Nagarajan et al. Mathematical Sciences 2013, 7:39 http://www.iaumath.com/content/7/1/39
OR IGINA L R ESEA R CH
Open Access
The generalized χ 2 sequence spaces over p- metric spaces defined by Musielak Subramanian Nagarajan1* , Saivaraju Nallswamy2 and Velmurugan Subramanian2
Abstract In this we introduce generalized χ 2 sequence spaces over p- metric spaces defined by Musielak function paper, f = fmn and study some topological properties. Keywords: Analytic sequence; Double sequences; χ 2 space; Difference sequence space; Musielak-modulus function; p- metric space; Duals MSC: 40A05; 40C05; 40D05
Introduction Throughout this paper, w, χ, and denote the classes of all, gai, and analytic scalar valued single sequences, respectively. We write w2 for the set of all complex sequences (xmn ), where m, n ∈ N, the set of positive integers. Then, w2 is a linear space under the coordinatewise addition and scalar multiplication. Some initial works on double sequence spaces is found in Bromwich [1]. Later on, they were investigated by Hardy [2], Moricz [3], Moricz and Rhoades [4], Basarir and Solankan [5], Tripathy [6], Turkmenoglu [7], and many others. We procure the following sets of double sequences: Mu (t) := (xmn ) ∈ w2 : supm,n∈N |xmn |tmn < ∞ , Cp (t) := (xmn ) ∈ w2 : p − limm,n→∞ |xmn −|tmn =1 for some ∈ C} , C0p (t) := (xmn ) ∈ w2 : p − limm,n→∞ |xmn |tmn = 1 , ∞ ∞ |xmn |tmn < ∞ , Lu (t) := (xmn ) ∈ w2 : m=1
Cbp (t) :=Cp (t)
n=1
Mu (t) and C0bp (t) = C0p (t)
Mu (t),
where t = (tmn ) is the sequence of strictly positive reals tmn for all m, n ∈ N and p − limm,n→∞ denotes the limit in the Pringsheim’s sense. In the case where tmn = *Correspondence:
[email protected] 1 Department of Mathematics, SASTRA University,Thanjavur, 613 401, India Full list of author information is available at the end of the article
1 for all m, n ∈ N, Mu (t), Cp (t), C0p (t), Lu (t), Cbp (t), and C0bp (t) reduce to the sets Mu , Cp , C0p , Lu , Cbp , and C0bp , respectively. Now, we may summarize the knowledge given in some document related to the double sequence spaces. G¨okhan and Colak [8,9] have proved that Mu (t) and Cp (t), Cbp (t) are complete paranormed spaces of double sequences and gave the α−, β−, γ − duals of the spaces Mu (t) and Cbp (t). Quite recently, in her PhD thesis, Zeltser [10] has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely [11], and Tripathy [6] have independently introduced the statistical convergence and Cauchy for double sequences and given the relation between statistical convergent and strongly Ces`aro summable double sequences. Altay and Ba¸Sar [12] have defined the spaces BS , BS (t), CS p , CS bp , CS r , and BV of double sequences consisting of all double series whose sequence of partial sums is in the spaces Mu , Mu (t), Cp , Cbp , Cr , and Lu , respectively, and also examined some properties of those sequence spaces and determined the α− duals of the spaces BS , BV , CS bp , and the β(ϑ)− duals of the spaces CS bp and CS r of double series. Basar and Sever [13] have introduced the Banach space Lq of double sequences corresponding to the well-known space q of single sequences and examined some properties of the space Lq . Quite recently, Subramanian and Misra [14] have studied the 2 space χM (p, q, u) of double sequences and gave some inclusion relations. The class of sequences which is strongly Ces`aro summable with respect to a modulus was introduced
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Nagarajan et al. Mathematical Sciences 2013, 7:39 http://www.iaumath.com/content/7/1/39
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by Maddox [15] as an extension of the definition of strongly Cesa` ro summable sequences. Connor [16] further extended this definition to a definition of strong A- summability with respect to a modulus, where A = an,k is a non-negative regular matrix, and established some connections between strong A- summability, strong Asummability with respect to a modulus, and A- statistical convergence. In [17], the notion of convergence of double sequences was presented by Pringsheim. Also, in [18,19], and [20],
the four-dimensional matrix transformation
∞ mn (Ax)k, = ∞ m=1 n=1 ak xmn was studied extensively by Hamilton. We need the following inequality in the sequel of the paper. For a, b, ≥ 0 and 0 < p < 1, we have (a + b)p ≤ ap + b p .
(1)
The double series ∞ m,n=1 xmn is called convergent if and only if the double sequence (smn ) is convergent, where
m,n x (m, n ∈ N). A sequence x = (xmn ) smn = ij i,j=1 is said to be double analytic if supmn |xmn |1/m+n < ∞. The vector space of all double analytic sequences will be denoted by 2 . A sequence x = (xmn ) is called double gai sequence if((m + n)! |xmn |)1/m+n → 0 as m, n → ∞. The double gai sequences will be denoted by χ 2 . Let φ = {all finite sequences}. Consider a double sequence x = (xij ). The (m, n)th section x[m,n] of the sequence is defined by x[m,n] =
m,n i,j=0 xij ij for all m, n ∈ N, where ij denotes the dou1 ble sequence whose only non-zero term is a i+j in the ( )! i, j th place for each i, j ∈ N. A Fréchet coordinate space (FK-space or a metric space) X is said to have an AK property if (mn ) is a Schauder basis for X, or equivalently x[m,n] → x. An FDK-space is a double sequence space endowed with a complete metrizable space, locally convex topology under which the coordinate mappings x = (xk ) → (xmn )(m, n ∈ N) are also continuous. Let M and be mutually complementary modulus functions. Then, we have (1) For all u, y ≥ 0, uy ≤ M(u) + (y), (Young’s inequality; see [21]). (2) (2) For all u ≥ 0, uη(u) = M(u) + (η(u)) .
(3)
(3) For all u ≥ 0 and 0 < λ < 1, M (λu) ≤ λM(u).
(4)
Lindenstrauss and Tzafriri [22] used the idea of Orlicz function to construct Orlicz sequence space
∞ |xk | < ∞, for some ρ > 0 . M M = x ∈ w : k=1 ρ The space M with the norm
∞ |xk | x = inf ρ > 0 : M ≤1 k=1 ρ becomes a Banach space which is called an Orlicz sequence space. For M(t) = t p (1 ≤ p < ∞), the spaces M coincide with theclassical sequence space p . A sequence f = fmn of modulus function is called a Musielak-modulus function. A sequence g = gmn defined by gmn (v) = sup |v| u − fmn (u) : u ≥ 0 , m, n = 1, 2, · · · is called the complementary function of a Musielakmodulus function f. For a given Musielak modulus function f, the Musielak-modulus sequence space tf and its subspace hf are defined, respectively, as follows: tf = x ∈ w2 : If (|xmn |)1/m+n → 0 as m, n → ∞ and
hf = x ∈ w2 : If (|xmn |)1/m+n → 0 as m, n → ∞ ,
where If is a convex modular defined by ∞ ∞ fmn (|xmn |)1/m+n , x = (xmn ) ∈ tf . If (x) = m=1
n=1
We consider that tf is equipped with the Luxemburg metric ∞ ∞ fmn d (x, y) = supmn inf m=1
n=1
|xmn |1/m+n × mn
≤1 .
If X is a sequence space, we give the following definitions:
dual of X ; (1) X = the continuous
(2) X α = a = (amn ) : ∞ m,n=1 |amn xmn | < ∞, for each x ∈ X};
(3) X β = a = (amn ) : ∞ m,n=1 amn xmn is convergent, for each x ∈ X};
M,N amn xmn < ∞, (4) X γ = a = (amn ) : supmn ≥ 1 m,n=1 for each x ∈ X}; (5) let X be φ, then an FK-space ⊃ X f = f (mn ) : f ∈ X ; (6) X δ = a = (amn ) : supmn |amn xmn |1/m+n < ∞, for each x ∈ X}, where X α , X β , and X γ are called α− (or K¨othe-Toeplitz) dual of X, β− (or generalized K¨othe-Toeplitz) dual of X, γ − dual of X, and δ− dual of X, respectively. X α is defined
Nagarajan et al. Mathematical Sciences 2013, 7:39 http://www.iaumath.com/content/7/1/39
by Kantham and Gupta [21]. It is clear that X α ⊂ X β and X α ⊂ X γ , but X β ⊂ X γ does not hold since the sequence of partial sums of a double convergent series needs not to be bounded. The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz [23] as follows: Z () = {x = (xk ) ∈ w : (xk ) ∈ Z} for Z = c, c0 and ∞ , where xk = xk − xk+1 for all k ∈ N. Here, c, c0 , and ∞ denote the classes of convergent, null, and bounded scalar valued single sequences, respectively. The difference sequence space bvp of the classical space p is introduced and studied in the case 1 ≤ p ≤ ∞ and in the case 0 < p < 1 by Altay and Ba¸Sar in [12]. The spaces c (), c0 (), ∞ (), and bvp are Banach spaces normed by x = |x1 | + supk≥1 |xk | and xbvp ∞ 1/p |xk |p = , (1 ≤ p < ∞) . k=1
Later on, the notion was further investigated by many others. We now introduce the following difference double sequence spaces defined by Z () = x = (xmn ) ∈ w2 : (xmn ) ∈ Z , where Z = 2 , χ 2 and xmn = (xmn − xmn+1 ) − (xm+1n − xm+1n+1 ) = xmn − xmn+1 − xm+1n + xm+1n+1 for all m, n ∈ N.
Definition and preliminaries Let n ∈ N and X be a real vector space of dimension w, where n ≤ w. A real valued function dp (x1 , . . . , xn ) = (d1 (x1 ), . . . , dn (xn ))p on X satisfying the following four conditions: (1) (d1 (x1 ), . . . , dn (xn ))p = 0 if and and only if d1 (x1 ), . . . , dn (xn ) are linearly dependent, (2) (d1 (x1 ), . . . , dn (xn ))p is invariant under permutation, (3) (αd1 (x1 ), . . . , dn (xn ))p = |α| (d1 (x1 ), . . . , dn (xn ))p , α ∈ R (4) dp ((x1 , y1 ), (x2 , y2 ) · · · (xn , yn )) = (dX (x1 , x2 , · · · xn )p +dY (y1 , y2 , · · · yn )p )1/p for1 ≤ p < ∞; (or) (5) d ((x1 , y1 ), (x2 , y2 ), · · · (xn , yn )) := sup dX (x1 , x2 , · · · xn ), dY (y1 , y2 , · · · yn ) , for x1 , x2 , · · · xn ∈ X, y1 , y2 , · · · yn ∈ Y which is called the p product metric of the Cartesian product of n metric spaces is the p norm of the n -vector of the norms of the n subspaces. A trivial example of the p product metric of the n metric space is the p norm space which is X = R equipped with the following Euclidean metric in the product space:
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(d1 (x1 ), . . . , dn (xn ))E = sup (|det(dmn (xmn ))|) = ⎞ ⎛ d11 (x11 ) d12 (x12 ) · · · d1n (x1n ) ⎜ d21 (x21 ) d22 (x22 ) · · · d2n (x1n ) ⎟ ⎟ ⎜ sup ⎜ ⎟ , .. ⎠ ⎝ . dn1 (xn1 ) dn2 (xn2 ) · · · dnn (xnn ) where xi = (xi1 , · · · xin ) ∈ Rn for each i = 1, 2, · · · n. If every Cauchy sequence in X converges to some L ∈ X, then X is said to be complete with respect to the p- metric. Any complete p- metric space is said to be p- Banach metric space. Let X be a linear metric space. A function w : X → R is called paranorm if (1) (2) (3) (4)
w(x) ≥ 0 for all x ∈ X; w (−x) = w(x) for all x ∈ X, w (x + y) ≤ w(x) + w(y) for all x, y ∈ X; If (σmn ) is a sequence of scalars with σmn → σ as m, n → ∞, and (xmn ) is a sequence of vectors with w (xmn − x) → 0 as m, n → ∞, then w (σmn xmn − σ x) → 0 as m, n → ∞.
A paranorm w for which w(x) = 0 implies x = 0 is called a total paranorm, and the pair (X, w) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [24], Theorem 10.4.2, p.183). The notion of λ− double gai and double analytic sequences is as follows: Let λ = (λmn )∞ m,n=0 be a strictly increasing sequence of positive real numbers tending to infinity, that is, 0 < λ0 < λ1 < · · · and λmn → ∞ as m, n → ∞ and that a sequence x = (xmn ) ∈ w2 is λ− convergent to 0, called a the λ− limit of x, if μmn (x) → 0 as m, n → ∞, where μmn (x) =
1 m∈σ ,σ ∈P n∈σ ,σ ∈Prs rs ϕrs × λm,n − λm,n+1 − λm+1,n + λm+1,n+1 |xmn |1/m+n .
The sequence x = (xmn ) ∈ w2 is λ− double analytic if supuv |μmn (x)| < ∞. If limmn xmn = 0 in the ordinary sense of convergence, then ⎛ limmn ⎝
1 ϕrs
λm,n − λm,n+1 − λm+1,n
m∈σ ,σ ∈Prs n∈σ ,σ ∈Prs
+λm+1,n+1 ((m + n) ! |xmn − 0|)1/m+n
= 0.
This implies that it yields limuv μmn (x) = 0, and 2 hence, x = (xmn ) ∈ w is λ− convergent to be a Musielak-modulus function, 0. Let f = f mn
X, (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))p be a p-metric space,
Nagarajan et al. Mathematical Sciences 2013, 7:39 http://www.iaumath.com/content/7/1/39
and q = (qmn ) be double analytic sequence of strictly positive real numbers.By w2 (p − X), we denote the space of all sequences as X, (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))p . The following inequality will be used throughout the paper. If 0 ≤ qmn ≤ supqmn = H, K = max 1, 2H−1 , then |amn + bmn |qmn ≤ K |amn |qmn + |bmn |qmn (5) q for all m, n and amn , bmn ∈ C. Also, |a| mn ≤ max 1, |a|H for all a ∈ C. In the present paper, we define the following sequence spaces:
2q χf μ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp = limmn fmn μmn (x), (d (x1 ) , d (x2 ) , · · · , qmn d (xn−1 ))p =0 ,
2q f μ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp = supmn fmn μmn (x), (d (x1 ) , d (x2 ) , · · · , qmn <∞ . d (xn−1 ))p
If we take fmn (x) = x, we get
2q χμ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp μmn (x), (d (x1 ) , d (x2 ) , · · · , = limmn
qmn =0 , d (xn−1 ))p
2q μ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp μmn (x), (d (x1 ) , d (x2 ) , · · · , = supmn
qmn d (xn−1 ))p <∞ .
If we take q = (qmn ) = 1
χf2μ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp = limmn fmn μmn (x), (d (x1 ) , d (x2 ) , · · · ,
d (xn−1 ))p =0 , 2f μ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp = supmn fmn μmn (x), (d (x1 ) , d (x2 ) , · · · ,
<∞ . d (xn−1 ))p
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In the present paper, we plan to study some topological properties and inclusion relation between the above 2q defined sequence spaces, χf μ , (d (x1 ) , d (x2 ) , · · · , 2q d (xn−1 ))ϕp and f μ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp , which we shall discuss in this paper.
Main results
Theorem 1. Let f = fmn be a Musielak-modulus function and q = (qmn ) be a double analytic sequence of strictly positive real numbers; the sequence spaces 2q 2q χf μ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp and f μ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp are linear spaces. Proof. It is routine verification. Therefore, the proof is omitted. Theorem 2. Let f = fmn be a Musielak-modulus function and q = (qmn ) be a double analytic sequence of strictly positive real numbers; the sequence space 2q ϕ χf μ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))p is a paranormed space with respect to the paranorm defined by g(x) = inf fmn μmn (x), (d (x1 ) , d (x2 ) , · · · , qmn 1/H d (xn−1 ))ϕp ≤1 , where H = max (1, supmn qmn < ∞). Proof. Clearly, g(x) ≥ 0 for x = (xmn ) ∈ 2q 2 χf μ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))V p . Since fmn (0) = 0, we get g(0) = 0. Conversely, suppose that g(x) = 0, then inf fmn (μmn (x), (d (x1 ) , d (x2 ) , · · · , qmn 1/H d (xn−1 ))ϕp ≤ 1 = 0. Suppose that μmn (x) = 0 for each m, n ∈ N. Then, μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp → ∞. It follows that fmn μmn (x) , (d ( x1 ) , d ( x2 ) , · · · , qmn 1/H 2 d (xn−1 ))V → ∞ which is a contradiction. p Therefore, μmn (x) = 0. Let
qmn 1/H ≤1 fmn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
and
fmn
μmn (y), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))
ϕ p
qmn 1/H
≤1.
Nagarajan et al. Mathematical Sciences 2013, 7:39 http://www.iaumath.com/content/7/1/39
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Proof. First, we observe that
Then, by using Minkowski’s inequality, we have
ϕ
qmn 1/H
fmn μmn (x+y) , (d (x1 ) , d (x2 ) , · · · , d (xn−1 )) p qmn 1/H ≤ fmn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp ϕ qmn 1/H + fmn μmn (y), (d (x1 ) , d (x2 ), · · ·, d (xn−1 )) p .
So, we have g (x + y) = inf
≤ inf
2q
χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
2q ⊂ f μ , μmn (x) , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
Therefore,
μmn (x + y) , (d (x1 ) , d (x2 ) , · · ·, qmn 1/H ϕ d (xn−1 ))p ≤1 fmn
2q
f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
fmn μmn (x), (d (x1 ) , d (x2 ) , · · · , qmn 1/H d (xn−1 ))ϕp ≤1
fmn μmn (y), (d (x1 ) , d (x2 ) , · · · , qmn 1/H d (xn−1 ))ϕp ≤1 .
But
2q = f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
2q β ⊂
f μ
Hence,
2q
f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
g (x + y) ≤ g(x) + g(y). Finally, to prove that the scalar multiplication is continuous, let λ be any complex number. By definition, g (λx) = inf fmn μmn (λx) , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))
ϕ qmn 1/H p
≤1 .
Then,
g (λx) = inf ((|λ| t)qmn /H : fmn μmn (x) , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp 1 |λ| .
qmn 1/H
≤1 ,
Since |λ|qmn ≤ max 1, |λ|suppmn , we have
g (λx) ≤ max 1, |λ|suppmn × inf t qmn /H : fmn μmn (λx) , (d (x1 ) , d (x2 ) , · · ·, qmn 1/H ≤1 . d (xn−1 ))ϕp
Theorem 3. The β− dual space of (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
d (x2 ) , · · · , d (xn−1 ))ϕp .
β (6) 2q ⊂ χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
Therefore,
where t =
β
β 2q ⊂ χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
+ inf
β
β
Next, we show that
2q
χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
2q ⊂ f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
2q Let y = (ymn ) ∈ χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , β
∞
d (xn−1 ))ϕp . Consider f (x) = ∞ m=1 n=1 xmn ymn with 2q x = (xmn ) ∈ χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp x = [(λmn − λmn+1 ) − (λm+1n − λm+1n+1 )] ⎛
0 ⎜0 ⎜ ⎜. = ⎜ .. ⎜ ⎝0 0 ⎛
2q χf μ , μmn (x),
2q = f μ , μmn (x), (d (x1 ) ,
β
0 0
...0 ...0
rs 0 . . . λmnϕ(m+n)! 0 ...0
0 ⎜0 ⎜ ⎜. − ⎜ .. ⎜ ⎝0 0
0 0
0 0 −ϕrs λmn (m+n)!
...0 ...0
rs 0 . . . λmnϕ(m+n)! 0 ...0
0
⎞ ... 0 . . . 0⎟ ⎟ ⎟ ⎟ ⎟ . . . 0⎠ ... 0
0 0 −ϕrs λmn (m+n)!
0
⎞ ... 0 . . . 0⎟ ⎟ ⎟ ⎟ ⎟ . . . 0⎠ ... 0
Nagarajan et al. Mathematical Sciences 2013, 7:39 http://www.iaumath.com/content/7/1/39
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fmn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp ⎛ 0 0 ...0 0 ⎜0 0 ...0 0, ⎜ ⎜. ⎜ .. ⎜ ⎜ ϕrs −ϕrs ⎜ = ⎜0 0 . . . fmn λmn (m+n)! fmn λmn (m+n)! ⎜ ⎜ ⎜ ϕrs rs ⎜0 0 . . . fmn λ −ϕ(m+n)! f mn λmn (m+n)! mn ⎜ ⎝ 0 0
...0
0,
Proof. We recall that ⎞ ... 0 . . . 0⎟ ⎟ ⎟ ⎟ ⎟ ⎟ . . . 0⎟ . ⎟ ⎟ ⎟ ⎟ . . . 0⎟ ⎟ ⎠ ... 0
Hence, it converges to zero. Therefore, [(λmn − λmn+1 ) − (λm+1n − λm+1n+1 )] 2q ∈ χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
Hence, d ((λmn − λmn+1 ) − (λm+1n − λm+1n+1 ) , 0) = 1. But ymn ≤ f d ((λmn − λmn+1 ) − (λm+1n − λm+1n+1 ) , 0) ≤ f ·1<∞ for each m, n. Thus, (ymn ) is a p- metric paranormed space of double analytic sequence and, hence, an p- metric double analytic sequence. 2q In other words. y ∈ f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , 2q d (xn−1 ))ϕp . But y = (ymn ) is arbitrary in χf μ , β μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .Therefore,
2q
χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp ⊂
2q f μ , μmn (x), (d (x1 ) , d (x2 ) , · · ·
β
, d (xn−1 ))ϕp
(7)
.
From (6) and (7), we get
2q
χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
2q Theorem 4. The dual space of χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp is 2q f μ , μmn (x), (d (x1 ) , d (x2 ) , 2q χf μ , μmn (x), · · · , d (xn−1 ))ϕp . In other words. (d (x1 ) , d (x2 ) , · · ·
d (x2 ) , · · · , d (xn−1 ))ϕp .
∗
=
2q f μ , μmn (x), (d (x1 ) ,
0 0
...0 ...0
⎞ 0 ... 0 . . .⎟ ⎟ ⎟ ⎟ ⎟ 0 . . .⎠
rs 0 . . . λmnϕ(m+n)! 0 ...0 0 ...
rs in the (m, n)th position and zeros elsewith λmnϕ(m+n)! where, 2q χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp ⎛ ⎞ 0. . . 0 ⎜. ⎟ ⎜ ⎟ ⎜. ⎟ ⎜ ⎟ ⎜. ⎟ =⎜ ⎟ 1/m+n ⎜ ⎟ rs ⎜ 0 f λ ϕ(m+n)! ⎟ . 0 mn ⎜ ⎟ th ⎝ ⎠ (m, n) 0 . . 0
which is a p- metric of double gai sequence. Hence, 2q x ∈ χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .f (x) =
∞ m,n=1
xmn ymn
2q with x ∈ χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp ∗ 2q and f ∈ χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp , ∗ 2q where χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp is 2q the dual space of χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . 2q Take x = (xmn ) ∈ χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . Then, ymn ≤ f d(ϕrs , 0) < ∞∀m, n.
β
2q = f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
, d (xn−1 ))ϕp
λmn
⎛ 0 ⎜0 ⎜ ⎜. = ⎜ .. ⎜ ⎝0 0
(8)
Thus, (ymn ) is a p- metric of the double analytic sequence and an p- metric of double analytic sequence. 2q In other words, y ∈ f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . Therefore,
2q
χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
∗
2q = f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . This completes the proof.
Nagarajan et al. Mathematical Sciences 2013, 7:39 http://www.iaumath.com/content/7/1/39
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Theorem 5. (1) If the sequence fmn satisfies uniform 2 − condition, then α 2q χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
2qμ
= χg
, μuv (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
2q = χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . Proof. Let the sequence condition; we get 2qμ
χg
2qμ = χg , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
(2) If the sequence gmn satisfies uniform 2 − condition, then α 2qμ χg , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
We are granted with (9) and (11) that α 2q χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
fmn satisfies uniform 2 −
, μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
(3) Similarly, one can prove that
2q
χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
Proposition 1. If 0 < qmn < pmn < ∞ for each m and m, then
α
2q
let a ∈ χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . 2q Then, for all {xmn } with (xmn ) ∈ χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp , we have ∞ ∞
|xmn amn | < ∞.
(10)
m=1 n=1
Since the sequence fmn satisfies the uniform 2 − condition and then 2q (ymn ) ∈ χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp , we get (ϕrs amn )
m=1
∞ ϕrs ymn amn n=1 λmn (m+n)! < ∞. by (10). Thus,
∈
χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
∞
2q
2qμ χg , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp , and 2qμ χg , μmn (x), (d (x1 ) , d (x2 ) , · · · , hence, (amn ) ∈ d (xn−1 ))ϕp . This gives that =
2q
χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
2q
f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
2p ⊆ f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
2q Proof. Let x = (xmn ) ∈ f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . We have
α
2qμ ⊂ χg , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp ,
α
if the sequence gmn satisfies the uniform 2 − condition.
α (9) 2q ⊂ χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
, μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
2q ⊂ χf μ , μmn (x) , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
To prove the inclusion
2qμ
χg
α
2qμ ⊂ χg , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
(11)
2q supmn f μ ,μmn (x),(d (x1), d(x2 ) , · · · , d (xn−1 ))ϕp < ∞. This implies that 2q f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp < 1 for sufficiently large value of m and n. Since fmn s are nondecreasing, we get 2p supmn f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp 2q ≤ supmn f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . Thus, x = (xmn ) ∈ d (xn−1 ))ϕp .
2p f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · ,
Proposition 2. (1) If 0 < infqmn ≤ qmn < 1, then
2q
f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
⊂ 2f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
(2) If 1 ≤ qmn ≤ supqmn < ∞, then 2f μ , μmn (x) , 2q (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp ⊂ f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
Nagarajan et al. Mathematical Sciences 2013, 7:39 http://www.iaumath.com/content/7/1/39
Page 8 of 13
Proof. Let x = (xmn ) ∈ 2f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . Since 0 < inf qmn ≤ 1, we have supuv 2f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp 2q ≤ f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp ,
and hence x = (xmn ) ∈ 2f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
(3) Let qmn for each (m, n) and supmn qmn < ∞. Let x = (xmn ) ∈ 2f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . Then, for each 0 < < 1, there exists a positive integer N such that
2q Proof. Let x = (xmn ) ∈ f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp , i.e., 2q supmn f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp < ∞.
Let (αmn ) be double sequence of scalars such that |αmn | ≤ 1 for all m, n ∈ N × N. Then, we get 2q supmn f μ , μmn (αx) , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
2q ≤ supmn f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
2q Proposition 6. The sequence space f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp is monotone.
supuv 2f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp ≤ < 1,
for all m, n ≥ N. This implies that 2q supmn f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp ≤ supmn 2f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
2q Thus, x = (xmn ) ∈ f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . and f = fmn be Proposition 3. Let f = fmn sequences of Musielak functions; we have 2q f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp 2q f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp × 2q × ⊆ f +f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
Proof. The proof is easy, so we omit it. Proposition 4. For any sequence of Musielak functions f = fmn and q = (qmn ) be double analytic sequence of strictly positive real numbers. Then,
2q
χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
2q ⊂ f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
Proof. The proof is easy, so we omit it. 2q Proposition 5. The sequence space f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp is solid.
Proof. The proof follows from Proposition 5. Proposition 7. If f = then
fmn is any Musielak function,
2q
f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
∗
∗∗ 2q ⊂ f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp ϕ∗
rs if and only if supr,s≥1 ϕ ∗∗ < ∞. rs
2q Proof. Let x ∈ f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , ∗ ∗ ϕrs d (xn−1 ))ϕp and N = supr,s≥1 ϕ ∗∗ < ∞. Then, we get rs ϕ ∗∗ 2q f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))prs ϕ∗ 2q = N f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))prs = 0.
ϕ Thus, x ∈ 2q f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))p
∗∗
.
Conversely, suppose that ∗ 2q f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp ∗∗ 2q ⊂ f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp ∗ 2q and x ∈ f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . ∗ 2q Then, f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp < ϕ∗
rs = ∞, then for every > 0. Suppose that supr,s≥1 ϕ ∗∗ rs there exists a sequence of members rsjk such that ∗ ϕjk 2q limj,k→∞ ϕ ∗∗ = ∞. Hence, we have f μ , μmn (x), jk ϕ∗ = ∞. Therefore, x ∈ / (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))prs ∗∗ 2q f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp , which is a contradiction.
Nagarajan et al. Mathematical Sciences 2013, 7:39 http://www.iaumath.com/content/7/1/39
Proposition 8. If f = then
Page 9 of 13
fmn is any Musielak function,
2q f μ , μmn (x), (d (x1 ) , d (x2 ) , · · ·
∗ , d (xn−1 ))ϕp
∗∗ 2q = f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
if and only if
∗ ϕrs supr,s≥1 ϕ ∗∗ rs
<
ϕ ∗∗ ∞, supr,s≥1 ϕrs∗ rs
> ∞.
In this section, we introduce the following sequence spaces: 2qA χf μ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp = limmn fmn Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))p
fmn Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))p
2q
Proposition 9. The sequence space χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp is not solid. Proof. The result follows from the following example. Consider ⎛
⎞ 1 1 ... 1 ⎜1 1 . . . 1⎟ ⎜ ⎟ x = (xmn ) = ⎜ . ⎟ ⎝ .. ⎠ 1 1 ... 1
= limmn
Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , qmn
=0 ,
2qA μ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp = supmn
Let
Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , qmn
<∞ .
If we take q = (qmn ) = 1,
⎞ ⎛ m+n −1m+n . . . −1m+n −1 ⎜−1m+n −1m+n . . . −1m+n ⎟ ⎟ ⎜ =⎜ . ⎟, ⎠ ⎝ .. m+n m+n m+n −1 . . . −1 −1
ϕ (d χf2A , d , · · · , d , (x ) (x ) (x )) 1 2 n−1 p μ = limmn
fmn Amn μmn (x), (d (x1 ) , d (x2 ) , · · · ,
2q / χf μ , μmn (x), (d (x1 ) , for all m, n ∈ N. Then, αmn xmn ∈ 2q d(x2 ) , · · · , d (xn−1 ))ϕp . Hence, χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp is not solid.
<∞ .
2qA χμ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
d (xn−1 ))p
αmn
qmn
If we take fmn (x) = x, we get
d (xn−1 ))p
2q ∈ χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
=0 ,
2qA f μ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp = supmn
Proof. It is easy to prove, so we omit it.
qmn
2q
Proposition 10. The sequence space χf μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp is not monotone. Proof. The proof follows from Proposition 9.
Generalized four-dimensional infinite matrix sequence spaces
be a four-dimensional infinite matrix Let A = amn k of complex numbers. Then, we have A(x) = (Ax)k =
∞ ∞ mn a x which converges for each k, . mn m=1 n=1 k
d (xn−1 ))p
=0 ,
ϕ (d , d , · · · , d 2A , (x ) (x ) (x )) 1 2 n−1 p fμ = supmn
fmn Amn μmn (x) , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))p
<∞ .
Theorem 6. For a Musielak-modulus function, f = 2qA fmn . Then, the sequence spaces χf μ , (d (x1 ) , 2qA d (x2 ) , · · · , d (xn−1 ))ϕp and f μ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp are linear spaces over the set of complex numbers C. Proof. It is routine verification. Therefore, the proof is omitted.
Nagarajan et al. Mathematical Sciences 2013, 7:39 http://www.iaumath.com/content/7/1/39
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Theorem 7. any Musielak-modulus func For tion f = fmn and a double analytic sequence the space q = (qmn ) of strictly positive real numbers, 2qA χf μ , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp is a topological linear space paranormed by
So, we have fmn Amn μmn (x+y),(d (x1 ), d (x2 ) , · · · , g(x + y) = inf d (xn−1 ))ϕp
g(x) = inf fmn Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
qmn 1/H
≤ inf
≤1 ,
2qA Proof. Clearly, g(x) ≥ 0 for x = (xmn ) ∈ χf μ , 2 (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))V p . Since fmn (0) = 0, we get g(0) = 0. Conversely, suppose that g(x) = 0, then inf fmn Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , qmn 1/H d (xn−1 ))ϕp ≤ 1 = 0. Suppose that Amn μmn (x) = 0 for each m, n ∈ N, then Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp → ∞.
+ inf
d (xn−1 ))ϕp
d (xn−1 ))ϕp
g (λx) = inf
(d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp Amn μmn (y), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))
ϕ p
qmn 1/H
ϕ qmn 1/H d (xn−1 ) p ≤ fmn Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp +
fmn
≤1 .
((|λ| t)qmn /H : d (x2 ) , · · ·
≤ 1.
where t =
1 |λ| .
≤1 .
fmn (Amn μmn (x), (d (x1 ) ,
, d (xn−1 ))ϕp
qmn 1/H
≤1 ,
Since |λ|qmn ≤ max 1, |λ|suppmn , we have
suppmn inf t qmn /H : fmn (Amn μmn (λx), g(λx) ≤ max 1,|λ| (d (x1 ) , d (x2 ) , · · ·
, d (xn−1 ))ϕp
qmn 1/H
≤1 .
qmn 1/H
Amn μmn (y), (d (x1 ) , d (x2 ) , · · · ,
d (xn−1 ))ϕp
qmn 1/H
qmn 1/H
Then, by using Minkowski’s inequality, we have fmn Amn μmn (x + y) , (d (x1 ) , d (x2 ) , · · · ,
≤1
Finally, to prove that the scalar multiplication is continuous, let λ be any complex number. By definition, fmn (Amn μmn (λx) , (d (x1 ) , d (x2 ) , · · · , g (λx) = inf
Therefore, Amn μmn (x) = 0. Let
≤ 1 and fmn
g (x + y) ≤ g(x) + g(y).
Then,
qmn 1/H
qmn 1/H
Therefore,
It follows that fmn Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , qmn 1/H 2 d (xn−1 ))V → ∞ which is a contradiction. p fmn (Amn μmn (x),
≤1
Amn μmn (y), (d (x1 ) , d (x2 ) , · · · ,
fmn
(12)
fmn Amn μmn (x), (d (x1 ) , d (x2 ) , · · · ,
d (xn−1 ))ϕp
where H = max (1, supmn qmn < ∞).
qmn 1/H
qmn 1/H
.
2qA Theorem 8. The β− dual space of χf μ , Amn μmn (x), β 2qA (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp = f μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
Nagarajan et al. Mathematical Sciences 2013, 7:39 http://www.iaumath.com/content/7/1/39
Page 11 of 13
Proof. First, we observe that
with 2qA x = (xmn ) ∈ χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) ,· · ·, d (xn−1 ))ϕp
β 2qA χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
x = [(λmn − λmn+1 ) − (λm+1n − λm+1n+1 )] ⎛
2qA ⊂ f μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
0 0 ···0 ⎜0 0 ···0 ⎜ ... =⎜ ⎜ ⎝ 0 0 · · · amn λϕrs(m+n)! mn k
0 0
Therefore,
β 2qA ⊂ χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
0 0 ···0 ⎜0 0 ···0 ⎜ ⎜. . −⎜ ⎜. ⎜ ϕ ⎝0 0 · · · amn λmnrs(m+n)!
2qA β ⊂
f μ
2qA = f μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
2qA f μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp β 2qA ⊂ χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
Next, we show that
2qA
χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
···0
⎞ ··· 0 · · · 0⎟ ⎟ ⎟ ⎟ ... 0⎠ ··· 0
0 0 −ϕrs amn k λmn (m+n)!
0
fmn Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
0 ⎜0 ⎜ ⎜. ⎜ ⎜. ⎜ ⎜. ⎜ =⎜ ⎜0 ⎜ ⎜ ⎜ ⎜ ⎜0 ⎜ ⎝ 0
⎞ ··· 0 · · · 0⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ... 0⎠ ··· 0
⎞ 0 0⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ϕrs −ϕrs 0 ...fmn amn λmn (m+n)! fmn amn λmn (m+n)! ... 0⎟ . ⎟ k k ⎟ ⎟ ⎟ −ϕrs ϕrs 0 ...fmn amn λmn (m+n)! fmn amn λmn (m+n)! ... 0⎟ ⎟ k k ⎠ 0 ...0 0, ... 0 0 0
...0 ...0
0 0,
... ...
Hence, converges to zero. Therefore,
Hence,
k
0 0
⎛
But
−ϕrs amn k λmn (m+n)!
0
⎛
β 2qA f μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
···0
0 0
β
2qA ⊂ f μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
[(λmn − λmn+1 ) − (λm+1n − λm+1n+1 )] 2qA × ∈ χf μ ,Amn μmn (x),(d (x1 ), d (x2 ) ,· · ·, d(xn−1))ϕp . Hence, d amn k (λmn - λmn+1 )−(λm+1n −λm+1n+1 ) , 0)=1. However, ymn ≤ f d amn k (λmn − λmn + 1 ) − (λm + 1n − λm+1n+1 ) , 0) ≤ f · 1 < ∞ for each m, n. Thus, (ymn ) is a p- metric paranormed space of double analytic sequence and, hence, an p- metric double analytic sequence. 2qA In other words, y ∈ f μ , μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . However, y = (ymn ) is arbitrary in β 2q χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . Therefore,
2qA Let y = (ymn ) ∈ χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) , β
∞ · · · , d (xn−1 ))ϕp . Consider f (x)= ∞ m=1 n=1 xmn ymn
2qA
χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
β
2qA ⊂ f μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
(13)
Nagarajan et al. Mathematical Sciences 2013, 7:39 http://www.iaumath.com/content/7/1/39
Page 12 of 13
From (12) and (13), we get
2qA
χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
β
2qA = f μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
2qA Theorem 9. The dual space of χf μ , Amn μmn (x), 2qA (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp is f μ , Amn μmn (x) , (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . In other words, ∗ 2qA χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp 2qA = f μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . Proof. We recall that ⎞ ⎛ 0 0 ···0 0 ··· ⎜0 0 ···0 0 · · ·⎟ ⎟ ⎜ ⎟ ⎜. .. ⎟ λmn = ⎜ ⎟ ⎜ ⎟ ⎜0 0 · · · mn ϕrs 0 · · · ⎠ ⎝ ak λmn (m+n)! 0 0 ···0 0 ··· ϕrs with amn λmn (m+n)! in the (m, n)th position and zero elsek where, 2qA χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
⎞ 0. . . 0 ⎟ ⎜ .. ⎟ ⎜. ⎟ ⎜ 1/m+n ⎟ ϕ =⎜ rs ⎟ ⎜ 0 f amn λ (m+n)! . 0 mn ⎟ ⎜ k ⎠ ⎝ th (m, n) ⎛
0
.
. 0
which is a p- metric of double gai sequence. Hence,
2qA x ∈ χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .f (x) ∞ = xmn ymn m,n=1
with
2qA x ∈ χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
and
∗ 2qA f ∈ χf μ ,Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp ,
where ∗ 2qA χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp 2qA is the dual space of χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
2qA ∈ χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . Then, ymn ≤ f d(ϕrs , 0) < ∞∀m, n. (14) Take x
=
(xmn )
Thus, (ymn ) is a p- metric of double analytic sequence and, hence, an p- metric of double analytic 2qA sequence. In other words, y ∈ f μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp . Therefore,
2qA
χf μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp
∗
2q = f μ , Amn μmn (x), (d (x1 ) , d (x2 ) , · · · , d (xn−1 ))ϕp .
Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally in introducing gai-2 sequence spaces generalized over p-metric defined by Musielak modulus function and in studying topological properties. All authors read and approved the final manuscript. Author details 1 Department of Mathematics, SASTRA University,Thanjavur, 613 401, India. 2 Department of Mathematics, Sri Angalamman College of Engineering and Technology, Trichirappalli, 621 105, India. Received: 1 April 2013 Accepted: 5 May 2013 Published: 12 August 2013 References 1. Bromwich, TJI’A: An Introduction to the Theory of Infinite Series. Macmillan, New York (1965) 2. Hardy, GH: On the convergence of certain multiple series. Proc. Camb. Phil. Soc. 19, 86–95 (1917) 3. Moricz, F: Extentions of the spaces c and c0 from single to double sequences. Acta. Math. Hung. 57(1-2), 129–136 (1991) 4. Moricz, F, Rhoades, BE: Almost convergence of double sequences and strong regularity of summability matrices. Math. Proc. Camb. Phil. Soc. 104, 283–294 (1988) 5. Basarir, M, Solancan, O: On some double sequence spaces. J. Indian Acad. Math. 21(2), 193–200 (1999) 6. Tripathy, BC: On statistically convergent double sequences. Tamkang J. Math. 34(3), 231–237 (2003) 7. Turkmenoglu, A: Matrix transformation between some classes of double sequences. J. Inst. Math. Comp. Sci. Math. Ser. 12(1), 23–31 (1999) 8. Gökhan, A, Çolak, R: The double sequence spaces cP2 (p) and cPB 2 (p). Appl. Math. Comput. 157(2), 491–501 (2004) ∞ 9. Gökhan, A, Çolak, R: Double sequence spaces 2 . ibid. 160(1), 147–153 (2005) 10. Zeltser, M: Investigation of double sequence spaces by soft and hard analytical methods. Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, University of Tartu (2001) 11. Mursaleen, M, Edely, OHH: Statistical convergence of double sequences. J. Math. Anal. Appl. 288(1), 223–231 (2003) 12. Altay, B, Ba¸Sar, F: Some new spaces of double sequences. J. Math. Anal. Appl. 309(1), 70–90 (2005) 13. Ba¸Sar, F, Sever, Y: The space Lp of double sequences. Math. J. Okayama Univ. 51, 149–157 (2009) 14. Subramanian, N, Misra, UK: The semi normed space defined by a double gai sequence of modulus function. Fasciculi Math. 45, 111–120 (2010) 15. Maddox, IJ: Sequence spaces defined by a modulus. Math. Proc. Cambridge Philos. Soc. 100(1), 161–166 (1986)
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