Abstract

We introduce the strong -asymptotically equivalent and strong -asymptotically lacunary equivalent sequences which are some combinations of the definitions for asymptotically equivalent, statistical limit, modulus function, -convergence, and lacunary sequences. Then we use these definitions to prove strong -asymptotically equivalent and strong -asymptotically lacunary equivalent analogues of Connor’s results in Connor, 1988, Fridy and Orhan’s results in Fridy and Orhan, 1993, and Das and Patel’s results in Das and Patel, 1989.

1. Introduction

Let , , and denote the spaces of all real sequences, bounded and convergent sequences, respectively. Any subspace of is called a sequence space.

Let be a mapping of the set of positive integers into itself. A continuous linear functional on , the space of real bounded sequences, is said to be an invariant mean or -mean if and only if(i) when the sequence has for all ;(ii), where ;(iii) for all , where . The mapping is one to one with for all positive integers and , where denotes the th iterate of the mapping at . Thus extends the limit functional on in the sense that for all . If , write .

Several authors including Bilgin [1], Mursaleen [2], Savas [3], Schaefer [4], and others have studied invariant convergent sequences.

The notion of modulus function was introduced by Nakano [5]. We recall that a modulus is a function from to such that (i) if and only if , (ii) for , (iii) is increasing, and (iv) is continuous from the right at 0. Hence must be continuous everywhere on . Kolk [6], Maddox [7], Öztürk and Bilgin [8], Pehlivan and Fisher [9], Ruckle [10], and others used a modulus function to construct sequence spaces.

Following Freedman et al. [11], we call the sequence lacunary if it is an increasing sequence of integers such that as . The intervals determined by will be denoted by and . These notations will be used throughout the paper. The sequence space of lacunary strongly convergent sequences was defined by Freedman et al. [11] as follows:

Lacunary convergent sequences have been studied by Bilgin [12], Das and Mishra [13], Das and Patel [14], Savas and Patterson [15], and others. Marouf presented definitions for asymptotically equivalent sequences and asymptotic regular matrices in [16]. Patterson extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices in [17].

Recently, the concept of asymptotically equivalent was generalized by Bilgin [12], Kumar and Sharma [18], Patterson and Savas [19], Savas and Basarr [20], and Patterson and Savas [15]. In this paper we introduce the strong -asymptotically equivalent and strong -asymptotically lacunary equivalent sequences which are some combinations of the definitions for asymptotically equivalent, statistical limit, modulus function, -convergence, and lacunary sequences.

In addition to these definitions, natural inclusion theorems will also be presented.

2. Definitions and Notations

Now we recall some definitions of sequence spaces (see [7, 12, 15–17, 21–25]).

Definition 1. A sequence is statistically convergent to if

Definition 2. A sequence is strongly (Cesàro) summable to if

Definition 3. Let be any modulus; the sequence is strongly (Cesàro) summable to with respect to a modulus if

Definition 4. Two nonnegative sequences and are said to be asymptotically equivalent if , (denoted by ).

Definition 5. Two nonnegative sequences and are said to be asymptotically statistical equivalent of multiple provided that for every and simply asymptotically statistical equivalent, if .

Definition 6. Two nonnegative sequences and are said to be strong asymptotically equivalent of multiple provided that and simply strong asymptotically equivalent, if .

Definition 7. Let be a lacunary sequence; the two nonnegative sequences and are said to be asymptotically lacunary statistical equivalent of multiple provided that for every and simply asymptotically lacunary statistical equivalent, if .

Definition 8. Let be a lacunary sequence; the two nonnegative sequences and are said to be strong asymptotically lacunary equivalent of multiple provided that and simply strong asymptotically lacunary equivalent, if .

Definition 9. Let be any modulus; the two nonnegative sequences and are said to be -asymptotically equivalent of multiple provided that and simply strong -asymptotically equivalent, if .

Definition 10. Let be any modulus; the two nonnegative sequences and are said to be strong -asymptotically equivalent of multiple provided that and simply strong -asymptotically equivalent, if .

Definition 11. Two nonnegative sequences and are said to be -asymptotically statistical equivalent of multiple provided that for every and simply -asymptotically statistical equivalent, if .

Definition 12. Let be a lacunary sequence; two nonnegative sequences and are said to be -asymptotically lacunary statistical equivalent of multiple provided that for every and simply -asymptotically lacunary statistical equivalent, if .

Definition 13. Let be a lacunary sequence; two nonnegative sequences and are said to be strong -asymptotically lacunary equivalent of multiple provided that and simply -asymptotically lacunary statistical equivalent, if .

Definition 14. Let be any modulus and let be a lacunary sequence; two nonnegative sequences and are said to be strong -asymptotically lacunary equivalent of multiple provided that and simply -asymptotically lacunary equivalent, if .

Definition 15. Let be any modulus and let be a lacunary sequence; two nonnegative sequences and are said to be strong -asymptotically equivalent of multiple provided that and simply -asymptotically equivalent, if .
For we write for . Hence the two nonnegative sequences and are said to be strong almost -asymptotically equivalent of multiple provided that and simply -asymptotically equivalent, if .
For for all we write for . Hence the two nonnegative sequences and are said to be strong almost asymptotically equivalent of multiple provided that and simply strong almost asymptotically equivalent, if .

Definition 16. Let be any modulus and let be a lacunary sequence; two nonnegative sequences and are said to be strong -asymptotically lacunary equivalent of multiple provided that and simply -asymptotically lacunary equivalent, if .
For we write for . Hence the two nonnegative sequences and are said to be strong almost -asymptotically lacunary equivalent of multiple provided that and simply strong almost -asymptotically lacunary equivalent, if .
For for all we write for . Hence the two nonnegative sequences and are said to be strong almost asymptotically lacunary equivalent of multiple provided that and simply strong almost asymptotically lacunary equivalent, if .

3. Main Theorems

We now give lemma to be used later.

Lemma 17. Let be any modulus. Suppose for given , there exist and such that Then .

Proof. Let be given. Chose and such that It is sufficient to prove that there exists such that for , since, taking , (23) holds for and for all , which gives the result. Once has been chosen, is fixed, so Now, taking and , we have Taking sufficiently large, we can make which gives (23) and hence the result.

The next theorems show the relationship between the strong -asymptotically equivalence and the strong -asymptotically lacunary equivalence.

Theorem 18. Let be any modulus. Then for every lacunary sequence .

Proof. Let . Then, given , there exist and such that for and . Let ; write , where ; is an integer. Since , . We have For , since , therefore,
Then by lemma implies . It is easy to see that implies for every .

Proposition 19. Let be any modulus. Then for every lacunary sequence .

Proof. It follows from Theorem 18 for for all .

Proposition 20. for every lacunary sequence .

Proof. It follows from Proposition 19 for for all .

Theorem 21. Let be any modulus. Then(1)if , then implies ;(2)if , then implies ;(3)if , then .

Proof. Part (1): let and . There exists such that for sufficiently large . We have, for sufficiently large , that . Then which yields that .
Part (2): if , then there exists such that for every .
Now suppose that and . There exists such that for every ,
We can also find such that for all . Let be any integer with where . Now write from which we deduce that .
Part (3): this immediately follows from and .

Proposition 22. Let be any modulus. Then(1)if , then implies ;(2)if , then implies ;(3)if , then .

Proof. It follows from Theorem 21 for for all .

Proposition 23. Consider the following:if , then implies ;if , then implies ;if , then .

Proof. It follows from Proposition 22 for for all .

In the following theorem we study the relationship between the strong -asymptotically lacunary equivalence and the strong -asymptotically lacunary equivalence.

Theorem 24. Let be any modulus. Then(1)if , then ;(2)if , then .

Proof. Part : let and . We choose such that for every with . We can write where the first summation is over and the second summation over with . By definition of , we have
Therefore, .
Part : if , then for all . Let ; clearly Therefore, . By using (23) the proof is complete.

Proposition 25. Let be any modulus. Then(1)if , then ;(2)if , then .

Proof. It follows from Theorem 24 for for all .

Finally we give the relation between -asymptotically lacunary statistical equivalence and strong -asymptotically lacunary equivalence. Also we give relation between -asymptotically lacunary statistical equivalence and strong -asymptotically equivalence.

Theorem 26. Let be any modulus. Then(1)if , then ;(2)if is bounded and , then ;(3)if is bounded, then .

Proof. Part : take and . Let denote the sum over and .
Then
Therefore, .
Part : suppose that is bounded and . Since is bounded, there exists an integer such that for all . Given ,
Therefore, .
Part : follows from and .
For we write for . Hence the two nonnegative sequences and are said to be -asymptotically lacunary statistical equivalent of multiple provided that for every and -asymptotically lacunary statistical equivalent, if . Hence we have the following.

Proposition 27. Let be any modulus. Then(1)if , then ;(2)if is bounded and , then ;(3)if is bounded, then .

Proof. It follows from Theorem 26 for for all .

Proposition 28. implies .

Proof. It follows from Proposition 27 for for all .

Theorem 29. Let be bounded; then implies for every lacunary sequence .

Proof. Let be any integer with ; then
Consider the first term on the right in (38):
Since is bounded and , it follows from Theorem 26 that
Hence
Consider the second term on the right in (38); since is bounded, there exists an integer such that for all . We split the sum for into sums over and . Therefore, we have for every that
Since is continuous from the right at and is arbitrary; the expression on left side of (42) tends to zero as , uniformly in . Hence (38), (41), and (42) imply that .

Proposition 30. Let be bounded; then implies for every lacunary sequence .

Proof. It follows from Theorem 29 for for all .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work is supported by the Presidency of Scientific Research Projects of Yüzüncü Yıl University (IECMSA-2013). The author is thankful to the anonymous referees for careful reading and useful suggestions.