Open Access

On some new sequence spaces defined by infinite matrix and modulus

Advances in Difference Equations20132013:274

https://doi.org/10.1186/1687-1847-2013-274

Received: 22 June 2013

Accepted: 12 August 2013

Published: 19 September 2013

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 matrix A. Also, we define A-statistical convergence.

MSC:40H05, 40C05.

Keywords

modulus function almost convergence lacunary sequence φ-function statistical convergence A-statistical convergence

1 Introduction and background

Let s denote the set of all real and complex sequences x = ( x k ) . By l and c, we denote the Banach spaces of bounded and convergent sequences x = ( x k ) normed by x = sup n | x n | , respectively. A linear functional L on l is said to be a Banach limit [1] if it has the following properties:
  1. (1)

    L ( x ) 0 if n 0 (i.e., x n 0 for all n),

     
  2. (2)

    L ( e ) = 1 , where e = ( 1 , 1 , ) ,

     
  3. (3)

    L ( D x ) = L ( x ) , where the shift operator D is defined by D ( x n ) = { x n + 1 } .

     
Let B be the set of all Banach limits on l . A sequence x is said to be almost convergent if all Banach limits of x coincide. Let c ˆ denote the space of almost convergent sequences. Lorentz [2] has shown that
c ˆ = { x l : lim m t m , n ( x )  exists uniformly in  n } ,
where
t m , n ( x ) = x n + x n + 1 + x n + 2 + + x n + m m + 1 .

By a lacunary θ = ( k r ) , r = 0 , 1 , 2 ,  , where k 0 = 0 , we shall mean an increasing sequence of non-negative integers with k r k r 1 as r . The intervals determined by θ will be denoted by I r = ( k r 1 , k r ] and h r = k r k r 1 .The ratio k r k r 1 will be denoted by q r .

The space of lacunary strongly convergent sequences N θ was defined by Freedman et al. [3] as follows:
N θ = { x = ( x k ) : lim r 1 h r k I r | x k l e | = 0  for some  l } .
There is a strong connection between N θ and the space w of strongly Cesàro summable sequences which is defined by
w = { x = ( x k ) : lim n 1 n k = 0 n | x k l e | = 0  for some  l } .

In the special case where θ = ( 2 r ) , we have N θ = w .

More results on lacunary strong convergence can be seen from [411].

Ruckle [12] used the idea of a modulus function f to construct a class of FK spaces
L ( f ) = { x = ( x k ) : k = 1 f ( | x k | ) < } .

The space L ( f ) is closely related to the space l 1 which is an L ( f ) space with f ( x ) = x for all real x 0 .

Maddox [13] introduced and examined some properties of the sequence spaces w 0 ( f ) , w ( f ) and w ( f ) defined using a modulus f, which generalized the well-known spaces w 0 , w and w of strongly summable sequences.

Recently Savaş [14] generalized the concept of strong almost convergence by using a modulus f and examined some properties of the corresponding new sequence spaces. Waszak [15] defined the lacunary strong ( A , φ ) -convergence with respect to a modulus function.

Following Ruckle, a modulus function f is a function from [ 0 , ) to [ 0 , ) such that
  1. (i)

    f ( x ) = 0 if and only if x = 0 ,

     
  2. (ii)

    f ( x + y ) f ( x ) + f ( x ) for all x , y 0 ,

     
  3. (iii)

    f increasing,

     
  4. (iv)

    f is continuous from the right at zero.

     

Since | f ( x ) f ( y ) | f ( | x y | ) , it follows from condition (iv) that f is continuous on [ 0 , ) .

By a φ-function we understood a continuous non-decreasing function φ ( u ) defined for u 0 and such that φ ( 0 ) = 0 , φ ( u ) > 0 for u > 0 and φ ( u ) as u .

A φ-function φ is called no weaker than a φ-function ψ if there are constants c , b , k , l > 0 such that c ψ ( l u ) b φ ( k u ) (for all large u) and we write ψ φ .

φ-functions φ and ψ are called equivalent and we write φ ψ if there are positive constants b 1 , b 2 , c, k 1 , k 2 , l such that b 1 φ ( k 1 u ) c ψ ( l u ) b 2 φ ( k 2 u ) (for all large u).

A φ-function φ is said to satisfy ( Δ 2 ) -condition (for all large u) if there exists a constant K > 1 such that φ ( 2 u ) K φ ( u ) .

In the present paper, we introduce and study some properties of the following sequence space that is defined using the φ-function and the generalized three parametric real matrix.

2 Main results

Let φ and f be a given φ-function and a modulus function, respectively. Moreover, let A = ( a n k ( i ) ) be the generalized three parametric real matrix, and let a lacunary sequence θ be given. Then we define
N θ 0 ( A , φ , f ) = { x = ( x k ) : lim r 1 h r n I r f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) = 0  uniformly in  i } .
If x N θ 0 ( A , φ , f ) , the sequence x is said to be lacunary strong ( A , φ ) -convergent to zero with respect to a modulus f. When φ ( x ) = x , for all x, we obtain
N θ 0 ( A , f ) = { x = ( x k ) : lim r 1 h r n I r f ( | k = 1 a n k ( i ) ( | x k | ) | ) = 0  uniformly in  i } .
If we take f ( x ) = x , we write
N θ 0 ( A , φ ) = { x = ( x k ) : lim r 1 h r n I r | k = 1 a n k ( i ) φ ( | x k | ) | = 0  uniformly in  i } .
If we take A = I and φ ( x ) = x respectively, then we have [16]
N θ 0 = { x = ( x k ) : lim r 1 h r n I r f ( | x k | ) = 0  uniformly in  i } .
If we define the matrix A = ( a n k ( i ) ) as follows: for all i,
a n k ( i ) : = { 1 n if  n k , 0 otherwise ,
then we have
N θ 0 ( C , φ , f ) = { x = ( x k ) : lim r 1 h r n I r f ( | 1 n k = 1 n φ ( | x k | ) | ) = 0  uniformly in  i } , a n k ( i ) : = { 1 n if  i k i + n 1 , 0 otherwise ,
then we have
N θ 0 ( c ˆ , φ , f ) = { x = ( x k ) : lim r 1 h r n I r f ( | 1 n k = i i + n φ ( | x k | ) | ) = 0  uniformly in  i } .

We are now ready to write the following theorem.

Theorem 2.1 Let A = ( a n k ( i ) ) be the generalized three parametric real matrix, and let the φ-function φ ( u ) satisfy the condition ( Δ 2 ) . Then the following conditions are true.
  1. (a)

    If x = ( x k ) w ( A , φ , f ) and α is an arbitrary number, then α x w ( A , φ , f ) .

     
  2. (b)

    If x , y w ( A , φ , f ) , where x = ( x k ) , y = ( y k ) and α, β are given numbers, then α x + β y w ( A , φ , f ) .

     

The proof is a routine verification by using standard techniques and hence is omitted.

Theorem 2.2 Let f be any modulus function, and let the generalized three parametric real matrix A and the sequence θ be given. If
w ( A , φ , f ) = { x = ( x k ) : lim m 1 m n = 1 m f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) = 0  uniformly in  i } ,
then the following relations are true.
  1. (a)

    If lim inf r q r > 1 , then we have w ( A , φ , f ) N θ 0 ( A , φ , f ) .

     
  2. (b)

    If sup r q r < , then we have N θ 0 ( A , φ , f ) w ( A , φ , f ) .

     
  3. (c)

    1 < lim inf r q r lim sup r q r < , then we have N θ 0 ( A , φ , f ) = w ( A , φ , f ) .

     
Proof (a) Let us suppose that x w ( A , φ , f ) . There exists δ > 0 such that q r > 1 + δ for all r 1 , and we have h r / k r δ / ( 1 + δ ) for sufficiently large r. Then, for all i,
1 k r n = 1 k r f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) 1 k r n I r f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) = h r k r 1 h r n I r f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) δ 1 + δ 1 h r n I r f ( | k = 1 a n k φ ( | x k | ) | ) .
Hence, x N θ 0 ( A , φ , f ) .
  1. (b)
    If lim sup r q r < , then there exists M > 0 such that q r < M for all r 1 . Let x N θ 0 ( A , φ , f ) and ε be an arbitrary positive number, then there exists an index j 0 such that for every j j 0 and all i,
    R j = 1 h j n I r f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) < ε .
     
Thus, we can also find K > 0 such that R j K for all j = 1 , 2 ,  . Now, let m be any integer with k r 1 m k r , then we obtain, for all i,
I = 1 m n = 1 m f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) 1 k r 1 n = 1 k r f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) = I 1 + I 2 ,
where
I 1 = 1 k r 1 j = 1 j 0 n I j f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) , I 2 = 1 k r 1 j = j 0 + 1 m n I j f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) .
It is easy to see that
I 1 = 1 k r 1 j = 1 j 0 n I j f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) = 1 k r 1 ( n I 1 f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) + + n I j 0 f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) ) 1 k r 1 ( h 1 R 1 + + h j 0 R j 0 ) 1 k r 1 j 0 k j 0 sup 1 i j 0 R i j 0 k j 0 k r 1 K .
Moreover, we have, for all i,
I 2 = 1 k r 1 j = j 0 + 1 m n I j f ( | k = 1 a n k φ ( | x k | ) | ) = 1 k r 1 j = j 0 + 1 m 1 h j n I j f ( | k = 1 a n k φ ( | x k | ) | ) h j ε 1 k r 1 j = j 0 + 1 m h j ε k r k r 1 = ε q r < ε M .

Thus I j 0 k j 0 k r 1 K + ε M . Finally, x w ( A , ψ , f ) .

The proof of (c) follows from (a) and (b). This completes the proof. □

We now prove the following theorem.

Theorem 2.3 Let f be a modulus function. Then N θ 0 ( A , φ ) N θ 0 ( A , φ , f ) .

Proof Let x N θ 0 ( A , φ ) . Let ε > 0 be given and choose 0 < δ < 1 such that f ( x ) < ε for every x [ 0 , δ ] . We can write
1 h r n I r f | k = 1 a n k ( i ) φ ( | x k | ) | = S 1 + S 2 ,
where S 1 = 1 h r n I r f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) , and this sum is taken over
| k = 1 a n k ( i ) φ ( | x k | ) | δ
and
S 2 = 1 h r n I r f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) ,
and this sum is taken over
| k = 1 a n k ( i ) φ ( | x k | ) | > δ .
By the definition of the modulus f, we have S 1 = 1 h r n I r f ( δ ) = f ( δ ) < ε and further
S 2 = f ( 1 ) 1 δ 1 h r n I r k = 1 a n k ( i ) φ ( | x k | ) .

Therefore we have x N θ 0 ( 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 [17] (see also Schoenberg [18]) as follows: If denotes the set of natural numbers and K N , then K ( m , n ) denotes the cardinality of the set K [ m , n ] . The upper and lower natural density of the subset K is defined by
d ¯ ( K ) = lim n sup K ( 1 , n ) n and d ̲ ( K ) = lim n inf K ( 1 , n ) n .

If d ¯ ( K ) = d ̲ ( K ) , then we say that the natural density of K exists and it is denoted simply by d ( K ) . Clearly, d ( K ) = lim n K ( 1 , n ) n .

A sequence ( x k ) of real numbers is said to be statistically convergent to L if for arbitrary ε > 0 , the set K ( ε ) = { k N : | x k 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 [19] and Šalát [20].

In another direction, a new type of convergence, called lacunary statistical convergence, was introduced in [21] as follows.

A sequence ( x k ) n N of real numbers is said to be lacunary statistically convergent to L (or S θ -convergent to L) if for any ε > 0 ,
lim r 1 h r | { k I r : | x k L | ε } | = 0 ,

where | A | denotes the cardinality of A N . In [21] the relation between lacunary statistical convergence and statistical convergence was established among other things. Moreover, Kolk [22] defined A-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 the φ-function φ ( u ) .

Let θ be a lacunary sequence, and let A = ( a n k ( i ) ) be the generalized three parametric real matrix; let the sequence x = ( x k ) , the φ-function φ ( u ) and a positive number ε > 0 be given. We write, for all i,
K θ r ( ( A , φ ) , ε ) = { n I r : k = 1 a n k ( i ) φ ( | x k | ) ε } .
The sequence x is said to be ( A , φ ) -statistically convergent to a number zero if for every ε > 0 ,
lim r 1 k r μ ( K θ r ( ( A , φ ) , ε ) ) = 0 uniformly in  n ,
where μ ( K θ r ( ( A , φ ) , ε ) ) denotes the number of elements belonging to K θ r ( ( A , φ ) , ε ) . We denote by S θ 0 ( A , φ ) the set of sequences x = ( x k ) which are lacunary ( A , φ ) -statistical convergent to zero. We write
S θ 0 ( A , φ ) = { x = ( x k ) : lim r 1 h r μ ( K θ r ( ( A , φ ) , ε ) ) = 0  uniformly in  i } .

Theorem 3.1 If ψ φ , then S θ 0 ( A , ψ ) S θ 0 ( A , φ ) .

Proof By assumption we have ψ ( | x k | ) b φ ( c | x k | ) and we have, for all i,
k = 1 a n k ( i ) ψ ( | x k | ) b k = 1 a n k ( i ) φ ( c | x k | ) L k = 1 a n k ( i ) φ ( | x k | )
for b , c > 0 , where the constant L is connected with the properties of φ. Thus, the condition k = 1 a n k ( i ) φ ( | x k | ) 0 implies the condition k = 1 a n k ( i ) φ ( | x k | ) ε , and finally we get
μ ( K θ r ( ( A , φ ) , ε ) ) μ ( K θ r ( ( A , ψ ) , ε ) )
and
lim r 1 h r μ ( K θ r ( ( A , φ ) , ε ) ) lim r 1 h r μ ( K θ r ( ( A , ψ ) , ε ) ) .

This completes the proof. □

We finally prove the following theorem.

Theorem 3.2 (a) If the matrix A, the sequence θ and functions f and φ are given, then
N θ 0 ( ( A , φ ) , f ) S θ 0 ( A , φ ) .
(b) If the φ-function φ ( u ) and the matrix A are given, and if the modulus function f is bounded, then
S θ 0 ( A , φ ) N θ 0 ( ( A , φ ) , f ) .
(c) If the φ-function φ ( u ) and the matrix A are given, and if the modulus function f is bounded, then
S θ 0 ( A , φ ) = N θ 0 ( ( A , φ ) , f ) .
Proof (a) Let f be a modulus function, and let ε be a positive number. We write the following inequalities:
1 h r n I r f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) 1 h r n I r 1 f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) 1 h r f ( ε ) n I r 1 1 1 h r f ( ε ) μ ( K θ r ( A , φ ) , ε ) ,
where
I r 1 = { n I r : k = 1 a n k ( i ) φ ( | x k | ) ε } .
Finally, if x N θ 0 ( ( A , φ ) , f ) , then x S θ 0 ( A , φ ) .
  1. (b)
    Let us suppose that x S θ 0 ( A , φ ) . If the modulus function f is a bounded function, then there exists an integer L such that f ( x ) < L for x 0 . Let us take
    I r 2 = { n I r : k = 1 a n k ( i ) φ ( | x k | ) < ε } .
     
Thus we have
1 h r n I r f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) 1 h r n I r 1 f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) + 1 h r n I r 2 f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) 1 h r M μ ( K θ r ( ( A , φ ) , ε ) ) + f ( ε ) .

Taking the limit as ε 0 , we obtain that x N θ 0 ( A , φ , f ) .

The proof of (c) follows from (a) and (b).

This completes the proof. □

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, Istanbul Ticaret University

References

  1. Banach S: Theorie des Operations Linearies. PWN, Warsaw; 1932.MATHGoogle Scholar
  2. Lorentz GG: A contribution to the theory of divergent sequences. Acta Math. 1948, 80: 167–190. 10.1007/BF02393648MathSciNetView ArticleMATHGoogle Scholar
  3. Freedman AR, Sember JJ, Raphel M: Some Cesaro-type summability spaces. Proc. Lond. Math. Soc. 1978, 37: 508–520.View ArticleMathSciNetMATHGoogle Scholar
  4. Das G, Mishra SK: Banach limits and lacunary strong almost convergence. J. Orissa Math. Soc. 1983, 2(2):61–70.MathSciNetMATHGoogle Scholar
  5. Li J: Lacunary statistical convergence and inclusion properties between lacunary methods. Int. J. Math. Math. Sci. 2000, 23(3):175–180. 10.1155/S0161171200001964MathSciNetView ArticleMATHGoogle Scholar
  6. Savaş E: On lacunary strong σ -convergence. Indian J. Pure Appl. Math. 1990, 21(4):359–365.MathSciNetMATHGoogle Scholar
  7. Savaş E, Karakaya V: Some new sequence spaces defined by lacunary sequences. Math. Slovaca 2007, 57(4):393–399. 10.2478/s12175-007-0034-3MathSciNetMATHGoogle Scholar
  8. Savaş E, Patterson RF: Double σ -convergence lacunary statistical sequences. J. Comput. Anal. Appl. 2009, 11(4):610–615.MathSciNetMATHGoogle Scholar
  9. Savaş E: Remark on double lacunary statistical convergence of fuzzy numbers. J. Comput. Anal. Appl. 2009, 11(1):64–69.MathSciNetMATHGoogle Scholar
  10. Savaş E, Patterson RF: Double σ -convergence lacunary statistical sequences. J. Comput. Anal. Appl. 2009, 11(4):610–615.MathSciNetMATHGoogle Scholar
  11. Savaş E: On lacunary statistical convergent double sequences of fuzzy numbers. Appl. Math. Lett. 2008, 21: 134–141. 10.1016/j.aml.2007.01.008MathSciNetView ArticleMATHGoogle Scholar
  12. Ruckle WH: FK Spaces in which the sequence of coordinate vectors in bounded. Can. J. Math. 1973, 25: 973–978. 10.4153/CJM-1973-102-9MathSciNetView ArticleMATHGoogle Scholar
  13. Maddox IJ: Sequence spaces defined by a modulus. Math. Proc. Camb. Philos. Soc. 1986, 100: 161–166. 10.1017/S0305004100065968MathSciNetView ArticleMATHGoogle Scholar
  14. Savaş E: On some generalized sequence spaces. Indian J. Pure Appl. Math. 1999, 30(5):459–464.MathSciNetMATHGoogle Scholar
  15. Waszak A: On the strong convergence in sequence spaces. Fasc. Math. 2002, 33: 125–137.MathSciNetMATHGoogle Scholar
  16. Pehlivan S, Fisher B: On some sequence spaces. Indian J. Pure Appl. Math. 1994, 25(10):1067–1071.MathSciNetMATHGoogle Scholar
  17. Fast H: Sur la convergence statistique. Colloq. Math. 1951, 2: 241–244.MathSciNetMATHGoogle Scholar
  18. Schoenberg IJ: The integrability of certain functions and related summability methods. Am. Math. Mon. 1959, 66: 361–375. 10.2307/2308747MathSciNetView ArticleMATHGoogle Scholar
  19. Fridy JA: On statistical convergence. Analysis 1985, 5: 301–313.MathSciNetView ArticleMATHGoogle Scholar
  20. Šalát T: On statistically convergent sequences of real numbers. Math. Slovaca 1980, 30: 139–150.MathSciNetMATHGoogle Scholar
  21. Fridy JA, Orhan C: Lacunary statistical convergence. Pac. J. Math. 1993, 160: 43–51. 10.2140/pjm.1993.160.43MathSciNetView ArticleMATHGoogle Scholar
  22. Kolk E: Matrix summability of statistically convergent sequences. Analysis 1993, 13: 77–83.MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Savaş; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.