ISSN: 2229-371X
Ab Hamid Ganie1*, Mobin Ahmad2 and Neyaz Ahmad Sheikh3 |
Corresponding author: Ab Hamid Ganie, E-mail: ashamidg@rediffmail.com |
Related article at Pubmed, Scholar Google |
Visit for more related articles at Journal of Global Research in Computer Sciences
The aim of the present paper is to introduce some new generalized difference sequence spaces with respect to modulus function involving strongly almost summable sequences. We give some topological properties and inclusion relations on these spacesa.
INTRODUCTION |
A sequence space is defined to be a linear space of real or complex sequences. Throughout the paper N, R and C denotes the set of non-negative integers, the set of real numbers and the set of complex numbers respectively. Letω denote the space of all sequences (real or complex). Let l∞ and c be Banach spaces of bounded and convergent sequences ![]() ![]() ![]() ![]() |
Lorentz, called a sequence {xn} almost convergent if all Banach limits of x, L(x), are same and this unique Banach limit is called F-limit of x [1]. In his paper, Lorentz proved the following criterian for almost convergent sequences. |
A sequence ![]() |
![]() |
where, ![]() |
We denote the set of almost convergent sequences by f. |
Several authors including Duran [2], Ganie et al. [3-7], King [8], Lorentz [1] and many others have studied almost convergent sequences. Maddox [9,10] has defined x to be strongly almost convergent to a number α if |
![]() |
By [f] we denote the space of all strongly almost convergent sequences. It is easy to see that ![]() |
The concept of paranorm is related to linear matric spaces. It is a generalization of that of absolute value. Let X be a linear space. A function P:x→R is called a paranorm, if [11,12]. |
![]() |
![]() |
![]() |
![]() |
![]() ![]() ![]() ![]() ![]() |
A paranorm p for which p(x)=0 implies x=0 is called total. It is well known that the metric of any linear metric space is given by some total paranorm [10]. |
The following inequality will be used throughout this paper. Let p=(pk) be a sequence of positive real numbers with ![]() ![]() ![]() |
![]() |
Nanda defined the following [13,14]: |
![]() |
![]() |
![]() |
The difference sequence spaces, |
![]() |
where X= ∞ l , C and C0, were studied by Kizmaz [15]. |
It was further generalized by Ganie et al. [5], Et and Colak [16], Sengonul [17] and many others. |
Further, it was Tripathy et al. [18] generalized the above notions and unified these as follows: |
![]() |
Where |
![]() |
and |
![]() |
Recently, M. Et [19] defined the following: |
![]() |
![]() |
![]() |
Following Maddox [20]and Ruckle [21], a modulus function g is a function from [0,∞) to [0,∞) such that |
(i) g(x)=0 if and only if x=0, |
(ii)![]() |
(iii) g is increasing, |
(iv) g if continuous from right at x=0. |
Maddox [10] introduced and studied the following sets: |
![]() |
![]() |
of sequences that are strongly almost convergent to zero and strongly almost convergent. |
Let p=(pk) be a sequence of positive real numbers with ![]() |
MAIN RESULTS: |
In the present paper, we define the spaces ![]() ![]() |
![]() |
![]() |
![]() |
Where (pk) is any bounded sequence of positive real numbers. |
Theorem 1: Let (pk) be any bounded sequence and g be any modulus function. Then ![]() ![]() |
Proof: We shall prove the result for![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() |
As n→∞, uniformly in m. This proves that ![]() |
Theorem 2: Let g be any modulus function. Then |
![]() |
Proof: We shall prove the result for ![]() ![]() |
Now, by definition of g, we have |
![]() |
![]() |
Thus, for any number L, there exists a positive integer KL such that ![]() |
![]() |
![]() |
Since, ![]() ![]() |
Theorem 3: ![]() |
![]() |
Proof: From Theorem 2, for each ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
which shows that ![]() |
![]() |
where, Sα is an integer such that α<Sα. Now, let α→0 for any fixed x with ![]() ![]() ![]() |
![]() |
As g is continuous, we have, for 1≤n≤N and by choosing α so small that (Equation 3) |
![]() |
Consequently, (2) and (3) gives that ![]() |
Theorem 4: Let X be any of the spaces [f,g], [f,g]0 and [f,g]∞. Then,![]() ![]() |
Proof: We give the proof for the space [f, g]∞ and others can be proved similarly. So, let ![]() ![]() |
Since, g is increasing function, we have |
![]() |
![]() |
![]() |
Thus, ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Theorem 5: ![]() |
Proof: The proof is obvious from Theorem 4 above. |
Theorem 6: Let g, g1 and g2 be any modulus functions. Then, |
(i) ![]() |
(ii) ![]() |
Proof: Let ε be given small positive number and choose δ with 0< δ<1 such that g(t)< ε for 0<t≤ δ. We put ![]() |
![]() |
where the first summation is over ![]() |
![]() |
and for yk+m> δ, we use the fact that |
![]() |
Now, by definition of g, we have for yk+m> δ that |
![]() |
Thus (Equation 5), |
![]() |
Consequently, we see from (4) and (5) that ![]() |
To prove (ii), we have from (1) that |
![]() |
Let ![]() ![]() |
Theorem 7: Let g, g1 and g2 be any modulus functions. Then, |
![]() |
![]() |
![]() |
![]() |
Proof: The follows as a routine verification as of the Theorem 6. |
Theorem 8: The spaces ![]() ![]() |
Proof: To show that the spaces ![]() ![]() |
Let pk=1 for all k and g(x)=x with r=1=n. Then, ![]() ![]() ![]() |
From above Theorem, we have the following corollary. |
Corollary 9: The spaces ![]() ![]() |
Theorem 10: The spaces ![]() ![]() |
Proof : To show that the spaces ![]() ![]() ![]() |
![]() |
Then, ![]() |
References |
|