R E S E A R C H
Open Access
On some new sequence spaces defined by
infinite matrix and modulus
Ekrem Sava¸s
**Correspondence:
[email protected] Department of Mathematics, Istanbul Ticaret University, Üsküdar, Istanbul, Turkey
Abstract
The goal of this paper is to introduce and study some properties of some sequence spaces that are defined using the
ϕ-function and the generalized three parametric
real matrixA. Also, we defineA-statistical convergence.MSC: Primary 40H05; secondary 40C05
Keywords: modulus function; almost convergence; lacunary sequence;
ϕ-function;
statistical convergence;A-statistical convergence1 Introduction and background
Letsdenote the set of all real and complex sequencesx= (xk). Byl∞andc, we denote the
Banach spaces of bounded and convergent sequencesx= (xk) normed byx=supn|xn|,
respectively. A linear functionalLonl∞is said to be a Banach limit [] if it has the following
properties:
() L(x)≥ifn≥(i.e.,xn≥for alln),
() L(e) = , wheree= (, , . . .),
() L(Dx) =L(x), where the shift operatorDis defined byD(xn) ={xn+}.
LetBbe the set of all Banach limits onl∞. A sequencex∈∞is said to be almost
con-vergent if all Banach limits ofxcoincide. Letcˆdenote the space of almost convergent sequences. Lorentz [] has shown that
ˆ c=
x∈l∞:lim
m tm,n(x) exists uniformly inn
,
where
tm,n(x) =
xn+xn++xn++· · ·+xn+m
m+ .
By a lacunaryθ= (kr),r= , , , . . . , wherek= , we shall mean an increasing sequence of non-negative integers withkr–kr–→ ∞asr→ ∞. The intervals determined byθwill be denoted byIr= (kr–,kr] andhr=kr–kr–.The ratiokkr–r will be denoted byqr.
The space of lacunary strongly convergent sequencesNθwas defined by Freedmanet al.
[] as follows:
Nθ=
x= (xk) :lim r
hr
k∈Ir
|xk–le|= for somel
.
There is a strong connection betweenNθand the spacewof strongly Cesàro summable
sequences which is defined by
w=
x= (xk) :lim n
n n
k=
|xk–le|= for somel
.
In the special case whereθ= (r), we haveN θ=w.
More results on lacunary strong convergence can be seen from [–].
Ruckle [] used the idea of a modulus functionf to construct a class of FK spaces
L(f) =
x= (xk) :
∞
k=
f |xk|
<∞
.
The spaceL(f) is closely related to the spacelwhich is anL(f) space withf(x) =xfor all realx≥.
Maddox [] introduced and examined some properties of the sequence spacesw(f),
w(f) andw∞(f) defined using a modulusf, which generalized the well-known spacesw,
wandw∞of strongly summable sequences.
Recently Savaş [] generalized the concept of strong almost convergence by using a modulusf and examined some properties of the corresponding new sequence spaces. Waszak [] defined the lacunary strong (A,ϕ)-convergence with respect to a modulus function.
Following Ruckle, a modulus functionf is a function from [,∞) to [,∞) such that (i) f(x) = if and only ifx= ,
(ii) f(x+y)≤f(x) +f(x)for allx,y≥, (iii) f increasing,
(iv) f is continuous from the right at zero.
Since|f(x) –f(y)| ≤f(|x–y|), it follows from condition (iv) thatfis continuous on [,∞). By aϕ-function we understood a continuous non-decreasing functionϕ(u) defined for
u≥ and such thatϕ() = ,ϕ(u) > foru> andϕ(u)→ ∞asu→ ∞.
Aϕ-functionϕis called no weaker than aϕ-functionψif there are constantsc,b,k,l> such thatcψ(lu)≤bϕ(ku) (for all largeu) and we writeψ≺ϕ.
ϕ-functionsϕandψare called equivalent and we writeϕ∼ψif there are positive con-stantsb,b,c,k,k,lsuch thatbϕ(ku)≤cψ(lu)≤bϕ(ku) (for all largeu).
Aϕ-functionϕis said to satisfy ()-condition (for all largeu) if there exists a constant
K> such thatϕ(u)≤Kϕ(u).
In the present paper, we introduce and study some properties of the following sequence space that is defined using theϕ-function and the generalized three parametric real ma-trix.
2 Main results
Letϕ andf be a givenϕ-function and a modulus function, respectively. Moreover, let A= (ank(i)) be the generalized three parametric real matrix, and let a lacunary sequence
θ be given. Then we define
Nθ(A,ϕ,f) =
x= (xk) :lim r
hr
n∈Ir
f
∞
k=
ank(i)ϕ |xk|
= uniformly ini
Ifx∈Nθ(A,ϕ,f), the sequencexis said to be lacunary strong (A,ϕ)-convergent to zero
then we have
Nθ(C,ϕ,f) =
then we have
Nθ(cˆ,ϕ,f) =
We are now ready to write the following theorem.
Theorem . LetA= (ank(i))be the generalized three parametric real matrix,and let the
ϕ-functionϕ(u)satisfy the condition().Then the following conditions are true.
(a) Ifx= (xk)∈w(A,ϕ,f)andαis an arbitrary number,thenαx∈w(A,ϕ,f).
(b) Ifx,y∈w(A,ϕ,f),wherex= (xk),y= (yk)andα,βare given numbers,then
αx+βy∈w(A,ϕ,f).
The proof is a routine verification by using standard techniques and hence is omitted.
Theorem . Let f be any modulus function,and let the generalized three parametric real matrix A and the sequenceθbe given.If
w(A,ϕ,f) =
≤
Moreover, we have, for alli,
I =
The proof of (c) follows from (a) and (b). This completes the proof.
We now prove the following theorem.
Theorem . Let f be a modulus function.Then N
By the definition of the modulusf, we haveS=hr
n∈Irf(δ) =f(δ) <εand further
S=f()
δ
hr
n∈Ir
∞
k=
ank(i)ϕ |xk|
.
Therefore we havex∈Nθ(A,ϕ,f).
This completes the proof.
3 A-Statistical convergence
The idea of convergence of a real sequence was extended to statistical convergence by Fast [] (see also Schoenberg []) as follows: IfNdenotes the set of natural numbers and
K⊂N, thenK(m,n) denotes the cardinality of the setK∩[m,n]. The upper and lower natural density of the subsetKis defined by
d(K) = lim
n→∞sup K(,n)
n and d(K) =nlim→∞inf K(,n)
n .
Ifd(K) =d(K), then we say that the natural density ofKexists and it is denoted simply by
d(K). Clearly,d(K) =limn→∞K(,nn).
A sequence (xk) of real numbers is said to be statistically convergent toLif for arbitrary
ε> , the setK(ε) ={k∈N:|xk–L| ≥ε}has natural density zero. Statistical convergence
turned out to be one of the most active areas of research in summability theory after the work of Fridy [] and Šalát [].
In another direction, a new type of convergence, called lacunary statistical convergence, was introduced in [] as follows.
A sequence (xk)n∈N of real numbers is said to be lacunary statistically convergent toL
(orSθ-convergent toL) if for anyε> ,
lim
r→∞
hr
k∈Ir:|xk–L| ≥ε= ,
where|A|denotes the cardinality ofA⊂N. In [] the relation between lacunary statistical convergence and statistical convergence was established among other things. Moreover, Kolk [] definedA-statistical convergence by using non-negative regular summability matrix.
In this section we define (A,ϕ)-statistical convergence by using the generalized three parametric real matrix and theϕ-functionϕ(u).
Letθ be a lacunary sequence, and letA= (ank(i)) be the generalized three parametric
real matrix; let the sequencex= (xk), theϕ-functionϕ(u) and a positive numberε> be
given. We write, for alli,
Kθr (A,ϕ),ε
=
n∈Ir:
∞
k=
ank(i)ϕ |xk|
≥ε
.
The sequencexis said to be (A,ϕ)-statistically convergent to a number zero if for every
ε> ,
lim
r
kr
μ Kθr (A,ϕ),ε
whereμ(Kr
θ((A,ϕ),ε)) denotes the number of elements belonging toKθr((A,ϕ),ε). We
de-note bySθ(A,ϕ) the set of sequencesx= (xk) which are lacunary (A,ϕ)-statistical
conver-gent to zero. We write
Sθ(A,ϕ) =
x= (xk) :lim r
hr
μ Kθr (A,ϕ),ε= uniformly ini
.
Theorem . Ifψ≺ϕ,then Sθ(A,ψ)⊂Sθ(A,ϕ).
Proof By assumption we haveψ(|xk|)≤bϕ(c|xk|) and we have, for alli,
∞
k=
ank(i)ψ |xk|
≤b
∞
k=
ank(i)ϕ c|xk|
≤L
∞
k=
ank(i)ϕ |xk|
forb,c> , where the constantLis connected with the properties ofϕ. Thus, the condition
∞
k=ank(i)ϕ(|xk|)≥ implies the condition
∞
k=ank(i)ϕ(|xk|)≥ε, and finally we get
μ Kθr (A,ϕ),ε
⊂μ Kθr (A,ψ),ε
and
lim
r
hr
μ Kθr (A,ϕ),ε
≤lim
r
hr
μ Kθr (A,ψ),ε
.
This completes the proof.
We finally prove the following theorem.
Theorem . (a)If the matrix A,the sequenceθ and functions f andϕare given,then
Nθ (A,ϕ),f
⊂Sθ(A,ϕ).
(b)If theϕ-function ϕ(u)and the matrix A are given,and if the modulus function f is bounded,then
Sθ(A,ϕ)⊂Nθ (A,ϕ),f
.
(c)If theϕ-functionϕ(u)and the matrix A are given, and if the modulus function f is bounded,then
Sθ(A,ϕ) =Nθ (A,ϕ),f
.
Proof (a) Letf be a modulus function, and letεbe a positive number. We write the fol-lowing inequalities:
hr
n∈Ir
f
∞
k=
ank(i)ϕ |xk|
≥ hr
n∈I
r
f
∞
k=
ank(i)ϕ |xk|
≥ hr
f(ε)
n∈Ir
≥ hr
f(ε)μ Kθr(A,ϕ),ε
,
where
Ir=
n∈Ir:
∞
k=
ank(i)ϕ |xk|
≥ε
.
Finally, ifx∈Nθ((A,ϕ),f), thenx∈Sθ(A,ϕ).
(b) Let us suppose thatx∈S
θ(A,ϕ). If the modulus functionf is a bounded function,
then there exists an integerLsuch thatf(x) <Lforx≥. Let us take
Ir=
n∈Ir:
∞
k=
ank(i)ϕ |xk|
<ε
.
Thus we have
hr
n∈Ir
f
∞
k=
ank(i)ϕ |xk|
≤ hr
n∈Ir
f
∞
k=
ank(i)ϕ |xk|
+
hr
n∈Ir
f
∞
k=
ank(i)ϕ |xk|
≤ hr
Mμ Kθr (A,ϕ),ε
+f(ε).
Taking the limit asε→, we obtain thatx∈N
θ(A,ϕ,f).
The proof of (c) follows from (a) and (b).
This completes the proof.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
This paper was presented during the ‘International Conference on the Theory, Methods and Applications of Nonlinear Equations’ held on the campus of Texas A&M University-Kingsville, Kingsville, TX 78363, USA on December 17-21, 2012, and submitted for conference proceedings.
Received: 22 June 2013 Accepted: 12 August 2013 Published: 19 September 2013 References
1. Banach, S: Theorie des Operations Linearies. PWN, Warsaw (1932)
2. Lorentz, GG: A contribution to the theory of divergent sequences. Acta Math.80, 167-190 (1948)
3. Freedman, AR, Sember, JJ, Raphel, M: Some Cesaro-type summability spaces. Proc. Lond. Math. Soc.37, 508-520 (1978)
4. Das, G, Mishra, SK: Banach limits and lacunary strong almost convergence. J. Orissa Math. Soc.2(2), 61-70 (1983) 5. Li, J: Lacunary statistical convergence and inclusion properties between lacunary methods. Int. J. Math. Math. Sci.
6. Sava¸s, E: On lacunary strongσ-convergence. Indian J. Pure Appl. Math.21(4), 359-365 (1990)
7. Sava¸s, E, Karakaya, V: Some new sequence spaces defined by lacunary sequences. Math. Slovaca57(4), 393-399 (2007)
8. Sava¸s, E, Patterson, RF: Doubleσ-convergence lacunary statistical sequences. J. Comput. Anal. Appl.11(4), 610-615 (2009)
9. Sava¸s, E: Remark on double lacunary statistical convergence of fuzzy numbers. J. Comput. Anal. Appl.11(1), 64-69 (2009)
10. Sava¸s, E, Patterson, RF: Doubleσ-convergence lacunary statistical sequences. J. Comput. Anal. Appl.11(4), 610-615 (2009)
11. Sava¸s, E: On lacunary statistical convergent double sequences of fuzzy numbers. Appl. Math. Lett.21, 134-141 (2008) 12. Ruckle, WH:FKSpaces in which the sequence of coordinate vectors in bounded. Can. J. Math.25, 973-978 (1973) 13. Maddox, IJ: Sequence spaces defined by a modulus. Math. Proc. Camb. Philos. Soc.100, 161-166 (1986) 14. Sava¸s, E: On some generalized sequence spaces. Indian J. Pure Appl. Math.30(5), 459-464 (1999) 15. Waszak, A: On the strong convergence in sequence spaces. Fasc. Math.33, 125-137 (2002) 16. Pehlivan, S, Fisher, B: On some sequence spaces. Indian J. Pure Appl. Math.25(10), 1067-1071 (1994) 17. Fast, H: Sur la convergence statistique. Colloq. Math.2, 241-244 (1951)
18. Schoenberg, IJ: The integrability of certain functions and related summability methods. Am. Math. Mon.66, 361-375 (1959)
19. Fridy, JA: On statistical convergence. Analysis5, 301-313 (1985)
20. Šalát, T: On statistically convergent sequences of real numbers. Math. Slovaca30, 139-150 (1980) 21. Fridy, JA, Orhan, C: Lacunary statistical convergence. Pac. J. Math.160, 43-51 (1993)
22. Kolk, E: Matrix summability of statistically convergent sequences. Analysis13, 77-83 (1993) doi:10.1186/1687-1847-2013-274
