The strongly generalized double difference sequence spaces defined by a modulus

In this paper we introduce the strongly generalized difference sequence spaces of modulus function and     , , i m n i i k l A a  is a non-negative four dimensional matrix of complex numbers and (pi(mn)) is a sequence of positive real numbers. We also give natural relationship between strongly generalized difference summable sequences with respect of modulus. We examine some topological properties of the above spaces and investigate some inclusion relations between these spaces.

A a 

Introduction
Throughout the paper w, x and Л denote the classes of all, gai and analytic scalar valued single sequences, respectively.We write w 2 for the set of all complex sequences (x mn ), where , m n   the set of positive integers.Then, w 2 is a linear space under the coordinate wise addition and scalar multiplication.
Some initial works on double sequence spaces are found in Bromwich (1965).Later on these were investigated by Hardy (1917), Moricz (1991), Moricz and Rhoades (1988), Basarir andSolancan (1999), Tripathy (2003), Turkmenoglu (1999) and many others.Quite recently Zeltser (2001) in her Ph.D. thesis, had essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences.Mursaleen and Edely (2004) have recently introduced the statistical convergence and Cauchy for double sequences and given the relation between statistical convergent and strongly Ces`aro summable double sequences.Subsequently Mursaleen (2004) and Mursaleen and Edely (2004), have defined the almost strong regularity of matrices for double sequences and applied these matrices to establish a core theorem and introduced the M-core for double sequences.
They have determined four dimensional matrices transforming every bounded double sequences x = (x mn ) into one whose core is a subset of the M-core of x.Recently, Altay and Basar (2005), 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 are in the spaces M u , Um(t), C p , C bp , C r and L u respectively, and also examined some properties of those sequence spaces and determined the α -duals of the spaces BS, CS bp and BV and the ﾟ     -duals of the spaces CS bp , CS r of double series.Quite recently Basar and Sever (2009), have introduced the Banach space L q of double sequences corresponding to the well-known space l q of single sequences and examined some properties of the space L q .Quite recently Das et al. (2008), Vakeel and Tabassum (2010, 2011aand b), Kumar (2007), Subramanian and Misra (2010), have studied the space   2 , , M p q u  of double sequences and gave some inclusion relations.
Spaces of strongly summable sequences were discussed by Kuttner (1946), Maddox (1979) and others.The class of sequences which are strongly Ces`aro summable with respect to a modulus was introduced by Maddox (1986), as an extension of the definition of strongly Ces`aro summable sequences.Cannor (1989) further extended this definition to the definition of stong A -summability, with respect to a modulus where A = (a n,k ) is a nonegative regular matrix and established some connection between strong A -summabilty with respect to a modulus and A -statistical convergence.The notion of double sequence was presented by Pringsheim (1900).The four dimensional matrix transformations was also studied extensively by Hamilton (1936Hamilton ( , 1938aHamilton ( and b, 1939)).In his work and throughout this paper, the four dimensional matrices and double sequences have real-valued entries unless specified otherwise.In this paper we extend a few results known in the literature for ordinary (single) sequence spaces to apply sequence spaces.A sequence x = (x i(mn) ) is said to be strongly (V 2 ,  2 ) summable to zero, if   be an infinite four dimensional matrix of complex numbers.We write Let p = (p mn ) be a sequence of positive real numbers with 0 < p mn < sup p mn = G and let D = max(1,2 G-1 ).Then, for   mn mn b a , ,the set of complex numbers , and for all   n m, we have The double series is said to be convergent if and only if the double sequence (S mn ) is convergent, where x    .The vector space of all double analytic sequences is denoted by Л 2 .
A sequence x = (x mn ) is said to be a double gai The set of all double gai sequences is denoted by x 2 .We denote  as the set of all finite sequences.The (m,n) th section, usually denoted by x [m,n] , of the sequence x = (x mn ) is defined by ; where ij  denotes the double sequence whose only non-zero term is   The difference sequence space (for single sequences), usually denoted by Z(), is defined as (KIZMAZ, 1981) , for all k , where w, c, c 0 and l  denote the class of all, convergent, null, and bounded scalar valued single sequences respectively.The above space is a Banach space normed by  .In this paper we define the difference double sequence space as follows: We also have, for all , it follows from (iv) that is continuous on [0, ).
A double sequence  2 = {(β r , u s )} is said to be a double  2 sequence if there exist two non-decreasing sequences of positive numbers tending to infinity such that β r+1 ≤ β r + 1, β 1 = 1 and u S+1 ≤ u s + 1, u 1 = 1.The generalized double de Vallee-Poussin mean is defined as

Let
 is an infinite four dimensional matrix of complex numbers and be a double analytic sequence of positive real numbers such that and f be a modulus.We define where: . In what follows in this paper we establish some of the topological properties of the above spaces and investigate inclusion relations between them.We prove: Theorem-1 Let f be a modulus function.Then  is a linear space over the complex field C.
. Then there exist integers D α and D u such that . By using (1) and the properties of modulus f, we have: This proves that Let be a modulus function.Then the inclusion , , , , , , , , , , , , holds .This completes the proof.
, where: x  .Hence we get g (0) = 0. Further since  and 1 M  , using Minkowski's inequality and definition of modulus f, for each (r, s), we have This follows that g (x) is sub-additive.Next, for any complex number α and the definition of modulus function, we have where Since f is modulus, we have x → 0 implies g (x) → 0. Similarly x → 0 and α → 0 implies g (αx) → 0. Thus we have for g x fixed and α → 0, g (αx) → 0. This completes the proof.
Proof: The proof is a routine verification.

Conclusion
We give natural relationship between strongly generalized difference spaces and investigate some inclusion relations between these spaces.


But Let 1 ≤ p i sup p i < .Then