We introduce the strong (σ,f)-asymptotically equivalent and strong (σ,f)-asymptotically lacunary equivalent sequences which are some combinations of the definitions for asymptotically equivalent, statistical limit, modulus function, σ-convergence, and lacunary sequences. Then we use these definitions to prove strong (σ,f)-asymptotically equivalent and strong (σ,f)-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 s, l∞, and c denote the spaces of all real sequences, bounded and convergent sequences, respectively. Any subspace of s is called a sequence space.

Let σ be a mapping of the set of positive integers into itself. A continuous linear functional ϕ on l∞, the space of real bounded sequences, is said to be an invariant mean or σ-mean if and only if

ϕ(x)≥0 when the sequence x=(xk) has xk≥0 for all k;

ϕ(e)=1, where e=(1,1,1,…);

ϕ(x)=ϕ(σx) for all x∈l∞, where σx=(xσ(n)). The mapping σ is one to one with σk(n)≠n for all positive integers n and k, where σk denotes the kth iterate of the mapping σ at n. Thus ϕ extends the limit functional on c in the sense that ϕ(x)=limx for all x∈c. If x=(xn), write Tx=Txn=(xσ(n)).

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 f is a function from [0,∞) to [0,∞) such that (i) f(x)=0 if and only if x=0, (ii) f(x+y)≤f(x)+f(y) for x,y≥0, (iii) f is increasing, and (iv) f is continuous from the right at 0. Hence f must be continuous everywhere on [0,∞). 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 θ=(kr) lacunary if it is an increasing sequence of integers such that k0=0,hr=kr-kr-1→∞ as r→∞. The intervals determined by θ will be denoted by Ir=(kr-1,kr] and qr=kr/kr-1. These notations will be used throughout the paper. The sequence space of lacunary strongly convergent sequences Nθ was defined by Freedman et al. [11] as follows:
(1)Nθ=x=xi∈s:hr-1∑i∈Irxi-s=0forsomes.

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 (σ,f)-asymptotically equivalent and strong (σ,f)-asymptotically lacunary equivalent sequences which are some combinations of the definitions for asymptotically equivalent, statistical limit, modulus function, σ-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 [x] is statistically convergent to L if
(2)limn1nthenumberofk≤n:xk-L≥ɛ=0foreveryɛ>0,denotedbyst-limx=L.

Definition 2.

A sequence [x] is strongly (Cesàro) summable to L if
(3)limn1n∑k=1nxk-L=0,denotedbyw-limx=L.

Definition 3.

Let f be any modulus; the sequence [x] is strongly (Cesàro) summable to L with respect to a modulus if
(4)limn1n∑k=1nfxk-L=0,(denotedbywf-limx=L).

Definition 4.

Two nonnegative sequences [x] and [y] are said to be asymptotically equivalent if limk(xk/yk)=1, (denoted by x∽y).

Definition 5.

Two nonnegative sequences [x] and [y] are said to be asymptotically statistical equivalent of multiple L provided that for every ɛ>0(5)limn1nthenumberofk≤n:xkyk-L≥ɛ=0,hhhhhhhhhhhhhhhhhhhhdenotedbyx∽Sy
and simply asymptotically statistical equivalent, if L=1.

Definition 6.

Two nonnegative sequences [x] and [y] are said to be strong asymptotically equivalent of multiple L provided that
(6)limn1n∑k=1nxkyk-L=0(denotedbyx∽wy)
and simply strong asymptotically equivalent, if L=1.

Definition 7.

Let θ be a lacunary sequence; the two nonnegative sequences [x] and [y] are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every ɛ>0(7)limr1hrthenumberofk∈Ir:xkyk-L≥ɛ=0,hhhhhhhhhhhhhhhhhhhhdenotedbyx∽Sθy
and simply asymptotically lacunary statistical equivalent, if L=1.

Definition 8.

Let θ be a lacunary sequence; the two nonnegative sequences [x] and [y] are said to be strong asymptotically lacunary equivalent of multiple L provided that
(8)limr1hr∑k∈Irxkyk-L=0denotedbyx∽Nθy
and simply strong asymptotically lacunary equivalent, if L=1.

Definition 9.

Let f be any modulus; the two nonnegative sequences [x] and [y] are said to be f-asymptotically equivalent of multiple L provided that
(9)limkfxkyk-L=0denotedbyx∽fy
and simply strong f-asymptotically equivalent, if L=1.

Definition 10.

Let f be any modulus; the two nonnegative sequences [x] and [y] are said to be strong f-asymptotically equivalent of multiple L provided that
(10)limn1n∑k=1nfxkyk-L=0denotedbyx∽wfy
and simply strong f-asymptotically equivalent, if L=1.

Definition 11.

Two nonnegative sequences [x] and [y] are said to be Sσ-asymptotically statistical equivalent of multiple L provided that for every ɛ>0(11)limn1nthenumberofk≤n:xσk(i)yσk(i)-L≥ɛ=0,hhuniformlyini=1,2,3,…,(denotedbyx∽Sσy)
and simply σ-asymptotically statistical equivalent, if L=1.

Definition 12.

Let θ be a lacunary sequence; two nonnegative sequences [x] and [y] are said to be Sσ,θ-asymptotically lacunary statistical equivalent of multiple L provided that for every ɛ>0(12)limr1hrthenumberofk∈Ir:xσk(i)yσk(i)-L≥ɛ=0,hhuniformlyini=1,2,3,…,(denotedbyx∽Sσ,θy)
and simply Sσ,θ-asymptotically lacunary statistical equivalent, if L=1.

Definition 13.

Let θ be a lacunary sequence; two nonnegative sequences [x] and [y] are said to be strong σ-asymptotically lacunary equivalent of multiple L provided that
(13)limr1hr∑k∈Irxσk(i)yσk(i)-L=0,uniformlyini=1,2,3,…,denotedbyx∽Nσ,θy
and simply σ-asymptotically lacunary statistical equivalent, if L=1.

Definition 14.

Let f be any modulus and let θ be a lacunary sequence; two nonnegative sequences [x] and [y] are said to be strong f-asymptotically lacunary equivalent of multiple L provided that
(14)limr1hr∑k∈Irfxkyk-L=0,denotedbyx∽Nθ,fy
and simply f-asymptotically lacunary equivalent, if L=1.

Definition 15.

Let f be any modulus and let θ be a lacunary sequence; two nonnegative sequences [x] and [y] are said to be strong (σ,f)-asymptotically equivalent of multiple L provided that
(15)limn1n∑k=1nfxσk(i)yσk(i)-L=0,uniformlyini=1,2,3,…,denotedbyx∽wσ,fy
and simply (σ,f)-asymptotically equivalent, if L=1.

For (i)=i+1 we write [wf] for wσ,f. Hence the two nonnegative sequences [x] and y are said to be strong almost f-asymptotically equivalent of multiple L provided that
(16)limn1n∑k=1nfxk+iyk+i-L=0,uniformlyini=1,2,3,…,denotedbyx∽[wf]y
and simply f-asymptotically equivalent, if L=1.

For f(x)=x for all x∈[0,∞) we write [w] for [wf]. Hence the two nonnegative sequences [x] and [y] are said to be strong almost asymptotically equivalent of multiple L provided that
(17)limn1n∑k=1nxk+iyk+i-L=0uniformlyini=1,2,3,…,denotedbyx∽[wf]y
and simply strong almost asymptotically equivalent, if L=1.

Definition 16.

Let f be any modulus and let θ be a lacunary sequence; two nonnegative sequences [x] and [y] are said to be strong (σ,f)-asymptotically lacunary equivalent of multiple L provided that
(18)limr1hr∑k∈Irfxσk(i)yσk(i)-L=0,uniformlyini=1,2,3,…,denotedbyx∽Nσ,θ,fy
and simply (σ,f)-asymptotically lacunary equivalent, if L=1.

For (i)=i+1 we write Mθ,f for Nσ,θ,f. Hence the two nonnegative sequences [x] and [y] are said to be strong almost f-asymptotically lacunary equivalent of multiple L provided that
(19)limr1hr∑k∈Irfxk+iyk+i-L=0,uniformlyini=1,2,3,…,denotedbyx∽Mθ,fy
and simply strong almost f-asymptotically lacunary equivalent, if L=1.

For f(x)=x for all x∈[0,∞) we write [wθ] for wθ,f. Hence the two nonnegative sequences [x] and [y] are said to be strong almost asymptotically lacunary equivalent of multiple L provided that
(20)limr1hr∑k∈Irxk+iyk+i-L=0,uniformlyini=1,2,3,…,denotedbyx∽Mθy
and simply strong almost asymptotically lacunary equivalent, if L=1.

3. Main Theorems

We now give lemma to be used later.

Lemma 17.

Let f be any modulus. Suppose for given ɛ>0, there exist no and io such that
(21)1n∑k=0n-1fxσk(i)yσk(i)-L<ɛ∀n≥no,i≥io.
Then x∽wσ,fy.

Proof.

Let ɛ>0 be given. Chose n1 and io such that
(22)1n∑k=0n-1fxσk(i)yσk(i)-L<ɛ2∀n≥n1,i≥io.
It is sufficient to prove that there exists n2 such that for n≥n2, io≥i≥0(23)1n∑k=0n-1fxσk(i)yσk(i)-L<ɛ
since, taking no=max(n1,n2), (23) holds for n>no and for all i, which gives the result. Once io has been chosen, io is fixed, so
(24)∑k=0iofxσk(i)yσk(i)-L=Rsay.
Now, taking io≥i≥0 and n>io, we have
(25)1n∑k=0n-1fxσk(i)yσk(i)-L=1n∑k=0io-1∑k=ion-1fxσk(i)yσk(i)-L≤Rn+ɛ2.
Taking n sufficiently large, we can make R/n+ɛ/2<ɛ which gives (23) and hence the result.

The next theorems show the relationship between the strong (σ,f)-asymptotically equivalence and the strong (σ,f)-asymptotically lacunary equivalence.

Theorem 18.

Let f be any modulus. Then x∽Nσ,θ,fy⇔x∽wσ,fy for every lacunary sequence θ.

Proof.

Let x∽Nσ,θ,fy. Then, given ɛ>0, there exist ro and L such that
(26)1hr∑k∈Irfxσk(i)yσk(i)-L<ɛ
for r≥ro and i=kr-1+1+v,v≥0. Let n≥hr; write n=mhr+p, where 0≤p≤hr; m is an integer. Since n≥hr, m≥0. We have
(27)1n∑k=0n-1fxσk(i)yσk(i)-L≤1n∑k=0(m+1)hr-1fxσk(i)yσk(i)-L=1n∑j=0m∑k=jhr(j+1)hr-1fxσkiyσki-L≤m+1hrɛn≤2mhrɛnform≥0.
For hr/n≤1, since mhr/n≤1, therefore,
(28)1n∑k=0n-1fxσk(i)yσk(i)-L≤2ɛ.

Then by lemma x∽Nσ,θ,fy implies x∽wσ,fy. It is easy to see that x∽wσ,fy implies x∽Nσ,θ,fy for every θ.

Proposition 19.

Let f be any modulus. Then x∽Mθ,fy⇔x∽[wf]y for every lacunary sequence θ.

Proof.

It follows from Theorem 18 for σ(i)=i+1 for all i=1,2,3,….

Proposition 20.

x∽Mθy⇔x∽[w]y for every lacunary sequence θ.

Proof.

It follows from Proposition 19 for f(x)=x for all x∈[0,∞).

Theorem 21.

Let f be any modulus. Then

if liminfrqr>1, then x∽wσ,fy implies x∽Nσ,θ,fy;

if limsuprqr<∞, then x∽Nσ,θ,fy implies x∽wσ,fy;

if 1<liminfrqr≤limsuprqr<∞, then x∽wσ,fy⇔x∽Nσ,θ,fy.

Proof.

Part (1): let x∽wσ,fy and liminfrqr>1. There exists δ>0 such that qr=(kr/kr-1)≥1+δ for sufficiently large r. We have, for sufficiently large r, that (hr/kr)≥δ/(1+δ). Then
(29)1kr-1∑k=1krfxσk(i)yσk(i)-L≥1kr-1∑k∈Irfxσk(i)yσk(i)-L=hrkr1hr∑k∈Irfxσk(i)yσk(i)-L≥δ1+δ1hr∑k∈Irfxσk(i)yσk(i)-L
which yields that x∽Nσ,θ,fy.

Part (2): if limsuprqr<∞, then there exists K>0 such that qr<K for every r.

Now suppose that x∽Nσ,θ,fy and ɛ>0. There exists m0 such that for every m≥m0,
(30)Hm=1hm∑k∈Imfxσk(i)yσk(i)-L<ɛ,∀i.

We can also find R>0 such that Hm≤R for all m. Let n be any integer with kr≥n>kr-1 where r>m0. Now write
(31)1n∑k=0n-1fxσkiyσki-L≤1kr-1∑k=1krfxσk(i)yσk(i)-L=1kr-1∑k∈I1fxσkiyσki-L+∑k∈I2fxσkiyσki-L+⋯+∑k∈Irfxσkiyσki-L=1kr-1k1k1∑k∈I1fxσkiyσki-L+k2-k1k2-k1×∑k∈I2fxσkiyσki-L+⋯+km0-km0-1km0-km0-1∑k∈Im0fxσk(i)yσk(i)-L+⋯+kr-kr-1kr-kr-1∑k∈Irfxσk(i)yσk(i)-L=k1kr-1H1+k2-k1kr-1H2+⋯+km0-km0-1kr-1Hm0+⋯+kr-kr-1kr-1Hr≤km0kr-1sup1<k<km0Hk+supk>km0Hk(kr-km0)kr-1<Rkm0kr-1+ɛK
from which we deduce that x∽wσ,fy.

Part (3): this immediately follows from (1) and (2).

Proposition 22.

Let f be any modulus. Then

if liminfrqr>1, then x∽[wf]y implies x∽Mθ,fy;

if limsuprqr<∞, then x∽Mθ,fy implies x∽[wf]y;

if 1<liminfrqr≤limsuprqr<∞, then x∽[wf]y⇔x∽Mθ,fy.

Proof.

It follows from Theorem 21 for σ(i)=i+1 for all i=1,2,3,….

Proposition 23.

Consider the following:

if liminfrqr>1, then x∽[w]y implies x∽Mθy;

if limsuprqr<∞, then x∽Mθy implies x∽[w]y;

if 1<liminfrqr≤limsuprqr<∞, then x∽[w]y⇔x∽Mθy.

Proof.

It follows from Proposition 22 for f(x)=x for all x∈[0,∞).

In the following theorem we study the relationship between the strong (σ,f)-asymptotically lacunary equivalence and the strong σ-asymptotically lacunary equivalence.

Theorem 24.

Let f be any modulus. Then

if x∽Nσ,θy, then x∽Nσ,θ,fy;

if limt→∞(ft/t)=β>0, then x∽Nσ,θy⇔x∽Nσ,θ,fy.

Proof.

Part (1): let x∽Nσ,θy and ɛ>0. We choose 0<δ<1 such that f(u)<ɛ for every u with 0≤u≤δ. We can write
(32)1hr∑k∈Irfxσk(i)yσk(i)-L=1hr∑1fxσk(i)yσk(i)-L+1hr∑2fxσkiyσki-L,
where the first summation is over xσk(i)/yσk(i)-L≤δ and the second summation over xσk(i)/yσk(i)-L>δ with k∈Ir. By definition of f, we have
(33)1hr∑k∈Irfxσk(i)yσk(i)-L≤ɛ+2f1δ-11hr∑k∈Irfxσkiyσki-L.

Therefore, x∽Nσ,θ,fy.

Part (2): if limt→∞(f(t)/t)=β>0, then f(t)≥βt for all t>0. Let x∽Nσ,θ,fy; clearly
(34)1hr∑k∈Irfxσk(i)yσk(i)-L≥1hr∑k∈Irβxσk(i)yσk(i)-L=β1hr∑k∈Irxσkiyσki-L.
Therefore, x∽Nσ,θy. By using (23) the proof is complete.

Proposition 25.

Let f be any modulus. Then

if x∽Mθy, then x∽Mθ,fy;

if limt→∞(f(t)/t)=β>0, then x∽Mθy⇔x∽Mθ,fy.

Proof.

It follows from Theorem 24 for σ(i)=i+1 for all i=1,2,3,….

Finally we give the relation between Sσ,θ-asymptotically lacunary statistical equivalence and strong (σ,f)-asymptotically lacunary equivalence. Also we give relation between Sσ,θ-asymptotically lacunary statistical equivalence and strong (σ,f)-asymptotically equivalence.

Theorem 26.

Let f be any modulus. Then

if x∽Nσ,θ,fy, then x∽Sσ,θy;

if f is bounded and x∽Sσ,θy, then x∽Nσ,θ,fy;

if f is bounded, then x∽Nσ,θ,fy⇔x∽Sσ,θy.

Proof.

Part (1): take ɛ>0 and x∽Nσ,θ,fy. Let ∑1 denote the sum over k∈Ir and xσk(i)/yσk(i)-L≥ɛ.

Then
(35)1hr∑k∈Irfxσk(i)yσk(i)-L≥1hr∑1fxσk(i)yσk(i)-L≥fɛthenumberofk∈Ir:xσkiyσki-L≥ɛ.

Therefore, x∽Sσ,θy.

Part (2): suppose that f is bounded and x∽Sσ,θy. Since f is bounded, there exists an integer T such that f(x)≤T for all x≥0. Given ɛ>0,
(36)1hr∑k∈Irfxσk(i)yσk(i)-L≤T1hrthenumberofk∈Ir:xσkiyσki-L≥ɛ+f(ɛ).

Therefore, x∽Nσ,θ,fy.

Part (3): follows from (1) and (2).

For (i)=i+1 we write [Sθ] for Sσ,θ. Hence the two nonnegative sequences [x] and [y] are said to be [Sθ]-asymptotically lacunary statistical equivalent of multiple L provided that for every ɛ>0(37)limr1hrthenumberofk∈Ir:xσkiyσki-L≥ɛ=0,uniformlyini=1,2,3,…,(denotedbyx∽[Sθ]y)
and [Sθ]-asymptotically lacunary statistical equivalent, if L=1. Hence we have the following.

Proposition 27.

Let f be any modulus. Then

if x∽Mθ,fy, then x∽[Sθ]y;

if f is bounded and x∽[Sθ]y, then x∽Mθ,fy;

if f is bounded, then x∽Mθ,fy⇔x∽[Sθ]y.

Proof.

It follows from Theorem 26 for σ(i)=i+1 for all i=1,2,3,….

Proposition 28.

x∽Mθy implies x∽[Sθ]y.

Proof.

It follows from Proposition 27(1) for f(x)=x for all x∈[0,∞).

Theorem 29.

Let f be bounded; then x∽Sσ,θy implies x∽wσ,fy for every lacunary sequence θ.

Proof.

Let n be any integer with n∈Ir; then
(38)1n∑k=1nfxσk(i)yσk(i)-L=1n∑p=1r-1∑k∈Ipfxσk(i)yσk(i)-L+1n∑k=1+kr-1nfxσkiyσki-L.

Consider the first term on the right in (38):
(39)1n∑p=1r-1∑k∈Ipfxσkiyσki-L≤1kr-1∑p=1r-1∑k∈Ipfxσk(i)yσk(i)-L=1kr-1∑p=1r-1hp1hp∑k∈Ipfxσkiyσki-L.

Since f is bounded and x∽Sσ,θy, it follows from Theorem 26(2) that
(40)1hp∑k∈Ipfxσk(i)yσk(i)-L⟶0.

Hence
(41)1kr-1∑p=1r-1hp1hp∑k∈Ipfxσkiyσki-L⟶0.

Consider the second term on the right in (38); since f is bounded, there exists an integer T such that f(x)≤T for all x≥0. We split the sum for kr-1<k≤n into sums over xσk(i)/yσk(i)-L≥ɛ and xσki/yσki-L<ɛ. Therefore, we have for every ɛ>0 that
(42)1n∑k=1+kr-1nfxσk(i)yσk(i)-L≤T1hrthenumberofk∈Ir:xσkiyσki-L≥ɛ+f(ɛ).

Since x∽Sσ,θy,f is continuous from the right at 0 and ɛ is arbitrary; the expression on left side of (42) tends to zero as r→∞, uniformly in i. Hence (38), (41), and (42) imply that x∽wσ,fy.

Proposition 30.

Let f be bounded; then x∽[Sθ]y implies x∽[wf]y for every lacunary sequence θ.

Proof.

It follows from Theorem 29 for σ(i)=i+1 for all i=1,2,3,….

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.

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