Open Access

On generalized sequence spaces via modulus function

Journal of Inequalities and Applications20142014:101

https://doi.org/10.1186/1029-242X-2014-101

Received: 25 September 2013

Accepted: 4 February 2014

Published: 4 March 2014

Abstract

In this paper, we introduce and study the concept of lacunary strongly ( A , φ ) -convergence with respect to a modulus function and lacunary ( A , φ ) -statistical convergence and examine some properties of these sequence spaces. We establish some connections between lacunary strongly ( A , φ ) -convergence and lacunary ( A , φ ) -statistical convergence.

MSC:40H05, 40C05.

Keywords

modulus function almost convergence lacunary sequence φ-function statistical convergence

1 Introduction

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 | = 0 , for some  l } .
In the special case where θ = ( 2 r ) (see [3]) we have N θ = w , which is defined by
w = { x = ( x k ) : lim n 1 n k = 0 n | x k l | = 0 , for some  l } .
Das and Mishra [4] have introduced the space AC θ of lacunary almost convergent sequences and the space | AC θ | of lacunary strongly almost convergent sequences as follows:
AC θ = { x = ( x k ) : lim r 1 h r k I r ( x k + n L ) = 0 , for some  L  uniformly in  n }
and
| AC θ | = { x = ( x k ) : lim r 1 h r k I r | x k + n L | = 0 , for some  L  uniformly in  n } .
Ruckle 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 .

In 1999, Savaş [5] generalized the concept of strong almost convergence by using a modulus f and p = ( p k ) is a sequence of strictly positive real numbers as follows:
[ c ˆ ( f , p ) ] = { x : lim n 1 n k = 1 n f ( | x k + m L | ) p k = 0 , for some  L , uniformly in  m }
and
[ c ˆ ( f , p ) ] 0 = { x : lim n 1 n k = 1 n f ( | x k + m | ) p k = 0 , uniformly in  m } .

More investigations in this direction and more applications of the modulus can be found in [612].

Following Ruckle [13], 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.

     

By a φ-function we understand 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 non-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 .

A φ-function φ and ψ are called equivalent 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) and we write φ ψ .

A φ-function φ is said to satisfy the condition ( Δ 2 ) (for all large u) if for some constant k > 1 there is satisfied the inequality φ ( 2 u ) k φ ( u ) (see [12, 14]).

In this paper, we introduce and study some properties of the following sequence space which is generalization of Savaş [14].

2 Main results

Let φ and f be a given φ-function and modulus function, respectively, and let p = ( p n ) be a sequence of positive real numbers. Moreover, let A = ( A i ) be the generalized three parametric real matrix with A i = ( a n , k ( i ) ) and a lacunary sequence θ be given. Then we define the following sequence spaces:
N θ 0 ( A , φ , f , p ) = { x = ( x k ) : lim r 1 h r n I r f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) p n = 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 , p ) = { x = ( x k ) : lim r 1 h r n I r f ( | k = 1 a n k ( i ) x k | ) p n = 0 , uniformly in  i } .
If we take f ( x ) = x , we write
N θ 0 ( A , φ , p ) = { x = ( x k ) : lim r 1 h r n I r | k = 1 a n k ( i ) φ ( | x k | ) | p n = 0 , uniformly in  i } .
If we take p k = p , for all k, we have
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 | ) | ) p = 0 , uniformly in  i } .
If we take A = I and φ ( x ) = x , respectively, then we have
N θ 0 = { x = ( x k ) : lim r 1 h r k I r f ( | x k | ) p n = 0 } .
If we define the matrix A = ( a n k ( i ) ) as follows:
a n k ( i ) : = { 1 n , if  n k , 0 , otherwise ,
then we have
N θ 0 ( C , φ , f , p ) = { x = ( x k ) : lim r 1 h r n I r f ( | 1 n k = 1 n φ ( | x k | ) | ) p n = 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 , p ) = { x = ( x k ) : lim r 1 h r n I r f ( | 1 n k = i i + n φ ( | x k | ) | ) p n = 0 , uniformly in  i } .

If x N θ 0 ( c ˆ , φ , f ) , the sequence x is said to be almost lacunary strong φ-convergent to zero with respect to a modulus f. In the next theorem we establish inclusion relations between w ( A , φ , f , p ) and N θ 0 ( A , φ , f , p ) . We now have the following.

Theorem 2.1 Let f be any modulus function and let there be a φ-function φ and a generalized three parametric real matrix A; let p = ( p n ) be a sequence of positive real numbers and the sequence θ be given. If
w ( A , φ , f , p ) = { x = ( x k ) : lim m 1 m n = 1 m f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) p n = 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 , p ) N θ 0 ( A , φ , f , p ) .

     
  2. (b)

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

     
  3. (c)

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

     
Proof (a) Let us suppose that x w ( A , φ , f , p ) . 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 | ) | ) p n 1 k r n I r f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) p n = h r k r 1 h r n I r f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) p n δ 1 + δ 1 h r n I r f ( | k = 1 a n k φ ( | x k | ) | ) p n .

Hence, x N θ 0 ( A , φ , f , p ) .

(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 , p ) and ε is 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 | ) | ) p n < 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 | ) | ) p n 1 k r 1 n = 1 k r f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) p n = 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 | ) | ) p n , I 2 = 1 k r 1 j = j 0 + 1 m n I j f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) p n .
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 | ) | ) p n = 1 k r 1 ( n I 1 f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) p n + + n I j 0 f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) p n ) 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 | ) | ) p n = 1 k r 1 j = j 0 + 1 m 1 h j n I j f ( | k = 1 a n k φ ( | x k | ) | ) p n 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 , p ) .

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

Theorem 2.2 Let f, f 1 , be modulus functions. Then we have
N θ 0 ( A , f 1 , φ , p ) ( A , φ , f o f 1 , p ) .

Proof This can be proved by using techniques similar to those used in the theorem of Savaş [14]. □

Recently Savaş [14] defined ( A , φ ) -statistical convergence as follows.

Let θ be a lacunary sequence, and let A = ( a n k ( i ) ) be the generalized three parametric real matrix, 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 h r μ ( K θ r ( A , φ , ε ) ) = 0 , uniformly in  i ,
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 and 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 } .

More investigations in this direction can be found in [1520].

We now establish inclusion relations between N θ 0 ( A , φ , f , p ) and S θ 0 ( A , φ ) .

In the following theorem we assume that 0 < h = inf p n p n sup p k H .

Theorem 2.3 (a) If the matrix A and the sequence θ and functions f and φ are given, then
N θ 0 ( A , φ , f , p ) 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 , p ) .
Proof (a) Let f be a modulus function and let ε be a positive numbers. We write the following inequalities, for all i,
1 h r n I r f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) p n = 1 h r n I r 1 f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) p n 1 h r n I r 1 [ f ( ε ) ] p n 1 h r n I r 1 min ( [ f ( ε ) ] inf p n , [ f ( ε ) ] H ) 1 h r μ ( K θ r ( A , φ , ε ) ) min ( [ f ( ε ) ] inf p n , [ f ( ε ) ] H ) ,
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 , p ) then x S θ 0 ( A , φ , f ) .
  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 K such that f ( x ) < K 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, for all i,
1 h r n I r f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) p n 1 h r n I r 1 f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) p n + 1 h r n I r 2 f ( | k = 1 a n k ( i ) φ ( | x k | ) | ) p n 1 h r n I r 1 max ( K h , K H ) + 1 h r n I r 2 [ f ( ε ) ] p n max ( K h , K H ) 1 h r μ ( K θ r ( A , φ , ε ) ) + max ( [ f ( ε ) ] h , [ f ( ε ) ] H ) .

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

This completes the proof. □

Declarations

Acknowledgements

I would like to express my gratitude to the referee of the paper for his useful comments and suggestions towards the quality improvement of the paper. This paper was presented during the ‘2nd International Eurasian Conference on Mathematical Sciences and Applications’ held on Sarajevo, Bosnia and Herzegovina on 26th to 29th August 2013, and it was submitted for the conference proceedings.

Authors’ Affiliations

(1)
Department of Mathematics, Istanbul Commerce University, Sütlüce

References

  1. Banach S: Theorie des Operations Lineaires. Panstwowe Wydawnictwo Naukove, Warszawa; 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. Savaş E: On some generalized sequence spaces defined by a modulus. Indian J. Pure Appl. Math. 1999,30(5):459-464.MathSciNetMATHGoogle Scholar
  6. Connor J: On strong matrix summability with respect to a modulus and statistical convergence. Can. Math. Bull. 1989,32(2):194-198. 10.4153/CMB-1989-029-3MathSciNetView ArticleMATHGoogle Scholar
  7. Maddox IJ: Sequence spaces defined by a modulus. Math. Proc. Camb. Philos. Soc. 1986, 100: 161-166. 10.1017/S0305004100065968MathSciNetView ArticleMATHGoogle Scholar
  8. Malkowsky E, Savaş E: Some λ -sequence spaces defined by a modulus. Arch. Math. 2000, 36: 219-228.MathSciNetMATHGoogle Scholar
  9. Nuray F, Savaş E: Some new sequence spaces defined by a modulus function. Indian J. Pure Appl. Math. 1993,24(11):657-663.MathSciNetMATHGoogle Scholar
  10. Pehlivan S, Fisher B: Some sequence spaces defined by a modulus. Math. Slovaca 1995,43(3):275-280.MathSciNetMATHGoogle Scholar
  11. Pehlivan S: Sequence space defined by a modulus function. Erc. Univ. J. Sci. 1989, 3: 875-880.Google Scholar
  12. Waszak A: On the strong convergence in sequence spaces. Fasc. Math. 2002, 33: 125-137.MathSciNetMATHGoogle Scholar
  13. Ruckle WH: FK Spaces in which the sequence of coordinate vectors is bounded. Can. J. Math. 1973, 25: 973-978. 10.4153/CJM-1973-102-9MathSciNetView ArticleMATHGoogle Scholar
  14. Savaş E: On some new sequence spaces defined by infinite matrix and modulus. Adv. Differ. Equ. 2013., 2013: Article ID 274 10.1186/1687-1847-2013-274Google Scholar
  15. Fast H: Sur la convergence statistique. Colloq. Math. 1951, 2: 241-244.MathSciNetMATHGoogle Scholar
  16. Šalát T: On statistically convergent sequences of real numbers. Math. Slovaca 1980, 30: 139-150.MathSciNetMATHGoogle Scholar
  17. Fridy JA: On statistical convergence. Analysis 1985, 5: 301-313.MathSciNetView ArticleMATHGoogle Scholar
  18. Fridy JA, Orhan C: Lacunary statistical convergence. Pac. J. Math. 1993, 160: 43-51. 10.2140/pjm.1993.160.43MathSciNetView ArticleMATHGoogle Scholar
  19. Savaş E: Strongly almost convergence and almost λ -statistical convergence. Hokkaido Math. J. 2000, 29: 63-68.Google Scholar
  20. Schoenberg IJ: The integrability of certain functions and related summability methods. Am. Math. Mon. 1959, 66: 361-375. 10.2307/2308747MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Savaş; licensee Springer. 2014

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 credited.