Abstract
Purpose
For a non-trivial ideal, a lacunary sequence θ = (k r ), and a modulus function f, the purpose of present paper is to introduce certain new notions:-asymptotically equivalent,-asymptotically equivalent, and-asymptotically equivalent sequences of numbers.
Methods
We use an analytic method to obtain our results.
Results
Certain theorems on generalized equivalent sequences by the use of ideals, lacunary sequences, and a modulus function are obtained.
Conclusions
We observe that if the modulus function f satisfies, then the notions and respectively coincide with the notions and. However, if f is bounded, the notions and coincides respectively with the notions and.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Introduction
The idea of statistical convergence for number sequences is introduced by Fast[1] and later developed by[2–6] and many others.
Definition 1.1
[1] A number sequence x = (x k ) is said to be statistically convergent to a number L (denoted by) provided that for every ε > 0,
where the vertical bars denote the cardinality of the enclosed set.
By a lacunary sequence, we mean an increasing sequence θ = (k r ) of positive integers such that k0 = 0 and as. Let, I r = (kr−1k r ] and. Using lacunary sequences, Fridy et al.[7] defined S θ -convergence, a generalized statistical convergence as follows.
Definition 1.2
[7] Let θ = (k r ) be a lacunary sequence. A sequence x = (x k ) of numbers is said to be lacunary statistically convergent to a number L (denoted by) if for each ε > 0,
In order to compare the rate of growth of two sequences, Marouf[8] defined asymptotically equivalent sequences of real numbers and studied its relations with certain matrix-transformed sequences. Patterson et al.[9] studied the asymptotically lacunary statistical equivalent analog of these ideas. Subsequently, many authors have shown their interest to solve different problems arising in this area (see[10–16]). In this work, we define asymptotically equivalent sequences using lacunary sequences, ideals and a modulus function and obtain some revelent connections between these notions.
Definition 1.3
[8] The two non-negative sequences x = (x k ) and y = (y k ) are said to be asymptotically equivalent of multiple L provided that
(denoted by x∼y) and is called simply asymptotically equivalent if L = 1.
For any non-empty set X, let denote the power set of X.
Definition 1.4
A family is said to be an ideal in X if
-
(1)
;
-
(2)
and
-
(3)
.
Definition 1.5
A non-empty family is said to be a filter in X if
-
(1)
;
-
(2)
and
-
(3)
.
An ideal is said to be non-trivial if. A non-trivial ideal is called admissible if it contains all the singleton sets.
Moreover, if is a non-trivial ideal on X, then is a filter on X and conversely. The filter is called the filter associated with the ideal.
Using ideals, Kostyrko et al.[17] defined-convergence, a stronger convergence in a metric space, whereas Dass et al.[18] unified this idea with statistical convergence for real sequences.
Definition 1.6
[17] Let be a non-trivial ideal in and (X ρ) be a metric space. A sequence x = (x k ) in X is said to be-convergent to ξ if for each ε > 0, the set
In this case, we write.
Definition 1.7
[18] A sequence x = (x k ) of numbers is said to be-statistical convergent or-convergent to L, if for every ε > 0 and δ > 0, we have
In this case, we write or.
Nakano[19] introduced the notion of a modulus function in 1953 as follows. By a modulus function, we mean a function f from to such that
-
(1)
f(x) = 0 if and only if x = 0;
-
(2)
f(x + y) ≤ f(x) + f(y) for all x ≥ 0,y ≥ 0;
-
(3)
f is increasing;
-
(4)
f is continuous from the right at 0.
It follows that f must be continuous on [0,1). A modulus may be bounded or unbounded. Many authors, including Connor[20], Kolk[21], Maddox[22], Öztürk et al.[23], Pehlivan et al.[24, 25] and many others used a modulus f to construct some sequence spaces.
Recently, Bilgin[26] used modulus function to define some notions of asymptotically equivalent sequences and studied some of their connections. We now consider some new kind of asymptotically equivalent sequences defined by ideals, lacunary sequences and a modulus function.
Methods
We use an analytic method to obtain our results.
Results and discussion
We now consider our main results. We begin with the following definitions.
Definition 2.1
Let be a non-trivial ideal in. The two non-negative sequences x = (x k ) and y = (y k ) are said to be strongly asymptotically equivalent of multiple L with respect to the ideal provided that for each ε > 0,
denoted by and simply strongly asymptotically equivalent with respect to the ideal, if L = 1.
Definition 2.2
Let be a non-trivial ideal in and θ = (k r ) be a lacunary sequence. The two non-negative sequences x = (x k ) and y = (y k ) are said to be asymptotically lacunary statistical equivalent of multiple L with respect to the ideal provided that for each ε > 0 and γ > 0,
denoted by and simply asymptotically lacunary statistical equivalent with respect to the ideal, if L = 1.
Definition 2.3
Let be a non-trivial ideal in and θ = (k r ) be a lacunary sequence. The two non-negative sequences x = (x k ) and y = (y k ) are said to be strongly asymptotically lacunary equivalent of multiple L with respect to the ideal provided that for each ε > 0,
denoted by and simply strongly asymptotically lacunary equivalent with respect to the ideal, if L = 1.
Definition 2.4
Let be a non-trivial ideal in and f be a modulus function. The two non-negative sequences x = (x k ) and y = (y k ) are said to be f -asymptotically equivalent of multiple L with respect to the ideal provided that for each ε > 0,
denoted by and simply f -asymptotically equivalent with respect to the ideal, if L = 1.
Definition 2.5
Let be a non-trivial ideal in and f be a modulus function. The two non-negative sequences x = (x k ) and y = (y k ) are said to be strongly f -asymptotically equivalent of multiple L with respect to the ideal provided that for each ε > 0,
denoted by and simply strongly f -asymptotically equivalent with respect to the ideal, if L = 1.
Definition 2.6
Let be a non-trivial ideal in, f be a modulus function and θ = (k r ) be a lacunary sequence. The two non-negative sequences x = (x k ) and y = (y k ) are said to be strongly f -asymptotically lacunary equivalent of multiple L with respect to the ideal provided that for each ε > 0,
denoted by and simply strongly f -asymptotically lacunary equivalent with respect to the ideal, if L = 1.
Lemma 2.1
[10, 25] Let f be a modulus function and let 0 < δ <1. Then for y ≠ 0 and each, we have.
Theorem 2.1
Let be a non-trivial ideal in, and f be a modulus function. Then,
-
(1)
if then and
-
(2)
, then .
Proof
-
(1)
Let and ε > 0 be given. Choose 0 < δ < 1 such that f(t) < ε for 0 ≤ t ≤ δ. We can write
where the first summation runs over, and the second summation on. Moreover, using the definition of the modulus function f , we have
Thus, for any η > 0,
Since, it follows the later set, and hence, the first set in above expression belongs to. This proves that. □
-
(2)
If , then we have f(t) ≥ αt for all t > 0. Suppose that . Since
it follows that for each ε > 0, we have
Since, it follows that the later set belongs to, and therefore, the theorem is proved.
Theorem 2.2
Let be a non-trivial ideal in, and f be a modulus function. Then,
-
(1)
if , then and
-
(2)
if f is bounded, then .
Proof
-
(1)
Suppose , and let ε > 0 be given, then we can write
□
Consequently, for any η > 0, we have
Since, it follows by Definition 2.5 that the later set belongs to, and therefore,.
-
(2)
Suppose f is bounded and . Since f is bounded, there exists a real number M such that supf(t) ≤ M. Moreover, for ε > 0, we can write
Now on applying the operators, the result follows similarly as in case of (1).
Theorem 2.3
Let be a non-trivial ideal in, θ = (k r ) be a lacunary sequence and f be a modulus function. If, then.
Proof
Suppose lim inf r q r > 1, then there exist δ > 0 such that. This implies that. Let. For a sufficiently large r, we obtain the following:
which gives for any ε > 0,
□
Since, it follows that the later set and, hence, the former set belongs to. This shows that.
Theorem 2.4
Let be a non-trivial ideal in, θ = (k r ) be a lacunary sequence and f be a modulus function. Then,
-
(1)
if , then ; and
-
(2)
, then .
Proof
The proof is similar to the proof of Theorem 2.1, so we omit it here. □
Theorem 2.5
Let be a non-trivial ideal in, θ = (k r ) be a lacunary sequence and f be a modulus function. Then,
-
(1)
if , then ;
-
(2)
if f is bounded, then .
Proof
-
(1)
Suppose , and let ε > 0 be given. Since
it follows that for any γ > 0, if we denote sets
then. Since, so. But then, by definition of an ideal,, and therefore,. □
-
(2)
Suppose that f is bounded, and let . Since f is bounded, there exists a positive real number M such that for all x ≥ 0. Further, using the fact
the proof can be obtained on the same lines as that for part (2) of Theorem 2.2.
Conclusions
We observe that if the modulus function f satisfies, then the notions and respectively coincide with the notions and. However, if f is bounded, the notions and coincides respectively with the notions and.
References
Fast H: Sur la convergence statistique. Colloq. Math 1951, 2: 241–244.
Conner JS: The statistical and strong p-Cesáro convergence of sequences. Analysis 1988, 8: 47–63.
Fridy JA: On statistical convergence. Analysis 1985, 5(4):301–313.
Rath D, Tripathy BC: On statistically convergent and statistically Cauchy sequences. Indian Jour. Pure Appl. Math 1994, 25(4):381–386.
Šalát T: On statistically convergent sequences of real numbers. Math. Slovaca 1980, 30: 139–150.
Schoenberg IJ: The integrability of certain functions and related summability methods. Amer. Math. Monthly 1959, 66: 361–375. 10.2307/2308747
Fridy JA, Orhan C: Lacunary statistical convergence. Pacific J. Math 1993, 160: 43–51. 10.2140/pjm.1993.160.43
Marouf M: Asymptotic equivalence and summability. Internat. J. Math. Math. Sci 1993, 16(4):755–762. 10.1155/S0161171293000948
Patterson RF, Savaş E: On asymptotically lacunary statistically equivalent sequences. Thai J. Math 2006, 4(2):267–272.
Başarir M, Altundaǧ S: On asymptotically equivalent difference sequences with respect to a modulus function. Ricerche Mat 2011, 60: 299–311. 10.1007/s11587-011-0106-0
Başarir M, Altundaǧ S:Onlacunary asymptotically equivalent sequences. Int. J. Math. Anal. (Ruse) 2008, 2(5–8):373–382.
Başarir M, Altundaǧ S: On Δ−lacunary statistical asymptotically equivalent sequences. Filomat 2008, 22(1):161–172. 10.2298/FIL0801161B
Kumar V, Sharma A: On asymptotically generalized statistical equivalent sequences via ideals. Tamkang J. Math 2011. in press in press
Li J: Asymptotic equivalence of sequences and summability. Internat J. Math. & Math. Sci 1997, 20(4):749–758. 10.1155/S0161171297001038
Patterson RF: On asymptotically statistically equivalent sequences. Demonstratio Math 2003, 36(1):149–153.
Pobyvanets IP: Asymptotic equivalence of some linear transformation defined by a non-negative matrix and reduced to generalized equivalence in the sense of Cesàro and Abel. Mat. Fiz 1980, 28(123):83–87.
Kostyrko P, Salat T, Wilczynski W:convergence. Real Anal. Exchange 2000/2001, 26(2):669–686.
Das P, Savas E, Ghosal S: On generalizations of certain summability methods using ideals. Appl. Math. Lett 2011, 24: 1509–1514. 10.1016/j.aml.2011.03.036
Nakano H: Concave modulars. J. Math. Soc. Japan 1953, 5: 29–49. 10.2969/jmsj/00510029
Connor JS: On strong matrix summability with respect to a modulus and statistical convergence. Canad. Math. Bull 1989, 32: 194–198. 10.4153/CMB-1989-029-3
Kolk E: On strong boundedness and summability with respect to a sequence of moduli, TartuUl. Toimetised 1993, 960: 41–50.
Maddox IJ: Sequence spaces defined by a modulus. Math. Proc. Camb. Phil. Soc 1986, 100: 161–166. 10.1017/S0305004100065968
Öztürk E, Bilgin T: Strongly summable sequence spaces defined by a modulus. Indian J. Pure Appl. Math 1994, 25(6):621–625.
Pehlivan S, Fisher B: On some sequence spaces. Indian J. Pure Appl. Math 1994, 25(10):1067–1071.
Pehlivan S, Fisher B: Some sequence spaces defined by a modulus. Math. Slovaca 1995, 45(3):275–280.
Bilgin T: f-Asymptotically lacunary equivalent sequences. Acta Univ. Apulensis 2011, 28: 271–278.
Acknowledgements
The authors are thankful to the reviewers of the paper for careful reading and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
Both authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed to the article equally. Both authors also read and approved the final manuscript for publication.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Kumar, V., Sharma, A. Asymptotically lacunary equivalent sequences defined by ideals and modulus function. Math Sci 6, 23 (2012). https://doi.org/10.1186/2251-7456-6-23
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/2251-7456-6-23