Generalised difference sequence space of non-absolute type 1

I t was Shiue [ 16] who have i ntroduced the Cesàro spaces of the type Cesp and Ces∞. I n view of Chiue, we shall i ntroduce and study some properties of generalised Cesàro difference sequence space. We also examine some of their basic properties viz., BK property and some i nclusions relations will be taken care of. Mathematical Subject Classification 2010: 40A05; 46A45; 40C05.


Introduction
By Π we shall denote the set of all sequences (real or complex) and any subspace of it is known as the sequence space. Also, let the set of non-negative integers, the set of real numbers and the set of complex numbers be denoted respectively by N, R and C. Let l ∞ , c and c 0 , respectively, denotes the space of all bounded sequences , the space of convergent sequences and the sequences converging to zero. Also, by bs, cs, l 1 and l p , we denote the spaces of all bounded, convergent, absolutely and p-absolutely convergent series, respectively (see ).
Suppose X is a vector space (real or complex) and H : X → R. We call (X , H) a paranormed space with H a paranorm for X provided : , and (iv) scalar multiplication is continuous, i.e., |β n − β| → 0 and H(s n − s) → 0 gives H(β n s n − βs) → 0 ∀ β ∈ R and s's in X , where θ represent zero vector in the space X .
Suppose A = (a mk ) be an infinite matrix with X, Y ⊂ Π. Then, matrix A represents the A-transformation from X into Y , if for b = (b k ) ∈ X the sequence Ab = {(Ab) m }, the A-transform of b exists and is in Y ; where (Ab) m = k a mk b k as can be seen in [24] A F K space Y is a complete metric sequence space with continuous coordinated p m : Y → C where p m (u) = u m for all u ∈ Y and m ∈ N. A normed F K space is called a BK space as defined in [26] and etc.
Let θ = (t j ) be increasing integer sequence. Then it will be called lacunary sequence if t 0 = 0 and t j = t j − t j−1 → ∞. By θ we will denote the intervals of the form I j = (t j−1 , t j ] and with q j we will denote the ratio t j t j−1 [4]. The spaces T ( ) where was introduced by Kizmaz [16] where T ∈ {l ∞ , c, c 0 } and u m = u m − u m−1 .
Next Tripathy and Esi [26] had studied it and considered it as follows. Consider the integer j ≥ 0. then Recently, in [27] we have the following: The Cesàro sequence spaces Ces p and Ces ∞ have been introduced by Shiue [25] and was further studied by several authors viz., Et [3], Orhan [20], Tripathy [27]. Ng and Lee [18] has introduced the Cesàro sequence spaces X p and X ∞ of non-absolute type and has shown that Ces p ⊂ X p is strict for 1 ≤ p ≤ ∞. Our aim in this paper is to bring out the spaces C (p) ( n m , θ) and C (p) [ n m , θ], where 1 ≤ p ≤ ∞ and study their various properties.
Proof : Let x j = (x j i ) i be any Cauchy sequence in C (p) ( m n , θ) for each j ∈ N. Therefore, we have Hence, m t=1 x i t − x j t → 0 and ∆ m n x i k − ∆ m n x j k → 0, as i, j → ∞ for each k ∈ N. Now, from, we have x i t − x j t → ∞ as i, j → ∞, for each k ∈ N. Therefore, (x j i ) i is a Cauchy sequence in C and hence converges since C is complete, and let lim i x i t = x t for each t ∈ N. Since x i is a Cauchy sequence, therefore for each > 0, we can find n = n 0 ( ) such that as r → ∞, we obtain x ∈ C (p) ( m n , θ). Therefore, C (p) ( m n , θ) is a Banach space. Since, C (p) ( m n , θ) is a Banach space with continuous co-ordinates, that is, x i − x ∆ θ p → 0 for each k ∈ N as i → ∞, consequently, it is a BK-space. Hence, the proof of the result is complete.
Proof : The proof is is similar to that of previous theorem and hence can be omitted. Proof : We only prove the result for C (∞) [ m n , θ] and rest can be proven in a similar fashion. So, to establish the result, we put n = p j = 1 for all j and θ = (2 r ). Then, is not solid. This proves the result.

Inclusion relations
In this section, we prove some basic inclusion relations for the given spaces.
Proof : We shall only prove (i). So, let x ∈ C (p) ( m−1 n , θ). Then, we have Hence, Thus, for each positive integer r 0 , we have To show inclusion is proper, we see that the sequence x = (k m−1 ) belongs to C (p) ( m n , θ) but does not belong to C (p) ( m−1 n , θ), for θ = (2 r ). This completes the proof.