We introduce and study new sequence spaces which arise from the notions of generalized de la Vallée-Poussin means, invariant means, and modulus functions.

1. Introduction

Let w be the set of all real or complex sequences and let l∞, c, and c0 be the Banach spaces of bounded, convergent, and null sequences x=(xk), respectively, with the usual norm ∥x∥=supn|xn|.

A sequence x=(xk)∈l∞ is said to be almost convergent if its Banach limit coincides. Let ĉ denote the space of all almost convergent sequences. Lorentz [1] proved that ĉ={x∈l∞:limmtmn(x)exist uniformly inn},
where tmn(x)=xn+xn+1+⋯+xn+mm+1.

The space [ĉ] of strongly almost convergent sequences was introduced by Maddox [2] as[ĉ]={x∈l∞:limmtmn(|x-le|)exist uniformly innfor somel∈C},
where e=(1,1,…).

Let σ be a one-to-one mapping from the set of positive integers into itself such that σm(n)=σm-1(σ(n)), m=1,2,3,…, where σm(n) denotes the mth iterate of the mapping σ in n, see [3]. A continuous linear functional φ on l∞ is said to be an invariant mean or a σ-mean, if and only if,

φ(x)≥0, when the sequence x=(xn) is such that xn≥0 for all n,

φ(e)=1, where e=(1,1,…),

φ(xσ(n))=φ(x), for all x∈l∞.

For a certain kind of mapping σ, every invariant mean φ extends the functional limit on the space c, in the sense that φ(x)=limx for all x∈c. Consequently, c⊂Vσ, where Vσ is the set of bounded sequences with equal σ-means. Schaefer [3] proved thatVσ={x∈l∞:limktkm(x)=L uniformlyinmfor someL=σ-limx},
where tkm(x)=xm+xσ(m)+⋯+xσk(m)k+1,t-1,m=0.
Thus we say that a bounded sequence x=(xk) is σ-convergent, if and only if, x∈Vσ such that σk(n)≠n for all n≥0, k≥1. Note that similarly as the concept of almost convergence leads naturally to the concept of strong almost convergence, the σ-convergence leads naturally to the concept of strong σ-convergence.

A sequence x=(xk) is said to be strongly σ-convergent (see, Mursaleen [4]), if there exists a number ℓ such that1k∑i=1k|xσi(m)-l|⟶0,
as k→∞ uniformly in m. We write [Vσ] to denote the set of all strong σ-convergent sequences and when (1.6) holds, we write [Vσ]-limx=ℓ. Taking σ(m)=m+1, we obtain [Vσ]=[ĉ]. Then the strong σ-convergence generalizes the concept of strong almost convergence. We also note that [Vσ]⊂Vσ⊂l∞.
It is also well known that the concept of paranorm is closely related to linear metric spaces. In fact, it is a generalization of absolute value. Let X be a linear space. A function p:X→ℝ is called a paranorm, if

p(0)≥0,

p(x)≥0, for all x∈X,

p(-x)=p(x), for all x∈X,

p(x+y)≤p(x)+p(y), for all x,y∈X (triangle inequality),

if (λn) is a sequence of scalars, with λn→λ(n→∞), and (xn) is a sequence of vectors with p(xn-x)→0(n→∞), then p(λnxn-λx)→0(n→∞) (continuity of multiplication by scalars).

A complete linear metric space is said to be a Fréchet space. A Fréchet sequence space X is said to be an FK space, if its metric is stronger than the metric of w on X, that is, convergence in the sequence space X implies coordinatewise convergence (the letters F and K stand for Fréchet and Koordinate, the German word for coordinate).

Note that, by Ruckle in [5], a modulus function f is a function from [0,∞) to [0,∞) such that

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

f(x+y)≤f(x)+f(y), for all x,y≥0,

f increasing,

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,∞). Furthermore, from condition (ii), we have f(nx)≤nf(x) for all n∈ℕ, and thus f(x)=f(nx1n)≤nf(xn),
hence 1nf(x)≤f(xn),∀n∈N.
In [5], Ruckle used the idea of a modulus function f in order to construct a class of FK spaces L(f)={x=(xk):∑k=1∞f(|xk|)<∞}.
From the definition, we can easily see that the space L(f) is closely related to the space l1, if we consider f(x)=x for all real numbers x≥0. Several authors study these types of spaces. For example, Maddox introduced and examined some properties of the sequence spaces w0(f), w(f) and w∞(f), defined by using a modulus f, which generalized the well-known spaces w0, w and w∞ of strongly summable sequences, see [6]. Similarly, Savaş in [7] generalized the concept of strong almost convergence by using a modulus f and examined some further properties of the corresponding new sequence spaces.

The generalized de la Vallé-Poussin mean is defined bytn(x)=1λn∑k∈Inxk,
where In=[n-λn+1,n] for n=1,2,…. Then a sequence x=(xk) is said to be (V,λ)- summable to a number L (see [8]), if tn(x)→L as n→∞, and we write[V,λ]0={x:limn1λn∑k∈In|xk|=0},[V,λ]={x:x-le∈[V,λ]0for somel∈C},[V,λ]∞={x:supn1λn∑k∈In|xk|<∞},
for the sets of sequences that are, respectively, strongly summable to zero, strongly summable, and strongly bounded by the de la Vallé-Poussin method. In the special case where λn=n, for n=1,2,3,…, the sets [V,λ]0, [V,λ], and [V,λ]∞ reduce to the sets w0, w, and w∞, which were introduced and studied by Maddox, see [6].

We also note that the sets of sequence spaces such as strongly σ-summable to zero, strongly σ-summable, and strongly σ-bounded with respect to the modulus function were defined by Nuray and Savaş in [9].

2. Main Results

Let p=(pk) be a sequence of real numbers such that pk>0 for all k, and supkpk<∞. This assumption is made throughout the rest of this paper. Then we now write[Vσ,λ,f,p]0={x:limn1λn∑k∈In{f(|xσk(m)|)}pk=0,uniformly inm},[Vσ,λ,f,p]={x:x-le∈[Vσ,λ,f,p]0for somel∈C},[Vσ,λ,f,p]∞={x:supn,m1λn∑k∈In{f(|xσk(m)|)}pk<∞}.
In particular, if we take pk=1 for all k, we have[Vσ,λ,f]0={x:limn1λn∑k∈Inf(|xσk(m)|)=0,uniformly inm},[Vσ,λ,f]={x:x-le∈[Vσ,λ,f]0for somel∈C},[Vσ,λ,f]∞={x:supn,m1λn∑k∈Inf(|xσk(m)|)<∞}.
Similarly, when σ(m)=m+1, then [Vσ,λ,f,p]0, [Vσ,λ,f,p] and [Vσ,λ,f,p]∞ are reduced to[V̂,λ,f,p]0={x:limn1λn∑k∈In{f(|xk+m|)}pk=0,uniformly inm},[V̂,λ,f,p]={x:x-le∈[V̂,λ,f,p]0for somel∈C},[V̂,λ,f,p]∞={x:supn,m1λn∑k∈In{f(|xk+m|)}pk<∞},respectively.
In particular, when pk=p for all k, then we have the spaces [V̂,λ,f,p]0=[V̂,λ,f]0,[V̂,λ,f,p]=[V̂,λ,f],[V̂,λ,f,p]∞=[V̂,λ,f]∞,
which were introduced and studied by Malkowsky and Savaş in [10]. Further, when λn=n, for n=1,2,3,…, the sets [V̂,λ,f]0 and [V̂,λ,f] are reduced to [ĉ(f)] and [ĉ0(f)] respectively, see [7]. Now, if we consider f(x)=x, then one can easily obtain [Vσ,λ,p]0={x:limn1λn∑k∈In|xσk(m)|pkuniformly inm},[Vσ,λ,p]={x:x-le∈[V̂σ,λ,p]0 for somel∈C},[Vσ,λ,p]∞={x:supn,m1λn∑k∈In|xσk(m)|pk<∞}.
If pk=1 for all k, then we can obtain the spaces [Vσ,λ]0, [Vσ,λ], and [Vσ,λ]∞. Throughout this paper, we use the notation f(|xk|)pk instead of {f(|xk|)}pk.

If p∈l∞, then it is clear that [Vσ,λ,f,p]0, [Vσ,λ,f,p], and [V,σ,λ,f,p]∞ are linear spaces over the complex field ℂ.

Lemma 2.1.

Let f be any modulus. Then
[Vσ,λ,f]∞=l∞σ(f)={x∈w:(f(|xσk(m)|))∈l∞}.

Proof.

Let x∈[Vσ,λ,f]∞. Then there is a constant M>0 such that
1λ1f(|xσk(m)|)≤supm,n1λn∑k∈Inf(|xσk(m)|)≤M,
for all m, and so (f(|xσk(m)|))∈l∞. Let x∈ℓ∞σ(f). Then there is a constant M>0 such that (f(|xσk(m)|))≤M for all k and m, and so
1λn∑k∈Inf(|xσk(m)|)≤M1λn∑k∈In1≤M,
for all m and n. Thus x∈[Vσ,λ,f]∞. This completes the proof.

If x∈[Vσ,λ,f,p], with (1/λn)∑k∈Inf(|xσk(m)-ℓe|)pk→0 as n→∞ uniformly in m, then we write xk→l[Vσ,λ,f,p].

The following well-known inequality ([11], page 190) will be used later.

If 0≤pk≤suppk=H and C=max(1,2H-1), then|ak+bk|pk≤C{|ak|pk+|bk|pk},
for all kand ak,bk∈ℂ.

In the following theorem, we prove xk→ℓ implies xk→ℓ∈[Vσ,λ,f,p] and we also prove the uniqueness of the limit ℓ. To prove the theorem, we need the following lemma.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B3">2</xref>]).

Let pk>0, qk>0. Then co(q)⊂c0(p), if and only if, limk→∞infpk/qk>0, where c0(p)={x:|xk|pk→0ask→∞}.

Note that no other relation between (pk) and (qk) is needed in Lemma 2.2.

Theorem 2.3.

Let limk→∞infpk>0. Then xk→ℓ implies xk→ℓ∈[Vσ,λ,f,p]. Let limk→∞pk=r>0. If xk→ℓ∈[Vσ,λ,f,p], then ℓ is unique.

Proof.

Let xk→ℓ. By the definition of modulus, we have f(|xk-ℓ|)→0. Since limk→∞infpk>0, it follows from the above lemma that f(|xk-ℓ|)pk→0 and consequently, xk→ℓ∈[Vσ,f,p].

Let limk→∞pk=r>0. Suppose that xk→ℓ1∈[Vσ,λ,f,p], xk→ℓ2∈[Vσ,λ,f,p] and |ℓ1-ℓ2|pk=a>0. Now, from (2.9) and the definition of modulus, we have
1λn∑k∈Inf(|l1-l2|)pk≤Cλn∑k∈Inf(|xσk(m)-l1|)pk+Cλn∑k∈Inf(|xσk(m)-l2|)pk.
Hence,
1λn∑k∈Inf(|l1-l2|)pk=0.
Further, f(|ℓ1-ℓ2|)pk→f(a)r as k→∞ and, therefore,
limn→∞1λn∑k∈Inf(|l1-l2|)pk=f(a)r.
From (2.11) and (2.12), it follows that f(a)=0 and by the definition of modulus, we have a=0. Hence ℓ1=ℓ2 and this completes the proof.

Theorem 2.4.

(i) Let 0<infkpk≤pk≤1. Then,
[Vσ,λ,f,p]⊂[Vσ,λ,f].

(ii) Let 0<pk≤supkpk<∞. Then,
[Vσ,λ,f]⊂[Vσ,λ,f,p].

Proof.

(i) Let x∈[Vσ,λ,f,p]. Since 0<infkpk≤1, we get
1λn∑k∈In{f(|xσk(m)-le|)}≤1λn∑k∈In{f(|xσk(m)-le|)}pk,
and hence x∈[Vσ,λ,f].

(ii) Let p≥1 for each k, and supkpk<∞. Let x∈[Vσ,λ,f]. Then, for each k, 0<ɛ<1, there exists a positive integer N such that
1λn∑k∈In{f(|xσk(m)-le|)}≤ɛ<1,
for all m≥N. This implies that
1λn∑k∈In{f(|xσk(m)-le|)}pk≤1λn∑k∈In{f(|xσk(m)-le|)}.
Therefore, x∈[Vσ,λ,f,p]. This completes the proof.

Finally, we conclude this paper by stating the following theorem. We omit the proof, since it involves routine verification and can be obtained by using standard techniques.

Theorem 2.5.

[Vσ,λ,f,p]0 and [Vσ,λ,f,p] are complete linear topological spaces, with paranorm g, where g is defined by
g(x)=supm,n(1λn∑k∈In{f(|xσk(m)|)}pk)M,
where M=max(1,{supkpk}).

Acknowledgment

The authors express their sincere thanks to the referee(s) for careful reading of the paper and several helpful suggestions.

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