A Note on Some Strongly Sequence Spaces

Abstract

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=(𝑥_𝑘), respectively, with the usual norm ‖𝑥‖=sup_𝑛|𝑥_𝑛|.

A sequence 𝑥=(𝑥_𝑘)∈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_∞∶lim_𝑚𝑡_𝑚𝑛(𝑥)existuniformlyin𝑛, (1.1) where 𝑡_𝑚𝑛(𝑥)=𝑥_𝑛+𝑥_𝑛+1+⋯+𝑥_𝑛+𝑚.𝑚+1 (1.2)

The space [𝑐]̂ of strongly almost convergent sequences was introduced by Maddox [2] as[𝑐]̂=𝑥∈𝑙_∞∶lim_𝑚𝑡_𝑚𝑛(|𝑥−ℓ𝑒|)existuniformlyin𝑛forsomeℓ∈ℂ, (1.3) where 𝑒=(1,1,…).

Let 𝜎 be a one-to-one mapping from the set of positive integers into itself such that 𝜎^𝑚(𝑛)=𝜎^𝑚−1(𝜎(𝑛)), 𝑚=1,2,3,…, where 𝜎^𝑚(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, (i)𝜑(𝑥)≥0, when the sequence 𝑥=(𝑥_𝑛) is such that 𝑥_𝑛≥0 for all 𝑛,(ii)𝜑(𝑒)=1, where 𝑒=(1,1,…),(iii)𝜑(𝑥_𝜎(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⊂𝑉_𝜎, where 𝑉_𝜎 is the set of bounded sequences with equal 𝜎-means. Schaefer [3] proved that𝑉_𝜎=𝑥∈𝑙_∞∶lim_𝑘𝑡_𝑘𝑚(𝑥)=𝐿uniformlyin𝑚forsome𝐿=𝜎−lim𝑥, (1.4) where 𝑡_𝑘𝑚(𝑥)=𝑥_𝑚+𝑥_𝜎(𝑚)+⋯+𝑥_{𝜎𝑘(𝑚)}𝑘+1,𝑡_−1,𝑚=0. (1.5) Thus we say that a bounded sequence 𝑥=(𝑥_𝑘) is 𝜎-convergent, if and only if, 𝑥∈𝑉_𝜎 such that 𝜎^𝑘(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=(𝑥_𝑘) is said to be strongly 𝜎-convergent (see, Mursaleen [4]), if there exists a number ℓ such that1𝑘^𝑘_𝑖=1𝑥_{𝜎𝑖(𝑚)}−ℓ⟶0, (1.6) as 𝑘→∞ uniformly in 𝑚. We write [𝑉_𝜎] to denote the set of all strong 𝜎-convergent sequences and when (1.6) holds, we write [𝑉_𝜎]−lim𝑥=ℓ. Taking 𝜎(𝑚)=𝑚+1, we obtain [𝑉_𝜎]=[𝑐]̂. Then the strong 𝜎-convergence generalizes the concept of strong almost convergence. We also note that 𝑉_𝜎⊂𝑉_𝜎⊂𝑙_∞. (1.7) 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:1)𝑝(0)≥0, (P:2)𝑝(𝑥)≥0, for all 𝑥∈𝑋, (P:3)𝑝(−𝑥)=𝑝(𝑥), for all 𝑥∈𝑋, (P:4)𝑝(𝑥+𝑦)≤𝑝(𝑥)+𝑝(𝑦), for all 𝑥,𝑦∈𝑋 (triangle inequality), (P:5) if (𝜆_𝑛) is a sequence of scalars, with 𝜆_𝑛→𝜆(𝑛→∞), and (𝑥_𝑛) is a sequence of vectors with 𝑝(𝑥_𝑛−𝑥)→0(𝑛→∞), then 𝑝(𝜆_𝑛𝑥_𝑛−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 (i)𝑓(𝑥)=0, if and only if, 𝑥=0, (ii)𝑓(𝑥+𝑦)≤𝑓(𝑥)+𝑓(𝑦), for all 𝑥,𝑦≥0, (iii)𝑓 increasing, (iv)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)1=𝑓𝑛𝑥𝑛𝑥≤𝑛𝑓𝑛, (1.8) hence 1𝑛𝑓(𝑥)𝑥≤𝑓𝑛,∀𝑛∈ℕ. (1.9) In [5], Ruckle used the idea of a modulus function 𝑓 in order to construct a class of 𝐹𝐾 spaces 𝐿𝑓=𝑥=𝑥_𝑘∶^∞_𝑘=1𝑓𝑥_𝑘<∞. (1.10) 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 by𝑡_𝑛(𝑥)=1𝜆_{𝑛𝑘∈𝐼𝑛}𝑥_𝑘, (1.11) where 𝐼_𝑛=[𝑛−𝜆_𝑛+1,𝑛] for 𝑛=1,2,…. Then a sequence 𝑥=(𝑥_𝑘) is said to be (𝑉,𝜆)- summable to a number 𝐿 (see [8]), if 𝑡_𝑛(𝑥)→𝐿 as 𝑛→∞, and we write[𝑉,𝜆]₀=𝑥∶lim_𝑛1𝜆_{𝑛𝑘∈𝐼𝑛}𝑥_𝑘,[]=[]=0𝑉,𝜆𝑥∶𝑥−ℓ𝑒∈𝑉,𝜆₀,[]forsomeℓ∈ℂ𝑉,𝜆_∞=𝑥∶sup_𝑛1𝜆_{𝑛𝑘∈𝐼𝑛}𝑥_𝑘<∞, (1.12) 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 𝜆_𝑛=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=(𝑝_𝑘) be a sequence of real numbers such that 𝑝_𝑘>0 for all 𝑘, and sup_𝑘𝑝_𝑘<∞. This assumption is made throughout the rest of this paper. Then we now write𝑉_𝜎,𝜆,𝑓,𝑝₀=𝑥∶lim_𝑛1𝜆_{𝑛𝑘∈𝐼𝑛}𝑓𝑥_{𝜎𝑘(𝑚)}^𝑝𝑘,𝑉=0,uniformlyin𝑚_𝜎,𝜆,𝑓,𝑝=𝑥∶𝑥−ℓ𝑒∈𝑉_𝜎,𝜆,𝑓,𝑝₀,𝑉forsomeℓ∈ℂ_𝜎,𝜆,𝑓,𝑝_∞=𝑥∶sup_𝑛,𝑚1𝜆_{𝑛𝑘∈𝐼𝑛}𝑓𝑥_{𝜎𝑘(𝑚)}^𝑝𝑘<∞. (2.1) In particular, if we take 𝑝_𝑘=1 for all 𝑘, we have𝑉_𝜎,𝜆,𝑓₀=𝑥∶lim_𝑛1𝜆_{𝑛𝑘∈𝐼𝑛}𝑓𝑥_{𝜎𝑘(𝑚)},𝑉=0,uniformlyin𝑚_𝜎,𝜆,𝑓=𝑥∶𝑥−ℓ𝑒∈𝑉_𝜎,𝜆,𝑓₀,𝑉forsomeℓ∈ℂ_𝜎,𝜆,𝑓_∞=𝑥∶sup_𝑛,𝑚1𝜆_{𝑛𝑘∈𝐼𝑛}𝑓𝑥_{𝜎𝑘(𝑚)}<∞. (2.2) Similarly, when 𝜎(𝑚)=𝑚+1, then [V_𝜎,𝜆,𝑓,𝑝]₀, [𝑉_𝜎,𝜆,𝑓,𝑝] and [𝑉_𝜎,𝜆,𝑓,𝑝]_∞ are reduced to𝑉,𝜆,𝑓,𝑝₀=𝑥∶lim_𝑛1𝜆_{𝑛𝑘∈𝐼𝑛}𝑓𝑥_𝑘+𝑚^𝑝𝑘,==0,uniformlyin𝑚𝑉,𝜆,𝑓,𝑝𝑥∶𝑥−ℓ𝑒∈𝑉,𝜆,𝑓,𝑝₀forsomeℓ∈ℂ,𝑉,𝜆,𝑓,𝑝_∞=𝑥∶sup_𝑛,𝑚1𝜆_{𝑛𝑘∈𝐼𝑛}𝑓𝑥_𝑘+𝑚^𝑝𝑘<∞,respectively. (2.3) In particular, when 𝑝_𝑘=𝑝 for all 𝑘, then we have the spaces 𝑉,𝜆,𝑓,𝑝₀=𝑉,𝜆,𝑓₀,𝑉,𝜆,𝑓,𝑝=𝑉,𝜆,𝑓,𝑉,𝜆,𝑓,𝑝_∞=𝑉,𝜆,𝑓_∞, (2.4) which were introduced and studied by Malkowsky and Savaş in [10]. Further, when 𝜆_𝑛=𝑛, for 𝑛=1,2,3,…, the sets [𝑉,𝜆,𝑓]₀ and [𝑉,𝜆,𝑓] are reduced to [𝑐(̂𝑓)] and [𝑐₀̂(𝑓)] respectively, see [7]. Now, if we consider 𝑓(𝑥)=𝑥, then one can easily obtain 𝑉_𝜎,𝜆,𝑝₀=𝑥∶lim_𝑛1𝜆_{𝑛𝑘∈𝐼𝑛}𝑥_{𝜎𝑘(𝑚)}^𝑝𝑘,𝑉uniformlyin𝑚_𝜎,𝜆,𝑝=𝑥∶𝑥−ℓ𝑒∈𝑉_𝜎,𝜆,𝑝₀,𝑉forsomeℓ∈ℂ_𝜎,𝜆,𝑝_∞=𝑥∶sup_𝑛,𝑚1𝜆_{𝑛𝑘∈𝐼𝑛}𝑥_{𝜎𝑘(𝑚)}^𝑝𝑘<∞. (2.5) If 𝑝_𝑘=1 for all 𝑘, then we can obtain the spaces [𝑉_𝜎,𝜆]₀, [𝑉_𝜎,𝜆], and [𝑉_𝜎,𝜆]_∞. Throughout this paper, we use the notation 𝑓(|𝑥_𝑘|)^𝑝𝑘 instead of {𝑓(|𝑥_𝑘|)}^𝑝𝑘.

If 𝑝∈𝑙_∞, then it is clear that [𝑉_𝜎,𝜆,𝑓,𝑝]₀, [𝑉_𝜎,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 𝑉_𝜎,𝜆,𝑓_∞=ℓ^𝜎_∞𝑓=𝑥∈𝑤∶𝑓𝑥_{𝜎𝑘(𝑚)}∈ℓ_∞. (2.6)

Proof. Let 𝑥∈[𝑉_𝜎,𝜆,𝑓]_∞. Then there is a constant 𝑀>0 such that 1𝜆₁𝑓𝑥_{𝜎𝑘(𝑚)}≤sup_𝑚,𝑛1𝜆_{𝑛𝑘∈𝐼𝑛}𝑓𝑥_{𝜎𝑘(𝑚)}≤𝑀, (2.7) for all 𝑚, and so (𝑓(|𝑥_{𝜎𝑘(𝑚)}|))∈𝑙_∞. Let 𝑥∈ℓ^𝜎_∞(𝑓). Then there is a constant 𝑀>0 such that (𝑓(|x_{𝜎𝑘(𝑚)}|))≤𝑀 for all 𝑘 and 𝑚, and so 1𝜆_{𝑛𝑘∈𝐼𝑛}𝑓𝑥_{𝜎𝑘(𝑚)}1≤𝑀𝜆_{𝑛𝑘∈𝐼𝑛}1≤𝑀, (2.8) for all 𝑚 and 𝑛. Thus 𝑥∈[𝑉_𝜎,𝜆,𝑓]_∞. This completes the proof.

If 𝑥∈[𝑉_𝜎,𝜆,𝑓,𝑝], with (1/𝜆_𝑛)∑_{𝑘∈𝐼𝑛}𝑓(|𝑥_{𝜎𝑘(𝑚)}−ℓ𝑒|)^𝑝𝑘→0 as 𝑛→∞ uniformly in 𝑚, then we write 𝑥_𝑘→𝑙[𝑉_𝜎,𝜆,𝑓,𝑝].

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

If 0≤𝑝_𝑘≤sup𝑝_𝑘=𝐻 and 𝐶=max(1,2^𝐻−1), then𝑎_𝑘+𝑏_𝑘^𝑝𝑘≤𝐶𝑎_𝑘^𝑝𝑘+𝑏_𝑘^𝑝𝑘, (2.9) for all 𝑘and 𝑎_𝑘,𝑏_𝑘∈ℂ.

In the following theorem, we prove 𝑥_𝑘→ℓ implies 𝑥_𝑘→ℓ∈[𝑉_𝜎,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 [2]). Let 𝑝_𝑘>0, 𝑞_k>0. Then 𝑐_𝑜(𝑞)⊂𝑐₀(𝑝), if and only if, lim_𝑘→∞inf𝑝_𝑘/𝑞_𝑘>0, where 𝑐₀(𝑝)={𝑥∶|𝑥_𝑘|^𝑝𝑘→0a𝑠𝑘→∞}.

Note that no other relation between (𝑝_𝑘) and (𝑞_𝑘) is needed in Lemma 2.2.

Theorem 2.3. Let lim_𝑘→∞inf𝑝_𝑘>0. Then 𝑥_𝑘→ℓ implies 𝑥_𝑘→ℓ∈[𝑉_𝜎,𝜆,𝑓,𝑝]. Let lim_𝑘→∞𝑝_𝑘=𝑟>0. If 𝑥_𝑘→ℓ∈[𝑉_𝜎,𝜆,𝑓,𝑝], then ℓ is unique.

Proof. Let 𝑥_𝑘→ℓ. By the definition of modulus, we have 𝑓(|𝑥_𝑘−ℓ|)→0. Since lim_𝑘→∞inf𝑝_𝑘>0, it follows from the above lemma that 𝑓(|𝑥_𝑘−ℓ|)^𝑝𝑘→0 and consequently, 𝑥_𝑘→ℓ∈[𝑉_𝜎,𝑓,𝑝].
Let lim_𝑘→∞𝑝_𝑘=𝑟>0. Suppose that 𝑥_𝑘→ℓ₁∈[𝑉_𝜎,𝜆,𝑓,𝑝], 𝑥_𝑘→ℓ₂∈[𝑉_𝜎,𝜆,𝑓,𝑝] and |ℓ₁−ℓ₂|^𝑝𝑘=𝑎>0. Now, from (2.9) and the definition of modulus, we have 1𝜆_{𝑛𝑘∈𝐼𝑛}𝑓ℓ₁−ℓ₂^𝑝𝑘≤𝐶𝜆_{𝑛𝑘∈𝐼𝑛}𝑓𝑥_{𝜎𝑘(𝑚)}−ℓ₁^𝑝𝑘+𝐶𝜆_{𝑛𝑘∈𝐼𝑛}𝑓𝑥_{𝜎𝑘(𝑚)}−ℓ₂^𝑝𝑘. (2.10) Hence, 1𝜆_{𝑛𝑘∈𝐼𝑛}𝑓ℓ₁−ℓ₂^𝑝𝑘=0. (2.11) Further, 𝑓(|ℓ₁−ℓ₂|)^𝑝𝑘→𝑓(𝑎)^𝑟 as 𝑘→∞ and, therefore, lim_𝑛→∞1𝜆_{𝑛𝑘∈𝐼𝑛}𝑓ℓ₁−ℓ₂^𝑝𝑘=𝑓(𝑎)^𝑟. (2.12) 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<inf_𝑘𝑝_𝑘≤𝑝_𝑘≤1. Then, 𝑉_𝜎,𝜆,𝑓,𝑝⊂𝑉_𝜎,𝜆,𝑓. (2.13)
(ii) Let 0<𝑝_𝑘≤sup_𝑘𝑝_𝑘<∞. Then, 𝑉_𝜎,𝜆,𝑓⊂𝑉_𝜎,𝜆,𝑓,𝑝. (2.14)

Proof. (i) Let 𝑥∈[𝑉_𝜎,𝜆,𝑓,𝑝]. Since 0<inf_𝑘𝑝_𝑘≤1, we get 1𝜆_{𝑛𝑘∈𝐼𝑛}𝑓𝑥_{𝜎𝑘(𝑚)}≤1−ℓ𝑒𝜆_{𝑛𝑘∈𝐼𝑛}𝑓𝑥_{𝜎𝑘(𝑚)}−ℓ𝑒^𝑝𝑘, (2.15) and hence 𝑥∈[𝑉_𝜎,𝜆,𝑓].
(ii) Let 𝑝≥1 for each 𝑘, and sup_𝑘𝑝_𝑘<∞. Let 𝑥∈[𝑉_𝜎,𝜆,𝑓]. Then, for each 𝑘, 0<𝜀<1, there exists a positive integer 𝑁 such that 1𝜆_{𝑛𝑘∈𝐼𝑛}𝑓𝑥_{𝜎𝑘(𝑚)}−ℓ𝑒≤𝜀<1, (2.16) for all 𝑚≥𝑁. This implies that 1𝜆_{𝑛𝑘∈𝐼𝑛}𝑓𝑥_{𝜎𝑘(𝑚)}−ℓ𝑒^𝑝𝑘≤1𝜆_{𝑛𝑘∈𝐼𝑛}𝑓𝑥_{𝜎𝑘(𝑚)}−ℓ𝑒. (2.17) Therefore, 𝑥∈[𝑉_𝜎,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. [𝑉_𝜎,𝜆,𝑓,𝑝]₀ and [𝑉_𝜎,𝜆,𝑓,𝑝] are complete linear topological spaces, with paranorm 𝑔, where 𝑔 is defined by 𝑔(𝑥)=sup_𝑚,𝑛1𝜆_{𝑛𝑘∈𝐼𝑛}𝑓𝑥_{𝜎𝑘(𝑚)}^𝑝𝑘𝑀, (2.18) where 𝑀=max(1,{sup_𝑘𝑝_𝑘}).

Acknowledgment

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