≤ k ≤ t} . A triple number sequence x = (xmnk ) is said to be (V3 , λ3 ) − summable to a number L if P − limrst prst = L. If λrst = rst, then then (V3 , λ3) − i(m,n) summability is reduced to (C, 1, 1, 1) − summability. Let A = ai(k,`) is

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an infinite four dimensional matrix of complex numbers and p = pi(mnk) be a triple analytic sequence of positive real numbers such that 0 < h = inf i pi(mnk) ≤ supi pi(mnk) = H < ∞ and f be a modulus. We define n V 3 Γ3 λ2 [A, ∆m , p, f ] = x = (xmnk ) ∈ w3 : ipi(mnk) o X h  1 −1 m m+n+k lim λrst f Ai (∆ xmnk ) =0 , r,s,t→∞

mnk∈Irst

n V 3 Λ3 λ3 [A, ∆m , p, f ] = x = (xmnk ) ∈ w3 : ipi(mnk) o X h  1